66 AG17 Abstracts

IP1 of more tools from algebraic geometry. In particular, the Uses of Algebraic Geometry and Representation building blocks of the constructions could include pieces Theory in Complexity Theory of real, affine algebraic curves, surfaces, and threefolds, or higher degree splines, or toric varieties. In this talk, I will I will discuss two central problems in complexity theory discuss various examples, drawing on results from the EU and how they can be approached geometrically. In 1968, networks GAIA and SAGA. Strassen discovered that the usual algorithm for multiply- ing matrices is not the optimal one, which motivated a vast Ragni Piene body of research to determine just how efficiently nxn ma- Department of Mathematics trices can be multiplied. The standard algorithm uses on University of Oslo the order of n3 arithmetic operations, but thanks to this re- [email protected] search, it is conjectured that asymptotically, it is nearly as easy to multiply matrices as it is to add them, that is that one can multiply matrices using close to n2 multiplications. IP4 In 1978 L. Valiant proposed an algebraic version of the fa- Polynomial Dynamical Systems, Toric Differential mous P v. NP problem. This was modified by Mulmuley Inclusions, and the Global Attractor Conjecture and Sohoni to a problem in algebraic geometry: Geomet- ric Complexity Theory (GCT), which rephrases Valiant’s Polynomial dynamical systems (i.e., systems of differen- question in terms of inclusions of orbit closures. I will tial equations with polynomial right-hand sides) appear discuss matrix multiplication, GCT, and the geometry in- in many important problems in mathematics and applica- volved, including: secant varieties, dual varieties, minimal tions. For example, Hilbert’s 16th problem involves count- free resolutions, and the Hilbert scheme of points. ing limit cycles of two-dimensional polynomial dynamical systems (and is still largely unsolved). Also, some of the Joseph M. Landsberg best known chaotic dynamical systems (such as the Lorenz Department of Mathematics system) are polynomial dynamical systems. In applica- Texas A&M University tions, biological interaction networks and chemical reac- [email protected] tion networks often give rise to polynomial dynamical sys- tems.The Global Attractor Conjecture was formulated in the 1970s, and claims that the solutions of a large class of IP2 polynomial systems (called toric dynamical systems) con- On the Stability of Solutions in Numerical Alge- verge to a globally attracting point. In particular, accord- braic Geometry ing to this conjecture, toric dynamical systems cannot give rise to limit cycles or chaotic dynamics. We introduce toric In the last years a number of results concerning the stability differential inclusions, which are piecewise constant differ- properties of the solutions to problems in Algebraic Geom- ential inclusions with right-hand side given by polyhedral etry have proved that (in different situations) the so-called cones, and discuss about how toric dynamical systems can condition number of a pair (problem instance, solution) be embedded into toric differential inclusions, and this em- can be expected to be quite small. The condition number bedding can be used to prove the Global Attractor Conjec- of such a pair measures the variation of the solution when ture. The proof uses n-dimensional polyhedral fans, poly- the problem instance is perturbed. This talk will use its hedral partitions and convex geometry. We will also discuss more geometrical definition which sheds light on its prop- about some connections with the Boltzmann equation, and erties and shows that it is often a measure of complexity. applications of these results to understanding robustness I will present some classical, some recent and some new and homeostasis in biological interaction networks. results on the expected value of powers of the condition number in different situations. In particular, I will men- Gheorghe Craciun tion new results concerning the stability of the polynomial Department of Mathematics, University of eigenvalue problem and that of sparse or lacunary polyno- Wisconsin-Madison mial solving. These results follow from the use of classical [email protected] Algebraic Geometry theorems combined with techniques from integration in manifolds. There will be no much need of previous knowledge on any of these areas to follow the IP5 conference. This work summarizes a considerable number of papers by different people, including my coauthors D. Stochastic Geometry with Topological Flavor Armentano, P. B¨urgisser, F. Cucker, J. P. Dedieu, J. G. Criado del Rey, A. Leykin, G. Malajovich, L. M. Pardo Mapping the simplices of a Delaunay mosaic to their radii, and M. Shub. we generically get a (generalized) discrete Morse function. Assuming a stationary Poisson point process, we extend Carlos Beltr´an classic results in stochastic geometry and compute the ex- Departamento de Matematicas pected number of critical simplices and intervals in the dis- Universidad de Cantabria crete gradient as functions of the maximum radius. We ob- [email protected] tain similar results for Delaunay mosaics under the Fisher information metric, for weighted Delaunay mosaics, and for other related complexes. IP3 Herbert Edelsbrunner Algebraic Geometry for Geometric Modeling Institute of Science and Technology Austria Geometric modeling studies mathematical methods for [email protected] constructing objects in computer graphics and computer aided design. From early on, low degree polynomials and simple algebro-geometric tools were used. The advances of IP6 speed and memory of computer systems now permit the use Curves with Complex Multiplication and Applica- AG17 Abstracts 67

tions to Cryptography in a cone and it is among the most fundamental results in Convex Geometry and it has seen many variations and ex- There are many advantages to using cryptosystems based tensions. I will review some variations of Carath´eodory’s on elliptic curves or genus 2 curves. However, implemen- theorem that have interesting applications in combinato- tation of these cryptosystems requires a curve over a finite rial optimization and game theory. E.g., given a system field F where the number of F-rational points on the Ja- Ax = b, x ≥ 0, what is the size of the sparsest solution inte- cobian is prime (or a prime times a small cofactor). I will ger? Integral versions of Carath´eodorys theorem improve explain how the problem of finding such curves leads to prior bounds and show some of the fascinating structure questions about reductions of curves with complex mul- of sparse integer solutions of Diophantine equations. An- tiplication and conversely, how structural results about other example will be the use of Hilbert bases in augmenta- curves with complex multiplication affect algorithms for tion algorithms for optimization. I will mention some open constructing cryptographically useful curves. problems too.

Bianca Viray Jes´us A. De Loera University of Washington Department of Mathematics [email protected] Univ. of California, Davis [email protected] IP7 Gaussian Graphical Models from an Algebraic, Ge- IP10 ometric and Combinatorial Perspective Talk Title To Be Announced (Invited Speaker J. Faugre) In this talk we will introduce probabilistic Gaussian graph- ical models as an interesting study object for applied al- Talk title and abstract will be posted here when available. gebraic geometers. In particular, we discuss maximum likelihood estimation for Gaussian graphical models, which leads to the problem of maximizing the determinant func- Jean-Charles Faug`ere tion over a spectrahedron, and to the positive definite ma- Salsa project, INRIA Paris-Rocquencourt trix completion problem. Missing edges in a Gaussian [email protected] graphical model correspond to conditional independence relations. We present a representation of conditional in- dependence relations by a generalized permutohedron. Fi- SP1 nally, we discuss extensions of Gaussian graphical models SIAG/AG Early Career Prize Lecture: A Brief to exponential families leading to the concept of exponen- History of Smale’s 17th Problem tial varieties. In the 80s, Steve Smale, soon joined by Mike Shub, studied Caroline Uhler from a theoretical point of view the complexity of numer- Massachusetts Institute of Technology ically solving polynomial systems. They produced many [email protected] new ideas to realize Newton’s method algorithmic poten- tial but they also left several open questions, one of which is Smale’s 17th problem. From the early results to the fi- IP8 nal touch, I will sketch some of the ideas that leads to a Open Problems in Finite Frame Theory solution to Smale’s 17th problem.

Finite frame theory is the study of collections of vectors Pierre Lairez with certain application-driven properties. This talk will Inria Saclay le-de-France discuss a few instances of finite frame theory. Consider the [email protected] problem of packing points in real or complex projective space so that the minimum distance is maximized. Such optimal line packings find applications in sparse decom- CP1 position and quantum information theory. Today, there Solving the Polynomial Equations by Algorithms are various bounds on the best packings, notably, Leven- of Power Geometry shtein’s bound arising from linear programming. Ensem- bles which achieve equality in Levenshtein’s bound are pro- New methods for computation of solutions of an algebraic jective t-designs with small angle sets, and their existence equation of three variables near a critical point are pro- is the subject of several open problems. posed. These methods are: Newton polyhedron, power transformations, new versions of the implicit function the- Dustin Mixon orem [Bruno A.D. Power Geometry in Algebraic and Dif- Air Force Institute of Technology ferential Equations. Elsevier, Amsterdam, 2000] and uni- [email protected] formization of a planar algebraic curve. We begin from a survey of the new methods of computation of solutions of an algebraic equation of one and of two variables by means IP9 of the Hadamard polygon [Hadamard J. Etude sur les pro- Carath´eodory Style Theorems for Discrete Opti- prietes des fonctions entieres et en particulier d’une fonc- mization and Game Theory tion consideree par Riemann// Journal de mathmatiques pures et appliques, (1893) 9:2, 171–216; Bruno A.D. Lo- Convex geometry has been an important pillar of the the- cal Methods in Nonlinear Differential Equations. Springer- ory of optimization algorithms (e.g., think of Ellipsoid Verlag, Berlin, 1989, P. I, Ch. IV, Sec. 2.1] and Hadamard method) and my talk will show this continues to be the case polyhedron [Bruno A.D. On solution of an algebraic equa- today by focusing one influential theorem, Carath´eodory’s tion. Preprint of KIAM, No. 70, Moscow, 2016 (in Rus- theorem from 1905. In its most basic form it describes the sian)]. That approach works for differential equations (or- size of a minimal linear combination representing a vector dinary and partial) as well. In the survey [Bruno A.D. 68 AG17 Abstracts

Asymptotic solution of nonlinear algebraic and differential of prolongations needed to guarantee the existence of a dif- equations // International Mathematical Forum, (2015) ferential solution to the system. This bound has several ap- 10:11, 535–564 http://dx.doi.org/10.12988/imf.2015.5974] plications to the algebraic theory of differential equations, there are several nontrivial applications. which we discuss.

Alexander D. Bruno Richard S. Gustavson Keldysh Institute of Applied Mathematics of RAS CUNY Graduate Center [email protected] [email protected]

Alexander B. Batkhin Omar Leon Sanchez Keldysh Institute of Applied Mathematics of RAS University of Manchester Moscow Institute of Physics and Technology [email protected] [email protected]

CP2 CP1 Multi-point AG Codes on the GK Maximal Curve Computing Real Equilibria of Kuramoto Model We investigate multi-point Algebraic-Geometric codes as- The Kuramoto model is used to describe synchronization sociated to the GK maximal curve, starting from a divisor behavior of a large set of oscillators. The equilibria of this which is invariant under a large automorphism group of model can be computed by solving a system of polynomial the curve. We construct families of codes with large auto- equations using algebraic geometry. Typical methods for morphism groups. solving such polynomial systems, e.g., Gr¨obner basis meth- ods and homotopy continuation, compute all complex equi- Daniele Bartoli librium points when only the real equilibrium points are of Department of Mathematics - University of Perugia physical interest. We develop an approach to compute only [email protected] the real equilibrium points by reducing down to solving a univarite polynomial and compare it with other computa- Maria Montanucci tional algebraic geometric approaches. Analyzing this uni- Universit`a degli Studi della Basilicata variate approach allows us to prove that, asymptotically, [email protected] the maximum number of real equilibrium points grows at the same rate as the number of complex equilibrium points. Giovanni Zini Although the maximum number of real equilibrium points Universit`a degli Studi di Firenze is still an open question for four or more oscillators, we con- [email protected] jecture an upper bound for any number of oscillators which generalizes the known cases and is obtained with explicitly provided natural frequencies. CP2 F Owen Coss 4[v]-Double Cyclic Codes NCSU 2 Let F4[v]=F4 + vF4,v = v. F4[v]-Double cyclic codes [email protected] are in fact generalized quasi-cyclic codes of index two. We determine the structure of these codes and their duals by Hoon Hong giving generators. We enumerate them by giving a mass North Carolina State University formula. Finally, we study some structural properties of [email protected] skew generalized quasi-cyclic codes of index two over F4 + vF4. Jonathan Hauenstein University of Notre Dame Srinivasulu Bathala Dept. of App. Comp. Math. & Stats. Indian Institute of Technology Roorkee [email protected] [email protected]

Daniel Molzahn Maheshanand Bhaintwal Argonne National Laboratory Indian Institute of Technoloyg Roorkee. [email protected] India - 247667 [email protected]

CP1 Effective Methods for Proving the Consistency of CP2 Differential Equations Layered Codes Through Multilinear Algebra

This talk is focused on the question of consistency of a We describe different algebra based coding solutions for dis- system of algebraic differential equations F =0,thatis, tributed storage. The purpose of distributed storage is to whether the system has a solution in some differential field provide easy access to data that is stored on multiple disks extension of the coefficient field of F .Onemethodofsolv- and to protect the data against disk failures and erasures. ing this problem is to look for certain algebraic solutions to Layered codes were introduced by Tian et al. Various au- the system F = 0, thought purely as a system of algebraic thors showed that layered codes provide optimal trade-offs equations, and then attempt to prolong those solutions to between low storage overhead and efficient erasure repair see if they form well-defined solutions to the original differ- for a wide range of parameters. We discuss the determinant ential system. We define a recursive algorithm that effec- code construction by Elyasi and Mohajer that extends the tively computes an upper bound for the minimum number use of layered codes from a given finite number of disks to AG17 Abstracts 69

any number of disks while preserving the optimal trade-off. ingredients. First, an F4 style algorithm to compute the Joint work with Xiao Li. Groebner basis is adopted, where a novel prolongation is designed such that the sizes of coefficient matrices under Iwan Duursma consideration are nicely controlled. Second, the complex- University of Illinois at Urbana-Champaign ity bound of the algorithm is achieved by an estimation Department of Mathematics for the degree and height bounds of the polynomials in the [email protected] generalized Hermite normal form. Third, fast algorithms to compute Hermite normal forms of matrices over Z[x] are used as the computational tool. CP2 On the Remaining Heuristics of the DLP Algo- Xiaoshan Gao rithm in Small Characteristic Academy of Mathematics and Systems Science Chinese Academy of Sciences Razvan Barbulescu, Pierrick Gaudry, Antoine Joux, Em- [email protected] manuel Thom´epresented a heuristic quasi-polynomial time algorithm to solve Discrete Logarithm Problems in small characteristic. This result has been one of the most impor- CP3 tant breakthrough in Cryptography in the last years. The Minimal Free Resolution of the Associated Graded algorithm is very efficient under some heuristic assump- Ring of Certain Affine Monomial Curves tions (which are in fact always verified in practice) but from a mathematical point of view it still does not ensure Motivated by the fact that deep information can be ob- that DLP is indeed quasi-polynomial time. In this talk I tained from the numerical invariants of the minimal free will describe the ones which remain to prove and provide resolution, we will give an explicit minimal free resolution a strategy to address them using effective Chebotarev for of the associated graded ring for certain monomial curves function fields. One of the results is a condition on an error in affine 4-space. We will also determine which proper- term which, if proved, will imply the main claim. ties such as being Cohen-Macaulay, Gorenstein, etc. of the coordinate ring of these certain monomial curves can Giacomo Micheli be carried to the associated graded ring. This is a joint University of Oxford work with Esra Emine Zengin. This work is supported by [email protected] Balikesir University, BAP 2017/108. Pinar Mete CP3 Department of Mathematics Noetherianity for Cubic Polynomials Balikesir University [email protected] ∞ Let P3(C ) be the space of complex cubic polynomials in infinitely many variables. We show that this space is GL∞- CP3 noetherian, meaning that any GL∞-stable Zariski closed subset is cut out by finitely many orbits of equations. Our Generalized Weierstrass Semigroups and their method relies on a careful analysis of an invariant of cubics Poincar Series introduced here called q-rank. This result is motivated by recent work in representation stability, especially the the- We provide a generating set for generalized Weierstrass ory of twisted commutative algebras. It is also connected semigroups at several points on algebraic curves defined to certain stability problems in commutative algebra, such over a perfect field. These generators are finitely deter- as Stillman’s conjecture. mined and codify the Riemann-Roch spaces of every divi- sor supported on the specific points and their basis, which Rob H. Eggermont is of interest to applications in algebraic-geometric codes. University of Michigan This characterization of the generalized Weierstrass semi- [email protected] groups furthermore allows us to interpret the correspond- ing Poincar´e series as invariants of the semigroups since Harm Derksen they carry essential information to recover entirely the University of Michigan semigroup associated. Department of Mathematics Wanderson Ten´orio [email protected] University of Campinas [email protected] Andrew Snowden University of Michigan [email protected] CP4 Resonance Set of a Real Polynomial and its Appli- cation CP3 A Polynomial-time Algorithm to Compute Gener- Let fn(x) be a monic polynomial of degree n with real coef- alized Hermite Normal Forms of Matrices over Z[x] ficients. A pair of roots ti, tj , i, j =1,...,n, i = j,offn(x) is called p : q-commensurable if ti : tj = p : q. Resonance In this talk, a polynomial time algorithm will be given to set Rp:q(fn)offn(x) is called the set of all points of the n compute the generalized Hermite normal form for a matrix coefficient space Π ≡ R at which fn(x) has at least a pair F over Z[x], or equivalently, the reduced Groebner basis of of p : q-commensurable roots. The condition on existence the Z[x]-module generated by the column vectors of F. The of p : q-commensurable roots is formulated using certain algorithm has polynomial bit size computational complex- generalization of subdiscriminant of fn(x). Any partition ity and is also shown to be practically more efficient than of degree n of fn(x) defines a certain structure of its p : q- existing algorithms. The algorithm is based on three key commensurable roots and it corresponds to some algebraic 70 AG17 Abstracts

Vi variety l , i =1,...,pl(n)ofdimensionl in Π. The num- and Applications ber of such varieties of dimension l is equal to pl(n)and total number of all varieties consisting the resonance set We consider multidimensional optimization problems for- Rp:q (fn)isequaltop(n) − 1. The main result: each vari- mulated in the tropical mathematics setting to minimize ety Vl of Rp:q (fn)forl>1 allows polynomial parametriza- or maximize functions defined on vectors over idempotent tion in Π, which can be obtained from the parametrization semifields, subject to linear equality and inequality con- of the corresponding variety Vl−1.ThevarietyV1 allows straints. We start with a brief overview of known tropical polynomial parametrization constructed from q-binomial optimization problems and solution approaches. Further- coefficients. Resonance set Rp:1(f3)forp =1, 4, 9, 16 can more, some new problems are presented with nonlinear be used for solving the problem of formal stability of a sta- objective functions calculated using multiplicative conju- tionary point of a Hamiltonian system with three degrees gate transposition of vectors, including problems of Cheby- of freedom. The corresponding software package for CAS shev approximation, problems of approximation in span Maple is provided. seminorm, and pseudo-quadratic problems. To solve these problems, we apply methods based on the reduction to Alexander B. Batkhin the solution of parametrized inequalities, matrix sparsifi- Keldysh Institute of Applied Mathematics of RAS cation, and other techniques. The methods offer direct so- Moscow Institute of Physics and Technology lutions represented in a compact explicit vector form ready [email protected] for further analysis and straightforward computation. We conclude with a short discussion of the application of the results obtained to practical problems in location analysis, CP4 project scheduling and decision making. Difference Sets and Grassmannian Packings Nikolai Krivulin It is often of interest to find subspaces which are optimally Saint Petersburg State Univ spread apart. For example, if one wants a set of vectors Faculty of Math and Mechanics (representing one dimensional subspaces) which have sim- [email protected] ilar properties to orthonormal bases, the vectors should as non-parallel as possible. Such collections yield optimally CP5 robust representations of certain classes of data and are referred to as Grassmannian (fusion) frames. There are On the Sn-Invariant F-conjecture a number of constructions possible using tools from com- By using classical invariant theory, we reduce the S - binatorial design theory: difference sets and their gener- n invariant F-conjecture into a feasibility problem in polyhe- alizations. In this talk, the connection between algebraic dral geometry. We show by computer that for n ≤ 19, ev- combinatorics and geometric packings will be presented, ery integral S -invariant F-nef divisor on the moduli space including brand new constructions of packings. n of genus zero stable pointed curves is semi-ample, over ar- ≤ Emily King bitrary characteristic. Furthermore, for n 16, we show Universitaet Bonn that for every integral Sn-invariant nef (resp. ample) di- [email protected] visor D on the moduli space, 2D is base-point-free (resp. very ample). As applications, we obtain the nef cone of the moduli space of stable curves without marked points, and CP4 the semi-ample cone that of the moduli space of genus 0 stable maps to Grassmannian for small numerical values. String Topological Robotics David Swinarski, Han-Bom Moon We claim here to link two well known theories; namely Fordham University the string topology (founded by M. Chas and D. Sullivan [email protected], [email protected] in 1999) and the topological robotics (founded by M. Far- ber some few years later, in 2003). For our purpose, we consider G a compact Lie group acting transitively on a CP5 M path connected n-manifold X.Ontheset LP (X)of Practical Semialgebraic Geometry for Computer- the so-called loop motion planning algorithms, we define assisted Proofs and discuss the notion of transversality. Firstly, we define an intersection loop motion planning product at level of Motivated by computer assisted proof techniques, we inves- M chains of LP (X). Secondly, we define a boundary oper- tigate questions of computational geometry in the semi- M ator on the chains of LP (X) and extend this intersection algebraic domain, such as the following. Given a basic product at level of homology to a string loop motion plan- semialgebraic set S, described by a list of polynomial in- ning product. Finally, we show that this string product equalities, and basic semialgebraic sets S1,...,Sm,isS induces on the shifted string loop motion planning homol- contained in the union S1 ∪···∪S ? If not, find a point H M M m ogy ∗( LP (X)) := H∗+2n( LP (X)) a structure of an in the difference. In theory, any problems of this type (in associative and commutative graded algebra (acga). By fixed dimension and bounded degree) are solved problems the end, we ask how one may extend this acga-structure to of effective real algebraic geometry and can be solved us- a structure of Gerstenhaber algebra or that of a Batalin- ing quantifier elimination. However, even the best available Vilkovisky algebra. Some ideas will be suggested. Cylindrical Algebraic Decomposition (CAD) implementa- tion, in Mathematica, quickly reaches its limits when the My Ismail I. Mamouni number m of set grows even if the dimension is very small. CRMEF Rabat, Morocco We investigate practical and efficient approaches. [email protected] Matthias Koeppe Dept. of Mathematics CP5 Univ. of California, Davis Tropical Optimization Problems: Recent Results [email protected] AG17 Abstracts 71

Yuan Zhou crypto. UC Davis [email protected] Tanja Lange Technische Universiteit Eindhoven Coding Theory and Cryptology, Dept. of Math. and CS. MS1 [email protected] Computation of Isogenies and Frobenius MS1 Computation of isogenies is a major problem in cryptog- Code-based Hybrid Encryption, Revisited raphy especially with post quantum crypto-system using it (such as DeFeo-Jao-Plut 2011). After a brief review Code-based cryptography is one of the most promising can- of the Couveignes’s algorithm of 96 we will show in the didates for post-quantum cryptography, in particular with wake of the works of DeFeo11 and DeFeo-Hugounenq-Plut- regards to the upcoming NISTs call for papers for new post- Schost16 how we take advantage of the structure of the field quantum primitives. In this work, I will briefly present the (and later of the elliptic curve) we are working on to gen- most recent take on code-based encryption. The scheme erate relations useful to the computation of the isogeny. follows the hybrid encryption (KEM-DEM) paradigm first Especially we will study the action of the Frobenius on the introduced by Cramer and Shoup, and it is a revisitation torsion points (Tate Module) of the elliptic curves. of the Hybrid Niederreiter scheme first introduced in 2013. The revisitation stems from a recent attack from Bernstein Cyril Hugounenq et al. Laboratoire PRiSM [email protected] Edoardo Persichetti Dakota State University [email protected] MS1 Efficient Compression of SIDH Public Keys MS2 Supersingular isogeny Diffie-Hellman (SIDH) is an attrac- Sparse Solutions of Integer Programs tive candidate for post-quantum key exchange, in large part due to its relatively small public key sizes. We propose We present structural results on solutions to the Diophan- y b y ∈ Zt a range of new algorithms and techniques that acceler- tine system A = , ≥0 with the smallest number of ate SIDH public-key compression by more than an order non-zero entries. Our tools are algebraic and number the- of magnitude, making the compression process roughly as oretic in nature and include Siegel’s Lemma, generating fast as a round of standalone SIDH key exchange, while functions, and commutative algebra. These results have further reducing the size of the compressed public keys by some interesting consequences in discrete optimization. approximately 13%. These improvements enable the prac- tical use of compression, achieving public keys of only 330 Iskander Aliev bytes for the concrete parameters used by Costello et al. Cardiff University in CRYPTO 2016 and further strengthening SIDH as a [email protected] promising post-quantum primitive. MS2 Craig Costello Microsoft Research Discrete Quantitative Helly Number [email protected] If a system of linear equatations in n integer variables has exactly k integer solutions that there is a subsystem of David Jao c(n, k) inequalities that already has k integer solutions. University of Waterloo The minimal choice of c(n, k), for which the above is true [email protected] is called quantitative Helly number for the n-dimensional integer lattice. Quantitative Helly numbers occurr natu- Patrick Longa, Michael Naehrig rally in various applied contexts (including integer opti- Microsoft Research mization). Determination of the asymptotic behaviour of [email protected], [email protected] c(n, k) and computation of c(n, k) for concrete choices of n or k is a difficult task. In my talk, I will report on the re- Joost Renes cent progress in the study of c(n, k). This is joint work with Radboud University Nijmegen Bernardo Gonz´alez Merino, Matthias Henze, Ingo Paschke [email protected] and Stefan Weltge. Gennadiy Averkov David Urbanik University of Magdeburg University of Waterloo [email protected] [email protected]

MS2 MS1 Understanding Deep Neural Networks with Recti- Practical Post-quantum Cryptography fied Linear Units

This talk will cover some aspects of bringing post-quantum In this paper we investigate the family of functions rep- cryptography to life. Possibilities are side-channel attacks resentable by deep neural networks (DNN) with rectified on lattice-based crypto; PKI issues for code-based cryptog- linear units (ReLU). We give the first-ever polynomial time raphy or attacks and constructive aspects of lattice-based (in the size of data) algorithm to train a ReLU DNN with 72 AG17 Abstracts

one hidden layer to global optimality. This follows from our interpreted as Newton–Okounkov bodies. Recently Gross, complete characterization of the ReLU DNN function class Hacking, Keel, and Kontsevich developed a new construc- whereby we show that a Rn → R function is representable tion of toric degenerations for cluster varieties using su- by a ReLU DNN if and only if it is a continuous piecewise perpotentials. They conjecture the specialization to many linear function. The main tool used to prove this char- toric degenerations from representation theory. A first step acterization is an elegant result from tropical geometry. towards proving this was made by Magee, he shows the Further, for the n = 1 case, we show that a single hidden Gelfand-Tsetlin polytope for the flag variety can be ob- layer suffices to express all piecewise linear functions, and tained using their methods. Our work proves this special- we give tight bounds for the size of such a ReLU DNN. We ization is true for all string polytopes. If time permits, I follow up with gap results showing that there is a smoothly will explain the connection to joint work with Sara Lam- parameterized family of R → R “hard” functions that lead boglia, Kalina Mincheva, and Fatemeh Mohammadi. We to an exponential blow-up in size, if the number of layers is compute toric degenerations of SLn/B for n =4, 5 explic- decreased by a small amount. An example consequence of itly using the tropicalized flag variety and compare them our gap theorem is that for every natural number N, there to those obtained from string polytopes. exists a function representable by a ReLU DNN with depth N 2 + 1 and total size N 3, such that any ReLU DNN with Lara Bossinger 1 N+1 − depth at most N + 1 will require at least 2 N 1 total University of Cologne nodes. These constructions utilize the theory of zonotopes [email protected] from polyhedral theory. Amitabh Basu, Raman Arora, Poorya Mianjy, Anirbit MS3 Mukherjee Computing Toric Degenerations of Flag Varieties Johns Hopkins University Arising from Tropical Geometry [email protected], [email protected], mi- [email protected], [email protected] I give an overview talk on toric degenerations of flag va- rieties arising from tropical geometry and representation theory. I will compare toric degenerations arising from MS2 string polytopes with those obtained from tropical cones of Using Linear Symmetries in Integer Convex Opti- flag varieties and will explain how the corresponding toric mization polytopes can be seen as Newton-Okounkov bodies for the valuations associated to each tropical cone. This is based We consider integer convex optimization problems which on a joint work with Lara Bossinger, Sara Lamboglia, and are invariant under a finite linear group, its linear symme- Kalina Mincheva. tries. Standard techniques often seem to work poorly on such symmetric problems. We survey the different types Fatemeh Mohammadi of symmetries and we present some ideas on how to ex- Technical University of Berlin ploit these symmetries. This is joint work with Achill [email protected] Sch¨urmann.

Frieder Ladisch MS3 Universitat Rostock, Germany [email protected] Newton-Okounkov Bodies and Khovanskii Bases for Applications

MS3 Newton-Okounkov bodies were introduced by Kaveh- Khovanskii and Lazarsfeld-Mustat¸˘a to extend the theory of Volumes of Newton-Okounkov Bodies of Hessen- Newton polytopes to functions more general than Laurent berg Varieties and their Cohomology Rings polynomials. This theory has at least two implications for Hessenberg varieties are subvarieties of the full flag vari- applications. First is that Newton-Okounkov bodies pro- ety. In this talk, I will concentrate on Lie type A,andI vide an approach to counting the number of solutions to will talk about a relation between the volume of certain systems of equations that arise in applications. Another is Newton-Okounkov body of a regular nilpotent Hessenberg that when the Newton-Okounkov body is an integer poly- variety and its cohomology ring which is a Poincar´e dual- tope (there is a Khovanskii basis), there is a degeneration ity algebra. More precisely, I will explain that the volume to a toric variety which in principal should give a numeri- polynomial of its cohomology ring in fact describes the hon- cal homotopy algorithm for computing the solutions. This est volume of its Newton-Okounkov body with respect to talk will sketch both applications. the Plucker embedding. This is a joint work with Lauren DeDieu, Federico Galetto, and Megumi Harada. Frank Sottile Texas A&M University Hiraki Abe [email protected] McMaster University [email protected] MS4 Real Solving of Bivariate Real Equation Systems MS3 Flag Varieties of Type A, String Polytopes, and We present a new algorithm to get approximately the real Superpotentials zeros of a bivariate real equation system. We provide an algorithm for the problem and implement it. The method This talk is based on joint work with Ghislain Fourier can be extended to real solving of general non-linear con- (arXiv:1611.06504). String polytopes are famous exam- tinuous function systems. Experiments show the effectivity ples of polytopes arising in representation theory. They of our method. It works for bivariate polynomial systems provide toric degenerations of the flag variety and can be with several hundred degrees. It is a joint work with Junyi AG17 Abstracts 73

Wen. Tensors

Jinsan Cheng Finding a simpler representation of a nonlinear multivari- Chinese Academy of Sciences ate vector function is of interest for reducing its parametric [email protected] complexity, or to obtain an insight into its internal work- ings. We will discuss a decomposition method to express a given multivariate vector function f(u):Rm → Rn into MS4 linear combinations of univariate functions gi(·) in linear Resultant Formulations for Structured Polynomials forms of the input variables u, formally and System Solving ⎡  ⎤ g1(v1 u) ⎢ ⎥ ⎣ . ⎦ In this talk we discuss the resultant of polynomial systems f(u)=W . . with structured supports, which fall under the general class  v u of multihomogeneous systems. Surprisingly, in many cases gr( r ) these systems admit optimal matrices expressing their re- The approach is based on the observation that the Jacobian sultant, i.e., the matrix formulation is free of extraneous matrix of f(u) can be written as factors. This is in contrast to homogeneous systems of equations, where such formulations are quite rare. More- ⎡   ⎤ g1(v1 u) over, we apply the u-resultant method for computing the ⎢ . ⎥  solutions of a structured (square) polynomial system us- J(u)=W ⎣ .. ⎦ V . ing eigenvalues and eigenvectors. In doing so, special care g (vu) is taken in order to preserve optimality of the resultant r r matrix. By evaluating the Jacobian matrix in a set of sampling points u(k),fork =1,...,N, the decoupling problem is Angelos Mantzaflaris reduced to simultaneous matrix diagonalization, which is RICAM solved using the tensor Canonical Polyadic Decomposition. Austrian Academy of Sciences We will highlight applications of the decoupling approach angelos.mantzafl[email protected] in system identification problems, connections to the War- ing decomposition for polynomials, and a link to related Elias Tsigaridas coefficient-based tensor decoupling methods. Inria Paris-Rocquencourt [email protected] Philippe Dreesen,MariyaIshteva Vrije Universiteit Brussel [email protected], [email protected] MS4 Computing the Annihilator of Sequences Johan Schoukens Vrije Universiteit Brussel (VUB) This talk is on the annihilator of multivariate sequences [email protected] that are defined by m−primary ideals. We report on a work in progress on efficient computation of a Gr¨obner basis for the annihilator in polynomial time in terms of the degree MS5 of the ideal. This is a joint work with Eric Schost. Symmetric Tensor Approximation

Hamid Rahkooy We will describe an iterative method for finding low rank University of Waterloo tensor approximation. The algorithm applies to nth order [email protected] tensors with full and partial symmetries. We will also its applications to signal processing.

MS4 Ramin Goudarzi Karim, Carmeliza Navasca Department of Mathematics Dedekind Zeta Functions and Complex Dimension University of Alabama at Birmingham Koiran proved around 1996 that deciding whether the di- [email protected], [email protected] mension of an algebraic set X is at least d (for some input polynomials whose zero set is X, and an input non-negative MS5 integer d) can be done in the polynomial hierarchy, as- suming the Generalized Riemann Hypothesis (GRH). We Tensor Decompositions and Related Problems isolate a much more plausible hypothesis on Dedekind zeta Tensor decompositions are becoming increasingly popu- functions called DZH and show that it still implies Koiran’s lar in different disciplines. The first part of the talk improved complexity bound. More importantly, DZH is briefly introduces tensors and their most popular decompo- more likely to be within reach of current analytic tech- sitions. The second part of the talk presents an overview niques. We present numerous examples along the way, and of the mini-symposium by positioning the talks and re- assume only some basic algebraic background. vealing links between (constrained) tensor decompositions, inverse source problems, nonlinear system identification, J. Maurice Rojas,YuyuZhu blind source separation, rational approximation and sig- Texas A&M University nal estimation, among others. This work was supported in [email protected], [email protected] part by the ERC Advanced Grant SNLSID under contract 320378 and by FWO projects G.0280.15N and G.0901.17N. MS5 Decoupling Multivariate Vector Functions using Mariya Ishteva 74 AG17 Abstracts

Vrije Universiteit Brussel up to four clonal expansion stages, and conjecture the re- [email protected] sults for any number of clonal expansion stages. We then consider the practical identifiability of these models, show- ing that even when the models are reparameterized to be MS6 structurally identifiable, there are often significant limits The Binary Jukes-Cantor Model with a Molecular in what we can practically estimate due to real-world data Clock limitations.

We prove results about the polytope associated to the toric Marisa Eisenberg ideal of invariants of the binary Jukes-Cantor model with University of Michigan a molecular clock on a rooted phylogenetic tree. For in- [email protected] stance, we show that this polytope’s number of vertices is a Fibonacci number and present a description of the facets of this polytope using the combinatorial structure of the MS7 underlying rooted tree. We also study the generating sets Structural Identifiability of Biological Models and Gr¨obner bases of the toric ideal of invariants associated to the model, with the goal of determining results about Structural identifiability concerns finding which unknown about the Ehrhart polynomials. parameters of a model can be recovered from data, an im- portant first step in the parameter estimation problem. In Jane Coons this talk, we introduce the differential algebra approach North Carolina State University to structural identifiability and demonstrate this approach [email protected] on some biological models taken from the areas of phar- macokinetics, epidemiology, and physiology. In particular, we present some unidentifiable models, i.e. models with MS6 parameters that cannot be determined given even perfect Recursive Linear Programming on the Balanced input-output data, and we discuss what to do with such Minimal Evolution Polytope models.

The Balanced Minimal Evolution (BME) polytope is the Nicolette Meshkat latest in a succession of polytopes which model the space of Santa Clara University phylogenetic trees. We will review the definitions and re- [email protected] cursive properties of the BME polytope, as well as the per- mutoassociahedron and the perfect matching polytope, if time permits. We have recently found inequalities for many MS7 facets of the BME polytope, allowing fast and apparently Geometric Classification of Structural and Practi- accurate recovery of phylogenetic trees from a measured cal Non-identifiability distance vector.

Stefan Forcey, William Sands, Logan Keefe A model is structurally unidentifiable if it makes the same University of Akron predictions for more than one parameterization. I present [email protected], [email protected], a geometric interpretation of structural non-identifiability [email protected] and how this leads to a natural classification of struc- tural non-identifiabilities. I next consider practical non- identifiability. I define practical non-identifiability to be MS6 when a confidence region intersects the boundary of the model manifold. I discuss the relationship between practi- Quartet Trees in Phylogenomics cal identifiability, sloppiness, and effective theories biology Four leaf, or quartet trees provide a building block for un- and physics. derstanding evolution. In this session we survey how the combinatorics of combining quartet trees into larger trees Mark K. Transtrum influence our understanding of species tree reconstruction. Department of Physics and Astronomy We also examine the role of algebraic or nearly algebraic Brigham Young University models of evolution in constructing these estimated quartet [email protected] histories.

Joseph P. Rusinko MS7 Hobart and William Smith Colleges Identifiability of a Generalized HodgkinHuxley [email protected] Model

The Hodgkin-Huxley (HH) model for action potential gen- MS7 eration is a significant model in neuroscience. I will dis- Structural and Practical Identifiability in Multi- cuss a generalized version of the HH model and its struc- stage Clonal Expansion Models of Cancer tural identifiability properties. For ”voltage clamp” data, the steady-state gating variables are not identifiable, but Multistage clonal expansion (MSCE) models of carcinogen- their product together with the conductance term forms esis are continuous-time Markov process models often used an identifiable combination. The time constants for each to relate cancer incidence to biological mechanism. In this gating variable are identifiable and can be estimated us- talk, we examine the structural identifiability of a subclass ing the same approach used to demonstrate identifiability. of these models using the differential algebra approach tra- A Groebner basis calculation demonstrates that the expo- ditionally used for deterministic ordinary differential equa- nents of the gating variables are identifiable in the two-gate tion (ODE) models. Additionally, we determine the iden- case. These results are compared to the model’s structural tifiable combinations of the generalized MSCE model with identifiability given ”voltage ramp” and ”action potential AG17 Abstracts 75

clamp” data. and extends work of Aholt, Thomas, and Sturmfels using purely geometric techniques. Olivia Walch Univeristy of Michigan Lucas van Meter [email protected] University of Washington [email protected] MS8 The Multidegree of the Multi-image Variety MS9 Equiangular Tight Frames from Association The multi-image variety models taking pictures with n ra- Schemes tional cameras. It can be described as a subvariety of n Gr(2, 4) . We compute its cohomology class in the coho- Association schemes are combinatorial objects that simul- n mology ring of Gr(2, 4) and its multidegree in the Pl¨ucker taneously generalize the notions of strongly regular graphs, 5 n embedding (P ) . Based on joint work with Allen Knutson. finite abelian groups, and Gelfand pairs of (possibly non- abelian) finite groups. We explain how an association scheme naturally produces a finite number of unit-norm Laura Escobar tight frames (FUNTFs). When our scheme is an abelian University of Illinois at Urbana-Champaign group, we precisely obtain its harmonic frames, but for gen- [email protected] eral association schemes, we get a much broader family of FUNTFs. Among them are equiangular tight frames which MS8 are not harmonic, and which do not obey the integrality conditions imposed by abelian groups. Distortion Varieties Joseph Iverson We present a new framework, developed jointly with Joe Department of Mathematics Kileel, Zuzana Kukelova and Tomas Pajdla, for formulat- University of Maryland ing and solving minimal problems for camera models with [email protected] image distortion. The distortions of a given projective va- riety are parametrized by duplicating coordinates and mul- tiplying them with monomials. We study their degrees and John Jasper defining equations. Our formulas involve Chow polytopes Department of Mathematical Sciences and Gr¨obner bases. University of Cincinnati [email protected] Bernd Sturmfels MPI Leipzig and UC Berkeley Dustin Mixon [email protected] Air Force Institute of Technology [email protected]

MS8 Generalized Camera Models and Multi-view Ge- MS9 ometry An Overview of Applications of Algebra and Ge- ometry to Frame Theory Very general imaging systems can be modeled geometri- 3 cally as mappings from P to a line congruence. This Algebraic, geometric, and algebro-geometric techniques viewpoint applies to many existing camera models, such have proven useful in answering several important ques- as two-slit, panoramic, catadioptric cameras, and many tions in frame theory while also motivating concrete con- more. The multi-view geometry of these systems can be structions of frames satisfying various desirable properties. studied using the concurrent lines variety, which consists In this talk we summarize some of the main approaches as 3 of n-tuples of lines in P that intersect at a point. By fur- well as open questions. ther specifying the mapping from visual rays to P2,several classical features of perspective cameras, such as intrinsic Nate Strawn and extrinsic parameters, can be defined in a more general Department of Mathematics and Statistics setting. Joint work with John Canny, Martial Hebert, Jean Georgetown University Ponce, and Bernd Sturmfels. [email protected]

Matthew Trager Inria / Ecole Normale Superieure de Paris MS9 [email protected] Classifying Complex Equiangular Tight Frames by Computational Algebraic Geometry

MS8 In this talk we give an overview of the computational al- Two Hilbert Schemes in Computer Vision gebraic geometric approach to construct and classify com- plex equiangular tight frames (ETF). As an application, We discuss the abstract moduli of multiview varieties in we show how to classify all ETFs in dimension 3, and ex- both the calibrated and uncalibrated cases. By carefully plore for the first time ETFs in dimension 4. The main phrasing the moduli problems, we get natural morphisms algebraic tools are nontrivial algebraic identities involving to Hilbert schemes – the usual Hilbert scheme in the un- complex numbers of modulus 1 discovered by U. Haagerup calibrated case and a flag Hilbert scheme in the calibrated in 1996; and the main computational tool is J.-C. Faugere’s case. Deformation theory shows that the images of these C library FGb for efficiently computing a Grobner basis. morphisms are open subschemes of single irreducible com- ponents of the relevant Hilbert schemes. This reproduces Ferenc Sz¨ollosi 76 AG17 Abstracts

Department of [email protected] [email protected]

MS10 MS9 New Bounds for Equiangular Lines and Spherical Extreme Diagonally and Antidiagonally Symmetric Two-distance Sets Alternating Sign Matrices of Odd Order

The set of points in a metric space is called an s-distance For each α ∈{0, 1, −1}, we count alternating sign matri- set if pairwise distances between these points admit only s ces that are invariant under reflections in the diagonal and distinct values. Two-distance spherical sets with the set of in the antidiagonal (DASASMs) of fixed odd order with scalar products {a, −a}, a in [0, 1), are called equiangular. a maximal number of α’s along the diagonal and the an- The problem of determining the maximal size of s-distance tidiagonal, as well as DASASMs of fixed odd order with sets in various spaces has a long history in mathematics. a minimal number of 0’s along the diagonal and the an- We determine a new method of bounding the size of an tidiagonal. In these enumerations, we encounter product s-distance set in two-point homogeneous spaces via zonal formulas that have previously appeared in plane partition spherical functions. This method allows us to prove that or alternating sign matrix counting, namely for the num- the maximum size of a spherical two-distance set in Rn is ber of all alternating sign matrices, the number of cyclically n(n +1)/2 with possible exceptions for some n =(2k + symmetric plane partitions in a given box, and the num- 1)2 − 3, where k is a natural number. We also prove the ber of vertically and horizontally symmetric ASMs. Our universal upper bound (2/3)n/a2 for equiangular sets and, proofs are non-bijective (which is the standard situation in employing this bound, prove a new upper bound on the size this field), and so each of our theorems gives rise to a new of equiangular sets in an arbitrary dimension. Finally, we open problem of finding a bijective proof. classify all equiangular sets reaching this new bound. Arvind Ayyer Wei-Hsuan Yu Indian Institute of Science Department of Mathematics [email protected] Michigan State University [email protected] Roger Behrend Cardiff University behrendr@cardiff.ac.uk MS10 Rational Solutions of Linear Mahler Equations Ilse Fischer University of Vienna Much as differential equations use derivation for their basic ilse.fi[email protected] operator, so do Mahler equations the substitution opera- tor f(x) → f(xb). These equations have been introduced by Kurt Mahler in the late 1920’s in order to prove tran- MS10 scendence results. They are also related to the complex- ity of divide-and-conquer algorithms or to combinatorics Can You Count Genus 2 Surfaces in 3-Manifolds? of words. Mahler’s method for transcendence recently led to the need of solving Mahler equations. In this talk, we We will discuss a counting of genus 2 surfaces (incompress- concentrate on the search for rational solutions with coef- ible, or fully compressible) in 3-manifolds, and a counting ficients in a computable number field. One difficulty in of genus 2 normal and almost normal surfaces in triangu- the study of the Mahler equations is the degrees blow- lated 3-manifolds, and show the matching of the two counts up when the Mahler operator is applied to polynomials. using an invariant from string theory: the 3Dindex. Joint Classical methods for the difference equations do not ap- wowrk with N. Dunfield, C. Hodgson and H. Rubinstein. ply because it is necessary to replace the linear ordering of integers by a tree structure. We show that it is possible Stavros Garoufalidis to build a rational resolution algorithm whose complexity Georgia Tech is polynomial with respect to the size of the result. Joint [email protected] work (https://arxiv.org/abs/1612.05518)withFr´ed´eric Chyzak (Inria), Thomas Dreyfus (Univ. Lyon 1) et Marc Mezzarobba (Univ. Paris 6). MS10 Fr´ed´eric Chyzak Desingularization in Several Variables INRIA and Ecole´ Normale Sup´erieuredeLyon [email protected] Desingularization is the process of constructing from a given linear differential operator a higher order operator Thomas Dreyfus which has as few singularities as possible. The singulari- Universit´eLyon1 ties which can be removed in this way are called apparent [email protected] singularities. Desingularization is a classical procedure in the univariate case, but not so in the case of several vari- Philippe Dumas ables. We propose a notion of apparent singularities for INRIA multivariate D-finite functions and show that the classi- [email protected] cal desingularization techniques generalize to this setting. This is joint work with Yi Zhang and Ziming Li. Marc Mezzarobba INRIA Manuel Kauers Universite Lyon, AriC, LIP Johannes Kepler University AG17 Abstracts 77

[email protected] algorithm for such varieties. We prove analogous theoreti- cal results for genus two Jacobians with real multiplication by maximal orders, and we discuss the challenges involved MS11 in the practical implementation, such as the computation Improving Complexity Bounds for Hyperelliptic of suitable modular ideals. Point-counting Sean F. Ballentine Counting points on (Jacobians of) curves over finite fields University of Maryland and computing their zeta functions are important routines [email protected] for effective number theorists as well as cryptologists. We propose an algorithm alaSchoof for hyperelliptic curves Aurore Guillevic and study its complexity in large characteristic and high Inria Nancy Grand Est genus. Our algorithm heavily relies on the computation [email protected] of the -torsion of the Jacobian for sufficiently many , which we perform by computing geometric resolutions of Elisa Lorenzo Garcia polynomial systems built from Cantor’s analogues to divi- Universite de Rennes 1 sion polynomials for hyperelliptic curves. Using complexity [email protected] bounds for the computation of geometric resolutions, this allows us to improve on previous complexity bounds by Chloe Martindale Adleman and Huang. Universiteit Leiden Simon Abelard [email protected] Universit´e de Lorraine [email protected] Maike Massierer University of New South Wales Pierrick Gaudry [email protected] CNRS [email protected] Benjamin Smith NRIA and Laboratoire dInformatique de lEcole´ Pierre-Jean Spaenlehauer polytechnique INRIA [email protected] [email protected] Jaap Top University of Groningen MS11 [email protected] Short Generators without Quantum Computers: TheCaseofMultiquadratics MS11 Finding a short element g of a number field, given the Accelerating Point-counting Algorithms using Ex- ideal generated by g, is a classic problem in computa- plicit Endomorphisms tional algebraic number theory. Solving this problem re- covers the private key in cryptosystems introduced by Following the work of Schoof and Kedlaya, the most use- Buchmann–Maurer–M¨oller, Gentry, Smart–Vercauteren, ful modern point-counting algorithms for curves over finite Gentry–Halevi, Garg–Gentry–Halevi, et al. Work over the fields involve computing the characteristic polynomial of last few years has shown that for some number fields this the Frobenius endomorphism, restricted to one or more problem has a surprisingly low post-quantum security level. vector spaces associated with the curve. If C is a curve The point of this talk is that for some number fields this over Fq, for example, then Schoof-type algorithms consider problem has a surprisingly low pre-quantum security level. the action of Frobenius on torsion subgroups of the Ja- This is joint work with Jens Bauch, Henry de Valence, cobian of C, while Kedlaya-type algorithms consider the Tanja Lange, and Christine van Vredendaal. action of Frobenius on a certain cohomology group derived from the differentials of C. But the Jacobians of many Daniel J. Bernstein curves that we encounter in number theory and cryptog- University of Illinois at Chicago raphy come equipped with efficiently computable special [email protected] endomorphisms. In this talk we will describe some point- counting algorithms which can exploit this, using special endomorphisms in conjunction with Frobenius to count MS11 points faster. Improvements to Point Counting on Hyperelliptic Curves of Genus Two Benjamin Smith NRIA and Laboratoire dInformatique de lEcole´ Schoof’s algorithm is a standard method for counting polytechnique points on elliptic curves defined over finite fields of large [email protected] characteristic, and it was extended by Pila to higher- dimensional abelian varieties. Improvements by Elkies and Atkin lead to an even faster method for elliptic curves, MS12 known as the SEA algorithm. This is the current state of Fast Submodular Function Minimization the art for elliptic curves. Motivated by the fact that Ja- cobians of hyperelliptic curves of genus two have recently A function f :2U → R is submodular if for any pair of been found to be good alternatives to elliptic curves in subsets S, T ⊆ U over a universe U of n elements, we have cryptography, we investigate the possibility of applying the improvements of Elkies and Atkin to Pila’s point counting f(S ∪ T )+f(S ∩ T ) ≤ f(S)+f(T ) 78 AG17 Abstracts

These function arise in multiple areas and form the discrete and Semialgebraic Proofs analog of convex functions. One paradigmatic problem is that of submodular function minimization. Recently, con- Inspired by the breakthroughs of the polyhedral method tinuous optimization techniques have given rise to fast al- for combinatorial optimization in the 1980s, generations gorithms for submodular function minimization (SFM). In of researchers have studied the facet structure of convex this talk we will describe the state of the art, and some hulls to develop strong cutting planes. We ask how much faster algorithms approximate and bounded submodular of this process can be automated: In particular, can we functions. use algorithms to discover and prove theorems about cut- ting planes? We focus on general integer and mixed integer Deeparnab Chakrabarty programming, rather than combinatorial optimization, and Dartmouth College use the framework of cut-generating functions (Conforti et [email protected] al. 2013). Our approach is pragmatic. Our theorems and proofs come from a metaprogramming trick, applied to a grid-free implementation of the Basu–Hildebrand–K¨oppe MS12 (2014) extremality test for cut-generating functions in the Some Cut-generating Functions for Second-order Gomory–Johnson (1972) model; followed by practical com- Conic Set putations with semialgebraic cell complexes. The correct- ness of our proofs depends on that of the underlying imple- We study cut generating functions for conic sets. Our first mentation. Using our implementation of this approach, we main result shows that if the conic set is bounded, then cut have verified various theorems from the literature, found generating functions for integer linear programs can easily corrections to several published theorems, and discovered be adapted to give the integer hull of the conic integer new families of cut generating functions and corresponding program. Then we introduce a new class of cut gener- theorems with automatic proofs. The plan is to produce ating functions which are non-decreasing with respect to such new theorems in quantity. second-order cone. We show that, under some minor tech- Matthias Koeppe nical conditions, these functions together with integer lin- Dept. of Mathematics ear programming-based functions are sufficient to yield the 2 Univ. of California, Davis integer hull of intersections of conic sections in R . [email protected] Santanu Dey Georgia Institute of Technology Yuan Zhou [email protected] UC Davis [email protected]

MS12 MS13 Ordered Sets in Discrete Optimization Gram Spectrahedra

For a compact convex set S, the polytope defined by the The sum-of-squares representations of a given real poly- maximal and minimal points under a total order is a valid ∩ Zn nomial in several variables are parametrized by a convex relaxation of S . We first extend a recent study of body, its Gram spectrahedron. We discuss results on sums the lexicographic (lex) polytope by studying the combi- of squares that fit naturally into this context and relate natorial properties and the graph of the graded lex and to the minimum length and exact computations of sum- graded reverse lex polytopes. Each of these is shown to be ≤ of-squares representations. (Joint work with Lynn Chua, a(d, 2d)-polytope with diameter 3 and whose graphs are Rainer Sinn, and Cynthia Vinzant) expanders. Second, we show that for any S ∩{0, 1}n which ∩{ }n is a independence system, S 0, 1 is equal to the inter- Daniel Plaumann section of all its lex-ordered sets. This generalizes what is TU Dortmund commonly known about matroid and knapsack polytopes [email protected] and yields new insights into the convex hull through the combinatorial interplay between different lex-ordered sets. We argue that in general, lex optima yield a tight approx- MS13 imation factor of 1/n to the optimum over S ∩ Zn, but for ⊆ n On the Complexity of Deciding Connectivity S [0, 1] , they provide the integer optimum. We further Queries. Algorithms and Implementations prove that the bound provided by an arbitrary subset of lex-ordered sets is NP-hard to compute in general and clas- In this talk, we will review recent results on algorithms for sify easy and hard cases of lexicographic integer optimiza- deciding connectivity queries in real algebraic sets, and dis- tion. Finally, we will also use lex optimal points to define cuss experimental implementations. Joint work with Mo- a new two-term disjunction for mixed-integer optimization hab Safey El Din. and separate cutting planes from it, along with strengthen- ing commonly-used cuts such as the Gomory mixed integer Eric Schost cut. Computer Science Department Univ. of Western Ontario Akshay Gupte [email protected] Department of Mathematical Sciences Clemson University [email protected] MS13 Computing Real Radicals of Polynomial Ideals MS12 Assume that V = V√C (I) is a smooth affine algebraic vari- Automated Discovery of Cutting Plane Theorems ety, we show that re I has a finite set of generators with AG17 Abstracts 79

degrees bounded by deg(V ). We extend the probabilistic [email protected] algorithm given by Blanco, Jeronimo√ and Solern´o[JSC, 2004] for computing√ generators of I to computing gener- re MS14 ators of I with degrees bounded by deg(V ). The com- plexity of the probabilistic algorithm is singly exponential Semidefinite Programming and Numerical Alge- in the√ number of variables. Some discussions on comput- braic Geometry ing re I when V is a singular algebraic variety will also be Traditional interior point methods for semidefinite pro- given. This is ongoing joint work with Mohab Safey El Din gramming track along the central path, which arises as and Zhihong Yang. a solution curve to a system of polynomial equations from first-order optimality conditions. We utilize numerical al- Zhihong Yang gebraic geometry to track along solution paths arising from University of Chinese Academy of Sciences families of semidefinite programming problems. Examples [email protected] are used to compare the techniques with classical methods for solving semidefinite programs. Lihong Zhi Academia Sinica Jonathan Hauenstein [email protected] University of Notre Dame Dept. of App. Comp. Math. & Stats. [email protected] MS13 Alan Liddell On the Complexity of Root Clustering University of Notre Dame Dept. Appl. Comp. Math. Stat. Root isolation for a complex polynomial F (z) is a classical [email protected] computational problem. When the coefficients of F (z)are integers, the global complexity of this problem is relatively Yi Zhang well-understood from the work of Sch¨onhage (1982) and University of Notre Dame Pan (1997-2002). When the coefficients of F (z)aregen- [email protected] eral complex numbers, possibly non-algebraic, the root iso- lation problem might not be computable, much less have complexity analysis. This talk we describe some recent MS14 work to get around this barrier. First, we need to gen- Polynomial System Solving and Analytic Combina- eralize root isolation to root clustering problem. Second, torics in Several Variables we must consider a computational model in which the co- efficients of F (z) are “number oracles’ that can return a The field of analytic combinatorics studies the asymptotic p-bit absolute approximation for any given integer p.In behaviour of sequences through analytic properties of their this setting, we recently show (ISSAC 2016) that it is pos- generating functions. In addition to the now classical uni- sible to achieve complexity bounds for root clustering that variate theory, recent work in the study of analytic combi- are “near-optimal’ (in the sense of Pan). Our algorithm is natorics in several variables (ACSV) has shown how to de- based on subdivision, has explicit error bounds and there- rive asymptotics for the coefficients of certain D-finite func- fore implementable. We will also discuss the extension of tions by representing them as diagonals of multivariate ra- this work to the case where F (z)isanalytic. tional functions. Ultimately, the theory as it has currently developed requires one to solve several algebraic and semi- Chee K. Yap algebraic problems on certain polynomial systems. This New York University talk will quickly survey ACSV from a computer algebra Courant Institute viewpoint, and discuss the computational challenges in- [email protected] volved. Stephen Melczer University of Waterloo MS14 [email protected] Univariate Homotopy Continuation via Interval Arithmetic MS14 Polynomial System Solving and Numerical Linear Homotopy continuation is a main computational method Algebra in numerical algebraic geometry, and it provides a power- ful tool for finding the solutions to systems of polynomial This talk is about the problem of solving zero-dimensional equations. In this talk, I will present a predictor-corrector systems of polynomial equations using numerical linear al- method based on interval arithmetic and Newton-bisection gebra techniques. Let k be an algebraically closed field for homotopy continuation applied to univariate polynomi- and let p1 = p2 = ... = pn = 0 define such a system in n als. This new method for homotopy continuation is (1) effi- k : pi ∈ k[x1,...,xn]. It is a well known fact that the va- cient in practice, (2) can provide certified results for various riety V (I)oftheidealI = p1,...,pn is finite if and only predictor steps (such as Euler steps), and (3) takes fewer if the quotient ring k[x1,...,xn]/I is finite dimensional as steps than other certified homotopy continuation methods. a k-vector space. Multiplication in k[x1,...,xn]/I by a The interval homotopy method is implemented as a library polynomial f is a linear operation. Choosing a basis B for in c++. k[x1,...,xn]/I, this multiplication can be represented by a matrix mf called a multiplication matrix. The eigenstruc- Michael A. Burr ture of these multiplication matrices reveals the solutions Clemson University of the system. Methods using Groebner bases to calculate 80 AG17 Abstracts

the multiplication matrices make an implicit choice for the urally in various important computational science and en- basis of the quotient ring. From a numerical point of view, gineering problems. The problem is known to be computa- this is often not a very good choice. Significant improve- tionally hard and suffering from potential ill-conditioning, ment can be made by using elementary numerical linear and so researchers have turned their attention to numer- algebra techniques on a Macaulay-type matrix, which is ical reformulations. Among others, we mention that the built up by the coefficients of the pi, to choose a basis for decomposition of (n + 1)-dimensional symmetric tensors k[x1,...,xn]/I. In this talk we will present this technique is closely related to exponential analysis (from signal pro- and show how the resulting method can handle challeng- cessing) and sparse interpolation (from computer algebra). ing systems, both dense and sparse with a large number of Binary tensor decomposition (n + 1 = 2) can also be re- (possibly multiple) solutions. formulated as a Pad´e approximation or moment matching problem (from approximation theory). Simon Telen KU Leuven Annie Cuyt, Wen-shin Lee Dept. of Computer Science University of Antwerp [email protected] [email protected], [email protected]

Marc Van Barel Department of Computer Science MS15 Katholieke Universiteit Leuven, Belgium [email protected] A Tensor Decomposition Method for Inverse Source Problems

MS15 In this talk, we consider an application of CP decompo- Computing Structured Matrix Factorizations with sition of a tensor to bio-electromagnetic inverse problems. Applications in Tensor Decompositions and Blind Magnetoencephalography (MEG) is one of imaging modal- Source Separation ities in which the neural current inside the human brain is inversely reconstructed from the magnetic field measured Let V ⊂ CN be an irreducible non-degenerate al- on the head surface. To this inverse problem, we have gebraic variety, v1,...,vR ∈ V and J(v1,...,vR)= proposed an algebraic reconstruction algorithm by using span{v1,...,vR}. The classical trisecant lemma guaran- Pronys method: the positions of equivalent current dipoles tees that, if R

[email protected] parative approach to codiversification. Using the graph spectra of networks of interacting organisms, we construct a similarity metric that permits the use of established clus- MS16 tering and machine learning techniques to classify interac- New Perspectives on Rooted Trees and Parameter tions that exhibit similar phylogenetic and ecological struc- Spaces tures. Using this method, we are able to propose roles for unidentified host-associated bacterial clades based on the We describe compact spaces that parametrize configura- similarity of their interactions with their hosts to ecological tions of either points or lines in projective space and which interactions where the roles are known. combinatorics are controlled by rooted trees. We also study the relation of these parameter spaces with configurations Russell Y. Neches of tropical points and phylogenetic trees. This talk covers UC Davis work with N. Giansiracusa, E. Routis, and K. Ascher. [email protected]

Patricio Gallardo University of Georgia MS17 [email protected] Algebraic Identifiability of Gaussian Mixtures

Noah Giansiracusa We study the problem of recovering the parameters that Swarthmore College define a Gaussian mixture via its moments up to a certain [email protected] order. In this setting, we say that mixtures of Gaussians are algebraically identifiable if the map from the model pa- rameters to the moments is generically finite-to-one. Geo- Kenny Ascher metrically, this translates into determining whether or not Brown University the corresponding secant moment varieties have the ex- kenneth [email protected] pected dimension. In our results, we show the contrast between the univariate case and the one for higher dimen- Evangelos Routis sional Gaussians. Based on joint work with Kristian Ranes- IPMU tad and Bernd Sturmfels. [email protected] Carlos Amendola Ceron Technical University Berlin MS16 [email protected] Efficient Quartet Systems

Quartet trees displayed by larger phylogenetic trees have MS17 long been used as inputs for species tree and supertree re- Identifiability of a Multispecies Coalescent Model construction. Computational constraints prevent the use for the Evolution of k-mer Distances on a Phyloge- of all displayed quartets in many practical problems with netic Tree large numbers of taxa. We introduce the notion of an Ef- ficient Quartet System (EQS) to represent a phylogenetic A widely-used heuristic in molecular phylogenetics involves tree with a subset of the quartets displayed by the tree. reconstructing a phylogenetic tree on n taxa before ob- We show mathematically that the set of quartets obtained taining a sequence alignment. To develop a statistically- from a tree via an EQS contains all of the combinatorial consistent method in this context, we construct a multi- information of the tree itself. Using performance tests on species coalescent model for the evolution of the k-mer dis- simulated datasets, we also demonstrate that using an EQS tances between sequences. We give conditions on n and k to reduce the number of quartets in both summary method for which there exist phylogenetic invariants that can be pipelines for species tree inference as well as methods for used to distinguish between species trees. We then suggest supertree inference results in only small reductions in ac- how our identifiability results can be used to develop an curacy. algorithm for phylogenetic tree reconstruction. MaLyn Lawhorn Chris Durden Winthrop University North Carolina State University [email protected] [email protected]

MS16 MS17 Predicting Ecological Function in the Microbiome Identifiability of Species Phylogenies under a Mod- using Spectral Properties of Interacting Clades ified Coalescent

As a single lineage evolves and branches into a group of Coalescent models of evolution account for incomplete lin- distinct new lineages, it is said to diversify. However, it is eage sorting by specifying a species tree parameter which rarely the case that this process occurs in isolation; evolv- determines a distribution on gene trees. It has been shown ing lineages interact with one another with varying inten- that the unrooted topology of the species tree parame- sity over time. When the histories of interacting lineages ter of the multispecies coalescent is generically identifiable. are inferred, their linked phylogenetic trees are sometimes Moreover, a statistically consistent reconstruction method described as “tangled trees.” Most existing methods for in- called SVDQuartets has been developed to recover this pa- ferring the evolutionary history of these systems fall into rameter. In this talk, we describe a modified multispecies two categories; either they test for codiversification by fit- coalescent model that allows for varying effective popula- ting data to an idealized model, or in cases where codi- tion size and violations of the molecular clock. We show versification is known to have occurred, they endeavor to that the unrooted topology of the species tree for these predict the most likely history. Here, we present a com- models is generically identifiable and that SVDQuartets 82 AG17 Abstracts

is still a statistically consistent method for inferring this (solvers) that solve specific classes of systems of polynomial parameter. equations. In this talk we will briefly discuss such method for creating efficient solvers of systems of polynomial equa- Colby Long tions. This method is based on Grbner bases and it uses Ohio State University the structure of the system of polynomial equations repre- [email protected] senting a particular problem to design an efficient specific solver for this problem. We will discuss several approaches for improving the efficiency of the final solvers. We will also MS18 introduce the automatic generator of Grbner basis solvers The Nearest Point to a Variety Problem, Near the which could be used even by non-experts to efficiently solve Variety problems resulting in systems of polynomial equations. Fi- nally, we will demonstrate the usefulness of the approach We consider the problem of finding the nearest point to an by presenting new, efficient and numerical stable solutions algebraic variety. This includes as special instances several to several important computer vision problems and prob- important estimation problems, such as the triangulation lems from robotics. problem, camera resectioning, low rank approximation and approximate GCD. Although these problems are typically Zuzana Kukelova nonconvex, tractable SDP relaxations can be derived. We Czech Technical University in Prague study sufficient conditions that guarantee that SDP relax- [email protected] ations are tight under low noise. As an application, we provide a novel SDP relaxation for the nearest point prob- lem to a linear fractional variety, and we show that the MS18 relaxation is tight under low noise. Multi-focal Tensors from Invariant Differential Diego Cifuentes Forms, and Line Envelopes for Structure-from- Massachusetts Institute of Technology Motion [email protected] I will explain a way to regard the n-focal tensors of the ma- chine vision literature as pointwise expressions of certain MS18 weighted GL(V )-invariant algebraic differential forms on a Minimal Problems for the Calibrated Trifocal Va- space Fn of n-tuples of frames for tangent spaces of a pro- riety jective space PV . The forms arise from GL(V ) invariants in Λp1 V ∨ ⊗···⊗Λpn V ∨. The construction provides natural In computer vision, 3D reconstruction is a fundamental embeddings of Fn//GL(V ) into a linear space, and hence task: starting from photographs of a world scene, taken natural coordinates for this configuration space. I will also by cameras with unknown positions and orientations, how present another application of general linear invariant the- can we best create a 3D model of that world scene? Al- ory to visual geometry: a formula for the focal envelope of gorithms that do this built Street View (Google) and are a line family, which is the basis for a new structure-from- instrumental in autonomous robotics. In 2004, David Nis- motion algorithm. ter (Tesla) used Grobner bases to build a solver for robust reconstruction given just two photographs. This is a key James Mathews routine in much larger-scale reconstructions today. In this Stony Brook University talk, I will discuss reconstruction given three photographs, [email protected] where efforts to replicate Nister have so far proven elusive. My approach relies on applied algebraic geometry. In par- ticular, I shall introduce an algebraic variety whose points MS19 are 3x3x3 tensors in correspondence with configurations of three calibrated cameras. Special linear sections of this va- On the Structure of Maximal 2nd Degree Leven- riety recover camera configurations from image data. The schtein Frames main result is the determination of the algebraic degree of minimal problems for this recovery. These comprise inter- In terms of Levenschtein’s framework for characterizing esting enumerative geometry problems; the solution is by certain Grassmannian frames, maximal equiangular tight way of homotopy continuation calculations. frames (eg, SIC-POVMs) are maximal in the sense that they are the Grassmannian frames of largest cardinality Joe Kileel (given a fixed dimension) which may be described with a UC Berkeley polynomial of degree one. In this talk, we consider maximal [email protected] second degree Levenschtein frames, the largest Grassman- nian frames for which optimal coherence may be proven with second degree polynomials. As with the maximal de- MS18 gree one case, such frames are difficult to construct, as they Fast Groebner Basis Solvers for Computer Vision must satisfy rigid geometric and reconstructive properties. Based on our observations of known examples, we enter- Many problems in computer vision, but also in other fields tain a conjecture that such frames partition into highly such as robotics, control design or economics, can be for- symmetric systems of orthonormal bases (akin to the de- mulated using systems of polynomial equations. For com- composition of mutually unbiased bases) and develop ex- puter vision problems, general algorithms for solving poly- istence and construction principles based on block designs nomial systems cannot be efficiently applied. The reasons which hold if the conjecture is true. are twofold - computer vision and robotic applications usu- ally require real time solutions, or they often solve systems John Haas of polynomial equations for millions of different instances. Department of Mathematics Several approaches based on algebraic geometry have been University of Missouri recently proposed for the design of very efficient algorithms [email protected] AG17 Abstracts 83

Dustin Mixon Theory Air Force Institute of Technology [email protected] Polytopes appear in the theory of finite frames as images of certain frame varieties under a map that associates to Nathaniel Hammen each frame a sequence of spectra of frame operators. The Department of Mathematics and Statistics appearing so called sequences of eigensteps have been char- Air Force Institute for Technology acterized in terms of linear inequalities and hence form a [email protected] polytope. We discuss two cases of this situation, where the appearing polytopes coincide with Gelfand-Tsetlin poly- topes, whose integral points enumerate bases for irreducible MS19 representations of the complex general linear Lie algebra. Equality Conditions for the Levenstein Bounds in Christoph Pegel Projective Space Department of Mathematics University of Bremen Optimally low coherence frames lead to good compressed [email protected] sensing matrices. Nearly all currently known optimal pack- ings satisfy equality with either the Welch or orthoplex bounds. For a frame in d dimensional space, this requires a MS20 maximum frame size of d(d+1) vectors. Beyond this region, the Levenstein and Delsarte bounds apply. By studying On the Enumeration of Compacted Trees equality conditions for the Levenstein or Delsarte bounds, we may find packings that achieve optimality beyond this A compacted tree is a directed acyclic graph in which every ceiling. subtree of a given tree is represented in such a way that repeatedly occurring subtrees are represented only once. John Haas Further occurrences are realized by pointers to already Department of Mathematics present representations. We are interested in the asymp- University of Missouri totic number of compacted binary trees of size n where [email protected] the size equals the number of internal nodes. The diffi- culties originate from the superexponential growth of the ∼ n Nathaniel Hammen counting sequence, cn O(n!4 ), and the fact that the Department of Mathematics and Statistics structures are unlabelled. We solve the counting problem Air Force Institute for Technology for the particular subclass of binary compacted trees where [email protected] the so-called right height is bounded. This is achieved by first deriving a calculus for exponential generating func- tions for this particular class of unlabeled combinatorial Dustin Mixon objects. This leads to a sequence of ordinary differential Air Force Institute of Technology equations whose pattern can be determined by a guess and [email protected] prove approach. The solutions turn out to have regular singularities which can be determnined by analysing the indicial polynoimials, which eventually leads to the asymp- MS19 totic number of compacted binary trees with bounded right Spark Deficient Gabor Frames height. This is joint work with Antoine Genitrini, Manuel Kauers, and Michael Wallner A set in a vector space of dimension n has full spark when every subset of n elements is linearly independent; oth- Bernhard Gittenberger erwise, it is called spark deficient. For a function f de- Institute for Discrete Mathematics and Geometry, TU fined on a finite Abelian group, the Gabor frame (or Weyl- Wien Heisenberg orbit) is the set of all shift-frequency translates [email protected] of f, and their number is N 2,whereN is the size of the group. We will show that when the underlying group is MS20 not cyclic, then every Gabor frame is spark deficient. In Multivariate Algebraic Generating Functions: the cyclic case, it was shown by the speaker that almost Asymptotics and Examples every f generates a full spark Gabor frame. Returning to the cyclic case, we will briefly mention the connection We find a formula for the asymptotics of the coefficients of the full spark property to another important aspect of −β of a generating function of the form, H(z1,z1, ..., zd) ,as Weyl-Heisenberg orbits, namely the equiangularity prop- the indices approach infinity in a fixed ratio. Then, we erty (SIC-POVM problem). In particular, if Zauner’s con- look at how these results can be applied to some generat- jecture is true (existence of equiangular Gabor frames in ing functions. This work relies on the techniques in multi- all dimensions), then such a frame is spark deficient in di- variate analytic combinatorics developed by Pemantle and mensions 6n +3. Wilson. We combine the multivariate Cauchy integral for- mula with explicit contour deformations to compute the Romanos D. Malikiosis asymptotic formula. A change of variables will allow us to Department of Mathematics break the Cauchy integral into a univariate Cauchy inte- Technical University of Berlin gral and a (d − 1)-dimensional Fourier Laplace integral. A [email protected] challenge of using the formula is correctly identifying the points which contribute to asymptotics.

MS19 Torin Greenwood Polytopes in Frame Theory and Representation Georgia Institute of Technology 84 AG17 Abstracts

[email protected] University of Neuchˆatel [email protected]

MS20 Elisa Gorla Congruences Connecting Modular Forms and University of Neuchatel Truncated Hypergeometric Series [email protected] We prove a supercongruence modulo p3 between the pth Fourier coefficient of a weight 6 modular form and a trun- MS21 cated 6F5-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to Security Considerations for Galois Non-dual ζ(3) to produce non-trivial harmonic sum identities and the RLWE Families reduction of the resulting congruences between harmonic sums via a congruence between the Ap´ery numbers and We explore further the hardness of the non-dual discrete another Ap´ery-like sequence. We will highlight the role of variant of the Ring-LWE problem for various number rings, computer algebra in this work, and give explicit examples give improved attacks for certain rings satisfying some ad- indicating the need for algorithmic approaches to certify- ditional assumptions, construct a new family of vulnerable ing A ≡ B. This is joint work with Robert Osburn and Galois number fields, and apply some number theoretic re- Wadim Zudilin. sults on Gauss sums to deduce the likely failure of these attacks for 2-power cyclotomic rings and unramified mod- Armin Straub uli. University of Illinois at Urbana-Champaign [email protected] Hao Chen,KristinLauter Microsoft Research [email protected], [email protected] MS20 3-Dimensional Lattice Walks and Related Groups Katherine Stange University of Colorado, Boulder AmodelS is a subset of {−1, 0, 1}3 \{0, 0, 0} and S-walk [email protected] denotes any walk which starts from the origin (0, 0, 0) and takes its steps in S. Bostan, Bousquet-M´elou, Kauers and Melczer investigated 3-dimensional lattice walks restricted MS21 N3 to the non-negative octant . They conjectured that Multivariate Public Key Cryptosystems - Candi- many 3-dimensional walks were associated with a group dates for the Next Generation Post-quantum Stan- of an infinity order. In this note, we confirm these conjec- dards tures via the fixed point method and valuation strategy. Multivariate public key cryptosystems (MPKC) are one of Ronghua Wang the four main families of post-quantum public key cryp- Institue for algebra, Johannes Kepler University, Austria tosystems. In a MPKC, the public key is given by a set [email protected] of quadratic polynomials and its security is based on the hardness of solving a set of multivariate polynomials. In this talk, we will give an introduction to the multivariate Qinghu Hou, kang Du public key cryptosystems including the main designs, the Tianjin University, China main attack tools and the mathematical theory behind in qh [email protected], [email protected] particular algebraic geometry. We will also present state of the art research in the area. MS21 Jintai Ding Solving Degree of Multivariate Polynomial Systems University of Cincinnati and Castelnuovo-Mumford Regularity [email protected] The security of several post-quantum cryptoschemes is based on the assumption that solving a system of MS21 (quadratic) polynomial equations p1 = ··· = pm =0 in several variables over a finite field is hard. A com- Cyclic SRP - Cryptanalysis of the SRP Encryption mon efficient strategy to solve such a system is computing Scheme a lexicographic Groebner basis of the ideal (p1,...,pm). The fastest known algorithms for this scope are algorithms Multivariate Public Key Cryptography (MPKC) is one of which transforms the polynomial solving process into sev- the main candidates for secure communication in a post- eral steps of solving linear systems, such as F4 and F5. quantum era. Recently, Yasuda and Sakurai proposed To understand the complexity of these algorithms, it is a new multivariate encryption scheme called SRP, which important to know the highest degree of the polynomials combines the Square encryption scheme with the Rainbow involved during the computations, the so-called solving de- signature scheme and the Plus modifier. In this paper we gree of the system. We prove that the solving degree is propose a new attack against the SRP scheme, which is controlled by the Castelnuovo-Mumford regularity of the based on the min-Q-rank property of the system. Our at- h h homogeneous ideal (p1 ,...,pm) obtained by homogeniz- tack is very efficient and allows us to break the recom- ing the input polynomials. We apply this result to obtain mended parameter sets within minutes. Our attack shows bounds on the solving degree of zero-dimensional ideals and that combining a weak scheme with a secure one does not of several instances of the MinRank problem. increase automatically the security of the weak one. This is a joint work with Ray Perlner and Daniel Smith Tone Alessio Caminata (National Institute of Standards and Technology, Gaithers- AG17 Abstracts 85

burg, Maryland, USA). makes evident an interaction between the involved lengths, in the sense that not all of them can be simultaneously Albrecht R. Petzoldt very small. In this talk, we will show that by considering Technische Universit¨at Darmstadt divided differences, the Davenport-Mahler bound can be [email protected] extended to arbitrary graphs with vertices on the set of roots of a given univariate polynomial, and we show some applications. MS22 Initial Degenerations of the Grassmannian Daniel Perrucci, Paula Escorcielo The tropical Grassmannian TGr(d, n) parameterizes initial Departamento de Matem´atica degenerations of Gr(d, n) on the locus where all Pl¨ucker co- Universidad de Buenos Aires ordinates are nonvanishing. This comes equipped with two [email protected], [email protected] fan structures: the Groebner fan determined by initial de- generaions and the secondary fan determined by matroid decompositions of the hypersimplex. In this talk, I will use MS23 these two fan structures to relate the initial degenerations of Gr(d, n) to fiber products of matroid realization spaces. Improved Bounds for Equivariant Betti Numbers As an application, I will discuss smoothness of initial de- of Symmetric Semi-algebraic Sets generations for Grassmannians for small values of d and n. Let R be a real closed field and S ⊂ Rk be a semi-algebraic set defined by symmetric polynomials whose degree is d. Dan Corey The quantitative study the equivariant rational cohomol- Yale University ogy groups of such symmetric semi-algebraic sets was ini- [email protected] tiated recently and it was shown that in sharp contrast to the ordinary Betti numbers, the equivariant Betti num- bers can be bounded by quantity that is polynomial in the MS22 number of variables k. Theses results were obtained using Introduction to Khovanskii Bases, Valuations and arguments from equivariant Morse-theory. In this talk we Newton-Okounkov Bodies present a different method which improves the previous re- sults. In particular, we show that the equivariant cohomol- We discuss basic concepts, constructions and results about ogy groups vanish in dimension d and larger and give new Khovanskii bases and Newton-Okounkov bodies. We start bounds on the equivariant Betti numbers for this setup. by reviewing basic facts about valuations. Khovanskii More importantly, this new approach yields an algorithms bases are a far generalization of the notion of a SAGBI with polynomially bounded (in k) complexity for comput- basis for a sublagebra of polynomials to the setting of al- ing these equivariant Betti numbers. This is in sharp con- gebras equipped with valuations. Similarly, the notion of trast to the situation of a general semi-algebraic set, where a Newton-Okounkov body is a far generalization of the no- not even an algorithm with a single exponential complexity tion of the Newton polytope of a Laurent polynomial (or is not known for computing all the Betti numbers. more generally a projective toric variety) to graded alge- bras equipped with valuations. Cordian Riener Kiumars Kaveh Aalto University University of Pittsburgh [email protected] [email protected] Saugata Basu Purdue University MS22 [email protected] Khovanskii Bases and Tropical Geometry

By work of Anderson and others, a variety can often be MS23 degenerated to a toric variety if its coordinate ring comes equipped with a finite Khovanskii basis. Such a degen- A Framework for Investigating Multistationarity in eration allows numerous combinatorial and computational Messi Biochemical Systems techniques to be used on both the coordinate ring and the variety itself. I’ll talk about a recent result with Kiumars Many dynamical systems arising in systems biology exhibit Kaveh, where we give a necessary and sufficient condition bistability (2 or more equilibria that are stable to small per- for a variety to have a finite Khovanskii basis framed in the turbations), but it is often difficult to ascertain the maxi- language of tropical geometry. mum possible number of such (stable) steady states. Even Chris Manon for a reaction network already known to admit multiple George Mason University steady states. Here, we present a heuristic method for [email protected] investigating this problem for so-called MESSI networks (MESSI means “modifications of type enzyme-substrate or swap with intermediates’), which are common in biological MS23 signalling pathways. We demonstrate in several examples that our approach works well, and develop the associated On the Davenport-Mahler Bound mathematics. The Davenport-Mahler bound is a lower bound for the product of the lengths of the edges on a graph whose ver- Xiaoxiao Tang tices are the complex roots of a univariate polynomial, un- Department of Mathematics der certain assumptions. Roughly speaking, this bound Texas A&M University 86 AG17 Abstracts

[email protected] introduce the concept of modules over FI-algebras, and prove that certain finitely generated modules are Noethe- rian. We also demonstrate examples where Noetherianity MS23 fails. When Noetherianity holds, it allows the construction Real Intersection Between a Low-degree Curve and of free-resolutions for FI-algebras. a Sparse Hypersurface Robert Krone Descartes’ rule of signs shows that a real univariate poly- Queen’s University nomial with t monomials has at most t-1 positive roots. Department of Mathematics and Statistics It was shown in a succession of works (Khovanskii,Bihan- [email protected] Sottile) that the number of real solutions of a system can be bounded by a value which does not depend on the degrees Jan Draisma of the polynomials but only on the number of monomials Eindhoven University of Technology, The Netherlands which appear in the system. However, all these bounds Department of Mathematics and Computer Science are exponential in the number of monomials, and it is an [email protected] open question to know the optimal one. On the other di- rection, bounds for small families of systems have also been Alexei Krasilnikov studied. Several results (Li-Rojas-Wang,Avedao,Bihan-El Departamento de Matem´atica Hilany) handle small cases (intersection between a sparse Universidade de Bras´ılia, Bras´ılia- DF, 70910-900, Brazil plane curve and a line or a trinomial curve). In a previous [email protected] work, Koiran, Portier and Tavenas found a bound (poly- nomial in t and d) for the intersection between a sparse plane curve (defines by a t-monomials polynomial) and a MS24 degree-d plane curve. We want to generalize the last result Fourier Transforms of Polytopes, Ehrhart and Solid in the n-dimensional case. We will show a (polynomial in Angle Sums for Real Polytopes, and Symmetric t and d) upper bound for the number of real intersections Simplices between a degree-d curve and a t-sparse hypersurface. This is joint work with M. Safey El Din. Given a real closed polytope P, we describe the Fourier transform of its indicator function by using iterations of Sebastien Tavenas Stokes theorem. We then use the ensuing Fourier trans- LAMA form formulations, together with the Poisson summation Universit´eSavoieMontBlanc formula, to give a new algorithm to count fractionally- [email protected] weighted lattice points inside the one-parameter family of all real dilates of P. The combinatorics of the face poset of P plays a central role in the description of the Fourier MS24 transform of P. This theory gives an account of an ana- A Tale of Two Cones with Symmetry logue of Ehrhart theory for real polytopes. We also obtain a closed form for the codimension-1 coefficient that appears The relationship between the cone of positive semidefinite in an expansion of this sum in powers of the real dila- (psd) real forms and its subcone of sum of squares (sos) of tion parameter t. This codimension-1 formula generalizes forms is of fundamental importance in real algebraic geom- some known results about the Macdonald solid-angle poly- etry and optimization. The study of this relationship goes nomial, which is the analogous expression to the Ehrhart back to the 1888 seminal paper of Hilbert, where he gave polynomial, traditionally obtained by requiring that t as- a complete characterisation of the pairs (n,2d) for which sumes only integer values. Although most of the present a psd n-ary 2d-ic form can be written as sos. In this talk methodology applies to all real polytopes, a particularly I will show how this relationship changes under the addi- nice application is to the study of all real dilates of integer tional assumptions of symmetry. I will present our recent (and rational) polytopes. Finally, we study how certain results giving the analogues of Hilbert’s characterisation for integer simplices that have symmetric properties can be symmetric and even symmetric forms respectively. Along classified by studying our conjecture that says the follow- the way, I will also discuss briefly how test sets for posi- ing: an integer simplex tiles Euclidean space by a Weil tivity of symmetric polynomials play an important role in group if and only if its (discrete) solid angle sum equals establishing these analogues. its (continuous) volume. Based on works with Nhat Le Charu Goel Quang, Ricardo Diaz, Romanos-Diogenes Malikiosis, and University of Konstanz Zhang Yichi. [email protected] Sinai Robins University of Sao Paulo Salma Kuhlmann Brasil University of Konstanz , Germany [email protected] [email protected]

Bruce Reznick MS24 University of Illinois at Urbana-Champaign Multivariate Symmetric Interpolation and Subre- [email protected] sultants

In 1853, Sylvester introduced double sum expressions for MS24 two finite sets of indeterminates and subresultants for uni- Modules over FI-algebras variate polynomials, showing the relationship between both notions in several but not all cases. Here we show how An FI-algebra is a object that describes a family of K- Sylvester’s double sums can be interpreted in terms of sym- algebras with compatible symmetric group actions. We metric multivariate Lagrange interpolation, allowing to re- AG17 Abstracts 87

cover in a natural way the full description of all cases. compositions over C, but only one of them is formed by We will also report on preliminary results on extensions real summands. Thus, in the open sets, tensors are not to symmetric multivariate Hermite interpolation, and ap- identifiable over C, but are identifiable over R.Wealso plications. This work is a collaboration with Teresa Krick provide examples of non trivial euclidean open subsets in and Marcelo Valdettaro. a whole space of symmetric tensors (of degree 7 and 8 in three variables) and of almost unbalanced tensors Segre Agnes Szanto Product (P2 × P4 × P9) whose elements have typical real North Carolina State University rank equal to the complex rank, and are identifiable over [email protected] R, but not over C.

Cristiano Bocci MS25 Universita di Siena Taking a Step from the Waring Rank Toward the [email protected] Chow Rank Elena Angelini The main goal of this talk is to present a result related Universit`adiSiena to a conjecture suggested by Catalisano, Geramita, and Italy Gimigliano in 2002, which claims that the secant vari- [email protected] eties of tangential varieties to Veronese varieties are non- defective modulo a few known exceptions. Luca Chiantini Hirotachi Abo Universita di Siena University of Idaho [email protected] [email protected]

Nick Vannieuwenhoven MS25 Department of Computer Science Points of High Rank KU Leuven [email protected] For a projective variety X ⊂ PV ,theX-rank of a point p ∈ PV is the minimal integer r such that p is in the linear span of r-distinct points of X. For special cases MS25 this notion includes tensor rank, Waring rank and many Identifiability for Skew Symmetric Tensors others. I will present constructions of polynomials with high Waring rank and describe the loci of points of X- We prove that the generic element of the fifth secant variety rank equal to r,wherer is greater than the generic X- P2 P9 ⊂ P 3 C10 σ5(Gr( , )) ( ) of the Grassmannian of planes rank. This talk is based on a joint work with Zach Teitler, P9 of has exactly two decompositions as sum of five projec- Massimiliano Mella, and Kangjin Han. tive classes of decomposable skew-symmetric tensors. We P3 P8 show that this, together with Gr( , ), are the only non- Jaroslaw Buczynski identifiable cases among the non-defective secant varieties University of Warsaw Pk Pn σs(Gr( , )) for any n<14. In the same range for n we [email protected] classify all the weakly defective and all tangentially weakly defective secants varieties of any Grassmannians. We also 2 7 ∨ show that the dual variety (σ3(Gr(P , P ))) of the vari- 2 7 MS26 ety of 3-secant planes of the Grassmannian of P ⊂ P is 2 7 σ2(Gr(P , P )) the variety of bi-secant lines of the same Polyhedral Complexes in Phylogenetics and Gene Grassmannian. Interaction Studies

Alessandra Bernardi Next-Generation Sequencing has made a great amount of University of Torino genomic data available at low cost with the data sets of- Mathematical Department, Giuseppe Peano ten consisting of thousands of whole genomes. Modern [email protected] computers are unable to process the data even using ex- isting mathematical models of evolution, to say nothing of Davide Vanzo more sophisticated modern models that are continually un- University o Firenze der development. This rapid progress in molecular biology Firenze, Italy has created a pressing demand in mathematical methods, [email protected] which are currently underdeveloped and struggle to keep up with biological progress. In this talk, I will present recent developments in the field of evolutionary biology MS25 along with mathematical obstacles put in the way of this Real Identifiability vs Complex Identifiability progress. I will demonstrate how the solution to a partic- ular problem from molecular evolution requires fundamen- Let T be a real tensor of (real) rank r. T is identifiable tal advances in (Gromov-like) geometric group theory, the when it has a unique decomposition in terms of rank 1 ten- theory of secondary polytopes, computational complexity, sors. There are cases in which the identifiability fails over and probability. This is a joint work with Alexei Drum- the complex field, for general tensors of rank r. 1. Of- mond (U of Auckland, NZ), Chris Whidden (Fred Hutch ten, the failure is due to the existence of an elliptic normal Cancer Research Center, Seattle), Erick Matsen (FHCRC), curve through general points of the corresponding Segre, Niko Beerenwikle (ETH Z¨urich), and Bernd Sturmfels (UC Veronese or Grassmann variety. In this talk we show the Berkeley). The talk is based on [Gavryushkin and Drum- existence of nonempty euclidean open subsets of some va- mond. J Theoretical Biology, 2016], [Gavryushkin, Whid- riety of tensors of rank r, whose elements have several de- den, and Matsen. bioRxiv, 2016], [Beerenwinkel, Pachter, 88 AG17 Abstracts

Sturmfels. Statistica Sinica, 2007]. [email protected]

Alex Gavryushkin MS26 ETH Zurich [email protected] Complexity of Models for Genome Rearrangement A genome can be represented by an edge-labelled, directed graph having maximum total degree two. We explore MS26 three models for genome rearrangement, a common mode of molecular evolution: reversal, single cut-or-join (SCJ), The Structure of the Branching Polytopes for RNA and double cut-and-join (DCJ). Even for moderate size in- Structures puts and regardless of the model, there are a tremendous number of optimal rearrangement scenarios, a sequence of mutations that transform one genome into another. In Like proteins, RNA assumes complex three-dimensional hypothesizing, giving one optimal solution might be mis- structures to perform specific roles and understanding this leading and cannot be used for statistical inference. With a structure helps our understanding of the ways the non- focus on the SCJ model, we summarize the state-of-the-art coding RNAs perform their regulatory functions. That is in computational complexity and uniform sampling ques- why the problem of finding methods that can quickly and tions for each model surrounding optimal scenarios and reliably identify the structure of a given RNA has been phylogenetic trees. an important problem in computational biology. However, the methods developed still vary widely in the prediction Istvan Miklos accuracy. An important component of this problem is pre- Renyi Institute dicting the secondary structure, which identifies both the [email protected] canonically base-paired regions (helices) and non-paired re- gions (loops). In this work we focus on understanding the Heather C. Smith effects of the parameters used for scoring the multibranch Georgia Institute of Technology loops in the nearest-neighbor thermodynamic model. For Atlanta, GA 30332 this purpose, for each RNA we built and analyzed a so [email protected] called branching polytope. In this talk I will present our findings. MS27 Svetlana Poznanovi´c The DTM-signature for a Geometric Comparison Clemson University of Metric-measure Spaces from Samples [email protected] In this talk, we introduce the notion of DTM-signature, Christine Heitsch, Fidel Barrera-Cruz a measure on R+ that can be associated to any metric- School of Mathematics measure space. This signature is based on the dis- Georgia Institute of Technology tance to a measure (DTM) introduced by Chazal, Cohen- [email protected], fi[email protected] Steiner and Mrigot. It leads to a pseudo-metric between metric-measure spaces, upper-bounded by the Gromov- Wasserstein distance. Under some geometric assumptions, we derive lower bounds for this pseudo-metric. Given two MS26 N-samples, we also build an asymptotic statistical test based on the DTM-signature, to reject the hypothesis of Polyhedra, Sampling Algorithms for Random Poly- equality of the two underlying metric-measure spaces, up gons, and Applications to Ring Polymer Models to a measure-preserving isometry. We give strong theoreti- cal justifications for this test and propose an algorithm for In statistical physics, the basic (and highly idealized) its implementation. model of a ring polymer like bacterial DNA is a closed ran- dom flight in 3-space with equal-length steps, often called Claire Brecheteau a random equilateral polygon. In this talk, I will describe INRIA the moduli space of random equilateral polygons, giving [email protected] a sense of how this fits into a larger symplectic and al- gebraic geometric story. In particular, the space of equi- MS27 lateral n-gons turns out to (almost) be a toric symplec- tic manifold, which implies that, from a measure-theory Pseudo-multidimensional Persistence and its Ap- perspective, we can think of the space of n-gonsasthe plications product of a torus and a convex polytope. Hence, under- The application of persistent homology through the simul- standing the convex polytope provides deep insight into taneous filtration of multiple variables presents itself as a the moduli space. In this talk I will describe some ex- technique particularly useful in the analysis of datasets nat- amples, including a very fast and surprisingly simple al- urally described by more than one parameter. However, gorithm for uniformly sampling the space of equilateral as Carlsson and Zomorodian indicated in 2009, no higher- polygons developed with Jason Cantarella (University of dimensional analogue of the one-dimensional persistence Georgia), Bertrand Duplantier (CEA/Saclay), and Erica barcodethat is, a complete discrete algebraic invariantex- Uehara (Ochanomizu University). ists for higher-dimensional filtrations. In 2015 Xia and Wei proposed stacking one-dimensional heatmaps of the Betti Clayton Shonkwiler number function for a single homological dimension, with Department of Mathematics each row of the stack created by varying a second param- Colorado State University eter, for example density or curvature. The aim of this AG17 Abstracts 89

visualization is to increase the discriminatory power of cur- terest. rent one-dimensional persistence techniques, especially for datasets that have features more readily captured by a sec- Benjamin Schweinhart ond parameter. We apply this practical approach to three Ohio State University data sets (where the first parameter is proximity and the [email protected] second parameter is given in parenthesis): sinus shapes (curvature), Lissajous knots (curvature), and Kuramoto- Sivashinsky partial differential equation (time). This ap- MS28 proach increases the discriminatory power of current one- The Pure Condition Through the Lens of the Al- dimensional persistence techniques and remains computa- gebraic Matroid tionally feasible. The rigidity matroid for bar-and-joint frameworks can also Giseon Heo be derived as the algebraic matroid of the Cayley-Menger University of Alberta variety. Bases of the algebraic matroid define finite-to-one [email protected] dominant maps onto coordinate subspaces. The discrimi- nant of this map dictates the sets of lengths that lead to infinitesimal motions in the framework, also known as the Catalina Betancourt pure condition. In this talk, we will introduce the alge- University of Iowa braic matroid perspective on rigidity. We will also discuss [email protected] partial results on combinatorial characterization of the dis- criminant. Based on prior research with F. Kiraly and L. Rachel Neville Theran, and ongoing research with J. Sidman, L. Theran, Colorado State University C. Vinzant. [email protected] ZviH.Rosen Matthew Pietrosa Department of Mathematics University of Alberta University of Pennsylvania [email protected] [email protected]

Mimi Tsuruga MS28 University of California, Davis [email protected] Algebraic Geometry and Mechanism Science: Re- cent Results and Open Problems

Isabel Darcy In our contribution we present a selection of recent results University of Iowa and open problems from mechanism science. 1) Classifi- [email protected] cation of overconstrained 6R linkages is a long-standing open problem and a complete solutions is still out of reach. We present attempts via bond theory that succeeded in MS27 the classification of certain sub-cases and also a rather The Consistency of Gibbs Posteriors and the Ther- strange conditional classification of linkages with helical modynamic Formalism joints, based on number theory and a hypothetical classi- fication on nR linkages. 2) A classic field of mechanism We consider the problem of inference dynamical systems science is the construction of linkages to accommodate cer- with observational noise. We propose a Bayesian proce- tain tasks (linkage synthesis). We report on recent results dure to infer a posterior distribution over a family of dy- in the synthesis of single and multiple loop linkages via namical systems given observations from a generating sys- the factorization theory of motion polynomials and nu- tem again with noise. We prove convergence results using meric results for the synthesis of open kinematic chains. tools from dynamical systems theory. Quantities such as Future challenges are a better understanding of factoriza- topological entropy and topological pressure will be cen- tion theory, its extension to higher dimension and state- tral in proving convergence. This talk will show there is a ments on the synthesis of open chains that go beyond nu- strong connection between Bayesian inference with model merics. 3) Finally, we talk about singularity-free assembly misspecification and the thermodynamic formalism. mode changes, a recent topic in mechanism science with a well-established mathematical foundation (monodromy Sayan Mukherjee and Galois theory) that apparently has not yet been fully Duke University exploited. It has been shown that the monodromy group in [email protected] the prototypical example of a planar multi-looped linkage is smaller than the full symmetry group so that non-trivial MS27 Galois groups can be expected. Topological Similarity of Random Cell Complexes Hans-Peter Schr¨ocker University of Innsbruck In this talk, I’ll introduce the method of swatches, which [email protected] describes the local topology of a cell complex in terms of probability distributions of local configurations. It allows a distance to be defined which measures the similarity of MS28 the local topology of cell complexes. Convergence in this Surveying Applications of Rigidity Theory distance is related to the notion of a Benjamini-Schramm graph limit. I will use this to state universality conjectures Structural properties of rigidity arise in many areas, from about the long-term behavior of graphs evolving under cur- mechanical engineering to swarm robotics to medicine. An vature flow, and to test these conjectures computationally. engineer designing a trailer tail to improve fuel economy of The application is of both mathematical and physical in- semi-trailer trucks expects Computer Aided Design (CAD) 90 AG17 Abstracts

software to verify that the allowed motion of a folded struc- is based on the adaptive Lasso with coefficient weights for ture can transition it to an unfolded structure that main- sparse computation in the tensor rank detection. We also tains it shape. A swarm of robots requires control algo- provide an algorithm for solving the adaptive Lasso model rithms to help it maintain a formation when collectively problem for tensor approximation. The method is applied transporting an object, such as a partially constructed air- to background extraction and video compression problems. plane wing. A bioengineer relies on understanding the flexibility of a protein when identifying candidate drug molecules to inhibit a virus function. In this talk, I will Carmeliza Navasca survey how rigidity theory can be applied to model and Department of Mathematics analyze situations such as these, highlighting open prob- University of Alabama at Birmingham lems along the way. [email protected] Audrey St. John Mount Holyoke College Xioafei Wang [email protected] School of Mathematics and Statistics Northeast Normal University [email protected] MS28 Grassmannians, Invariant Theory and the Bracket Algebra in Rigidity Theory MS29

The goal of this talk is to give an introduction to Grass- A New Generation of Brain-computer Interfaces mannians, Invariant Theory and the Bracket Algebra as Driven by Discovery of Latent EEG-fMRI Linkages they relate to Rigidity Theory. For example, these ideas using Tensor Decomposition can be used to give a geometric interpretation of the van- ishing of bracket polynomials in order to better understand when a generically rigid framework admits nontrival inter- A Brain-Computer Interface (BCI) is a setup permitting nal motions. the control of external devices by decoding brain activity. Electroencephalography (EEG) has been used for decod- Will Traves ing brain activity since it is non-invasive, cheap, portable, US Naval Academy and has high temporal resolution to allow real-time oper- [email protected] ation. Owing to its poor spatial specificity, BCIs based on EEG can require extensive training and multiple trials to Jessica Sidman decode brain activity. On the other hand, BCIs based on Mount Holyoke College functional magnetic resonance imaging (fMRI) are more [email protected] accurate owing to its superior spatial resolution and sen- sitivity to underlying (local) neuronal processes. However, due to its relatively low temporal resolution, high cost, and MS29 lack of portability, fMRI is unlikely to be used for routine A Rectangular Relaxation for Multi-D Filter Bank BCI. We propose a new approach for transferring the ca- Design using Laurent Polynomials pabilities of fMRI to EEG, which includes simultaneous EEG/fMRI sessions for finding a mapping from EEG to A method for constructing multidimensional nonredundant fMRI, followed by a BCI run from only EEG data, but wavelet filter banks using Quillen-Suslin Theorem is pro- driven by fMRI-like features obtained from the mapping vided by Hur, Park, and Zheng in their recent paper. They identified previously. Our novel data-driven method, based provide an algorithm to construct these filter banks from on tensor decomposition, is likely to discover latent linkages a single lowpass filter with positive accuracy. In this talk, between electrical and hemodynamic signatures of neural we show that their construction works in a much more gen- activity hitherto unexplored using model-driven methods, eral sense to construct potentially redundant wavelet filter and is likely to serve as a template for a novel multi-modal banks. We show that their construction works to construct strategy, leading to a new generation of EEG-based BCIs. multidimensional wavelet filter banks even if you do not require finding an invertible square matrix with Laurent polynomial entries as one of its steps but only require that Gopikrishna Desphande a left-invertible matrix is found, square or rectangular. Auburn University [email protected] Cameron L. Farnsworth Department of Mathematics Yonsei University D Rangaprakash [email protected] University of California Los Angeles [email protected] Youngmi Hur Yonsei University Luke Oeding [email protected] Auburn University [email protected]

MS29 Andrzej Cichocki Low Rank Tensor Approximation via Sparse Opti- RIKEN Brain Science Institute mization for Video Processing [email protected]

In this talk, we propose a novel framework for finding a Xiaoping P. Hu low rank approximation of a given tensor. This framework University of California Riverside AG17 Abstracts 91

[email protected] walks in higher dimensions, with non-small steps, or with weighted steps. Particular attention will be given to the asymptotic enumeration of such walks using representa- MS29 tions of the generating functions as diagonals of rational Global Optimality in Matrix and Tensor Factoriza- functions, through the theory of analytic combinatorics in tion several variables.

Recently, convex solutions to low-rank matrix factoriza- Stephen Melczer tion problems have received increasing attention in ma- University of Waterloo chine learning. However, in many applications the data [email protected] can display other structures beyond simply being low-rank. For example, images and videos present complex spatio- temporal structures, which are largely ignored by current MS30 low-rank methods. This talk will explore a matrix factor- The Method of Differential Approximation in Enu- ization technique suitable for large datasets that captures merative Combinatorics additional structure in the factors by using a projective ten- sor norm, which includes classical image regularizers such In enumerative combinatorics, it is quite common to have as total variation and the nuclear norm as particular cases. in hand a number of known initial terms of a combinatorial Although the resulting optimization problem is not convex, sequence whose behavior you’d like to study. In this talk we show that under certain conditions on the factors, any we’ll describe the Method of Differential Approximation, a local minimizer for the factors yields a global minimizer for technique that can be used to shed some light on the nature their product. Examples in biomedical video segmentation of a sequence using only some known initial terms. While and hyperspectral compressed recovery show the advan- these methods are, on the face of it, experimental, they of- tages of our approach on high-dimensional datasets. ten guide the way toward rigorous proofs. We’ll exhibit the usefulness of this method through a variety of combinato- Rene Vidal rial topics, including chord diagrams, permutation classes, Johns Hopkins University and inversion sequences. [email protected] Jay Pantone Dartmouth College MS30 [email protected] Hermite Reduction in Logarithmic and Exponen- tial Extensions MS30 A hyperexponential function is a nonzero function such that its logarithmic derivative is a rational function in C(x). Walks, Difference Equations and Elliptic Curves Hyperexponential Hermite reduction decomposes a hyper-  In the recent years, the nature of the generating series of exponential function t(x)ast =(st) + rt with s, r ∈ C(x) such that tdx is hyperexponential if and only if r =0. the walks in the quarter plane has attracted the atten- A function t(x) is said to be a monomial over C(x)ift tion of many authors. The main questions are: are they is transcendental over C(x)andt belongs to C(x)[t]. A algebraic, holonomic (solutions of linear differential equa- monomial t is said to be logarithmic if t is a logarithmic tions) or at least hyperalgebraic (solutions of algebraic dif- derivative of a rational function, and t is exponential if t/t ferential equations)? In a seminal paper Bousquet- Mlou is a derivative of a rational function. Let t be either a loga- and Mishna attach a group to any walk in the quarter rithmic or an exponential monomial over C(x). We present plane and make the conjecture that a walk has a holonomic  generating series if and only if the associated group is fi- an algorithm that decomposes f ∈ C(x, t)asf = g +r with g,r ∈ C(x, t) such that fdx belongs to C(x, t)ifandonly nite. They proved that, if the group of the walk is finite, if r = 0. The algorithm does not need to solve any Risch’s then the generating series is holonomic, except, maybe, equation. We will also discuss whether such a reduction in one case, which was solved positively by Bostan, van can be extended to a field K(t), where K is a differential Hoeij and Kauers. In the infinite group case, Kurkova and field extension of C(x). This is joint work with Shaoshi Raschel proved that if the walk is in addition non singular, Chen and Hao Du. then the corresponding generating series is not holonomic. This work is very delicate, and relies on the explicit uni- Ziming Li formization of a certain elliptic curve. Recently, Bernardi, Key Laboratory of Mathematics Mechanization Bousquet-Mlou, and Raschel proved that 9 of the 51 such AMSS, Chinese Academy of Sciences walks have a generating series which is hyperalgebraic. In [email protected] this talk, I will discuss how difference Galois theory can be used to show that the remaining 42 walks have a gen- erating series which is not hyperalgebraic, reproving and MS30 generalizing the results of Kurkova/Raschel and also dis- New Advances in Enumerating Lattice Walks cuss a Galois theoretic proof of the recent work of Bernardi, through Rational Diagonals Bousquet-Mlou, and Raschel. This is joint work with T. Dreyfus, C. Hardouin and Julien Roques. The problem of enumerating lattice paths with a fixed set of allowable steps and restricted endpoint has a long his- Michael Singer tory dating back at least to the 19th century. For several North Carolina State University reasons, much research on this topic over the last decade [email protected] has focused on two dimensional lattice walks restricted to the first quadrant, whose allowable steps are ”small” (that is, each step has coordinates ±1 or 0). In this talk we re- MS31 lax some of these conditions and discuss recent work on Emergent Dynamics from Network Connectivity: 92 AG17 Abstracts

A Minimal Model networks (TBNs), initially developed to study neural net- works, have recently been used to model a variety of gene Many networks in the brain display internally-generated regulatory networks. In a TBN, the future state of each patterns of activity – that is, they exhibit emergent dynam- node is determined based on a threshold and a linear com- ics that are shaped by intrinsic properties of the network bination of the current states of its neighbors. In this talk, rather than inherited from an external input. While a com- we present an algorithm for identifying all threshold net- mon feature of these networks is an abundance of inhibi- work models that reproduce a given Boolean dataset. Our tion, the role of network connectivity in pattern generation method is rooted in algebraic combinatorics. remains unclear. In this talk I will introduce Combinato- rial Threshold-Linear Networks (CTLNs), which are simple Abdul Salam Jarrah “toy models” of recurrent networks consisting of threshold- American University in Sharjah linear neurons with binary inhibitory interactions. The [email protected] dynamics of CTLNs are controlled solely by the structure of an underlying directed graph. By varying the graph, we observe a rich variety of emergent patterns including: MS31 multistability, neuronal sequences, and complex rhythms. These patterns are reminiscent of population activity in Algebraic Methods for the Control of Boolean Net- cortex, hippocampus, and central pattern generators for works locomotion. I will present some theorems about CTLNs, and explain how they allow us to predict features of the dy- Many problems in biomedicine and other areas of the life namics by examining properties of the underlying graph. sciences can be characterized as control problems, with the Finally, I’ll show examples illustrating how these mathe- goal of finding strategies to change a disease or otherwise matical results guide us to engineer complex networks with undesirable state of a biological system into another, more prescribed dynamic patterns. desirable, state through an intervention, such as a drug or other therapeutic treatment. The identification of such Carina Curto strategies is typically based on a mathematical model of the Department of Mathematics process to be altered through targeted control inputs. This The Pennsylvania State University talk focuses on processes at the molecular level that deter- [email protected] mine the state of an individual cell, involving signaling or gene regulation. The mathematical model type considered is that of Boolean networks. The potential control targets MS31 can be represented by a set of nodes and edges that can Algebraic Geometry of K-canalizing Functions be manipulated to produce a desired effect on the system. This talk presents a method for the identification of po- Discrete models of gene regulatory networks have gained tential intervention targets in Boolean molecular network popularity in computational systems biology over the last models using algebraic techniques. The approach exploits dozen years. However, not all discrete network models re- an algebraic representation of Boolean networks to encode flect the behaviors of real biological systems. In this talk, the control candidates in the network wiring diagram as we focus on k-canalizing functions, a class of biologically the solutions of a system of polynomials equations, and relevant functions that is generalized from nested canal- then uses computational algebra techniques to find such izing functions. We extend results on nested canalizing controllers. Additionally, a formula, based on the prop- functions and derived a unique extended monomial form erties of Boolean canalization, for estimating the number of arbitrary Boolean functions. This gives us a stratifica- of changed transitions in the state space of the system as tion of the set of n-variable Boolean functions by canaliz- a result of an edge deletion in the wiring diagram will be ing depth. We obtain closed formulas for the number of discussed. n-variable Boolean functions with depth k,whichsimul- taneously generalizes enumeration formulas for canalizing, David Murrugarra and nested canalizing functions. We characterize the set University of Kentucky 2n of k-canalizing functions as an algebraic variety in F2 . [email protected] Next, we propose a method for the reverse engineering of networks of k-canalizing functions using techniques from computational algebra, based on our parametrization of k- MS32 canalizing functions. Optimization with Invariance Constraints Qijun He Virginia Tech We introduce a new class of optimization problems which [email protected] have constraints imposed by trajectories of a dynamical systems. As a concrete example, we consider the problem of optimizing a linear function over a subset of a convex set MS31 which remains invariant under the action of a linear, or a Reverse-engineering Threshold Networks From switched linear, dynamical system. We identify interesting Boolean Data settings where this problem can be solved in polynomial time by linear programming, or approximated in polyno- The inference of biological networks from experimental mial time by semidefinite programming. data is a key goal of systems biology. Given experimen- tal data (time-course, input-output, or steady-state), the Amir Ali Ahmadi objective is to identify the structure of the network as well Princeton University as the rules of interaction among the agents in the net- a a [email protected] work. However, even within the Boolean network modeling framework, there usually are many Boolean networks that Oktay Gunluk explain the available data. The so-called threshold Boolean IBM, Watson, New York AG17 Abstracts 93

[email protected] sults.

Timo de Wolff MS32 Texas A&M University dewolff@math.tamu.edu On Coercivity of Polynomials

We analyse the relationship between the coercivity prop- MS33 erty of multivariate polynomials, their Newton polytopes The Method of Gauss-Newton to Compute Power and Lojasiewicz exponents at infinity. Better under- Series Solutions of Polynomial Homotopies standing of coercivity property of multivariate polynomials might reveal more insight into the problem of global dif- We consider the extension of the method of Gauss-Newton feomorphism property of polynomial maps which we also from complex floating-point arithmetic to the field of trun- briefly discuss. cated power series with complex floating-point coefficients. With linearization we formulate a linear system where the Tomas Ba jbar coefficient matrix is instead a series with matrix coeffi- Goethe University Frankfurt cients. We show that in the regular case, the solution of the [email protected] linear system satisfies the conditions of the Hermite inter- polation problem. In general, we solve a Hermite-Laurent interpolation problem, via a lower triangular echelon form MS32 on the coefficient matrix. To set up the matrix series sys- Symmetric Sums of Squares tem without symbolic evaluation, we apply techniques of algorithmic differentiation. We demonstrate the applica- We consider the problem of finding sum of squares (sos) tion to polynomial homotopy continuation with a few il- expressions to establish the non-negativity of a symmetric lustrative examples. polynomial over a discrete hypercube whose coordinates are indexed by k-element subsets of [n]. We develop a vari- Nathan Bliss,JanVerschelde ant of the Gatermann-Parrilo symmetry-reduction method University of Illinois at Chicago tailored to our setting that allows for several simplifica- Chicago, Illinois USA tions and a connection to Razborov’s flag algebras. We [email protected], [email protected] show that every symmetric polynomial that has a sos ex- pression of a fixed degree also has a succinct sos expression MS33 whose size depends only on the degree and not on the num- ber of variables. This is joint work with James Saunderson, Hybridization of Bertini and the RegularChains Mohit Singh and Rekha Thomas. Maple Library The RegularChains Maple library is a symbolic implemen- Annie Raymond tation of constructible sets and procedures on them like University of Washington complement, intersection, containment, union, and image [email protected] under rational maps. A Maple package interfacing with Bertini is in progress which seeks to be a numerical imple- James Saunderson mentation of constructible sets with these same procedures, Massachusetts Institute of Technology as well as a procedure to turn symbolic constructible sets [email protected] into numerical ones. This is done by getting pseudowit- ness sets for X and Y for each quasiprojective algebraic Mohit Singh set X \ Y making up the constructible set. Turning sym- H. Milton Stewart School of Industrial & Systems bolic constructible sets into numerical constructible sets Engineering in this way lets us test numerical algorithms against the Georgia Institute of Technology RegularChains test library. [email protected] Jesse W. Drendel Rekha Thomas Colorado State University University of Washington [email protected] [email protected] MS33 MS32 Numerical Irreducible Decomposition for Likeli- hood Equations Current Key Problems on Sums of Nonnegative Circuit Polynomials The maximum likelihood degree (ML degree) is a topologi- cal invariant of an algebraic variety. In particular, it is the In 2014, Iliman and I introduced a new nonnegativity cer- degree of the projection of the likelihood correspondence tificate based on sums of nonnegative circuit polynomials to the space of data. The fiber over a general data point (SONC),whichareindependent of sums of squares. Ini- consists of critical points of the likelihood function with re- tially, we applied SONCs to global nonnegativity prob- spect to that data. So the ML degree counts the number of lems via geometric programming. Recently, Dressler, Il- complex (real and non real) critical points. Under reason- iman, and I proved a Positivstellensatz for SONCs, which able hypothesis, the maximum likelihood estimate will be provides a converging hierarchy of lower bounds for con- one of these critical points giving a measure of complex- strained polynomial optimization problems. These bounds ity for maximum likelihood estimation. In this talk, we can be computed efficiently via relative entropy program- discuss numerical irreducible decomposition for likelihood ming. In this talk I will give an overview about key ques- equations. Our equations define the likelihood correspon- tions regarding SONCs that follow from these recent re- dence as well as residual components. The goal of this 94 AG17 Abstracts

talk is to present algorithms that yield considerable im- of the real locus of the invariants for small n. provement for decomposing these equations over standard methods of numerical irreducible decomposition. Paul G¨orlach Max Planck Institute for Mathematics in the Sciences Jose I. Rodriguez [email protected] Department of Statistics University of Chicago Evelyne Hubert [email protected] INRIA M´editerran´ee [email protected]

MS33 Least Squares Modeling in Numerical Algebraic MS34 Computation Symmetry Preserving Lagrange-hermite Interpola- tion Linear and nonlinear least squares problems with empirical data arise frequently numerical algebraic geometry compu- We review basics of multivariate interpolation in order to tation. Particularly, such problems are often hypersensi- take into account symmetry. We provide an approach that tive to data perturbations and can be regularized as well- preserves the equivariance of maps. We provide explicit posed least squares problems. In this talk, we establish a constructions in the case of a reflexion group of symmetry. general regularization theorem for nonlinear least squares problems with empirical data parameters along with a sen- sitivity measure. We shall also provide case studies in least Evelyne Hubert squares modeling in numerical algebraic computation and INRIA M´editerran´ee the strategy of auxiliary equations that ensures the condi- [email protected] tions of the regularization theorem are satisfied. Further- more, we shall demonstrate the software package NAClab Cordian Riener that provide an intuitive interface for solving the linear Aalto University and nonlinear least squares problems arising in numerical [email protected] algebraic computation.

Zhonggang Zeng MS34 Northeastern Illinois University Local Invariant Sets of Polynomial and Analytic Department of Mathematics Vector Field [email protected] In the theory of autonomous ordinary differential equations invariant sets play an important role. In particular, we are MS34 interested in local analytic invariant sets near stationary Separating Invariants and Applications points. Invariant sets of a differential equation correspond to invariant ideals of the associated derivation in the power If one is interested in the classification of certain objects series algebra. In stability theory, invariant sets are used up to symmetry given by a group action, then rather than to obtain reduced differential equations. Poincar´e-Dulac considering the whole ring of invariants, one can consider normal forms are very useful in studying semi-invariants a separating set, that is, a set of invariants whose elements and invariant ideals. We prove that an invariant ideal with separate any two points that can be separated by some respect to a vector field, given in normal form, is already in- invariant. We explain how separating sets naturally arises variant with respect to the semisimple part of its Jacobian in applications, in particular when studying the structural at the stationary point. This generalizes a known result identifiability of mathematical models. about semi-invariants, that is invariant sets of codimen- sion 1. Moreover, we give a characterization of all ideals Emilie Dufresne which are invariant with respect to the semisimple part of University of Oxford the Jacobian. As an application, we consider polynomial [email protected] systems and we provide a sharp bound of the total degree of possible polynomial semi-invariants under some generic conditions. MS34 Niclas Kruff Rational Invariants for Orthogonal Equivalence of RWTH Aachen Homogeneous Polynomials niclas.kruff@rwth-aachen.de The orthogonal group O(n) acts naturally on the space d n ∗ Vd := Sym (R ) of homogeneous polynomials in n vari- MS35 ables of even degree d. We study the Rational Invariant Multi-tensor Decompositions for Personalized Can- Theory of this action from a computational perspective by cer Diagnostics and Prognostics O(n) constructing generators for the field K(Vd) of invariant rational functions on Vd. Geometrically this corresponds I will, first, briefly review our matrix and tensor mod- to describing general real hypersurfaces up to distance- eling of large-scale molecular biological data, which, as preserving transformations. We specify generating ratio- we demonstrated, can be used to correctly predict previ- nal invariants in terms of their restriction to a character- ously unknown physical, cellular, and evolutionary mech- izing subspace, and we describe a procedure for obtaining anisms that govern the activity of DNA and RNA. Sec- their explicit expressions in a nested way. The algorithmic ond, I will describe our recent generalized singular value problems of fast evaluation, rewriting and reconstruction decomposition (GSVD) and tensor GSVD comparisons of will be addressed and we give a semialgebraic description the genomes of tumor and normal cells from the same sets AG17 Abstracts 95

of astrocytoma brain and, separately, ovarian cancer pa- how to define TNS by graphs and we will define tensor net- tients, which uncovered patterns of DNA copy-number al- work ranks which can be used to measure the complexity of terations that are correlated with a patient’s survival and TNS. We will see that the notion of tensor network ranks is response to treatment. Third, I will present our higher- an analogue of tensor rank and multilinear rank. We will order GSVD (HO GSVD) and tensor HO GSVD, the only discuss basic properties of tensor network ranks and the mathematical frameworks that can create a single coher- comparison among tensor network ranks, tensors rank and ent model from, i.e., simultaneously find similarities and multilinear rank. If time permits, we will also discuss the dissimilarities across multiple two- or higher-dimensional dimension of tensor networks and the geometry of TNS. datasets, by extending the GSVD from two to more than This talk is based on papers joined with Lek-Heng Lim. two matrices or tensors. Ke Ye,Lek-HengLim Orly Alter University of Chicago Scientific Computing and Imaging Institute [email protected], [email protected] University of Utah [email protected] MS36 Genome-wide Phylogenomic Analysis Method Us- MS35 ing Likelihood Ratios On a Proof of the Set-theoretic Version of the Salmon Conjecture - Part II Theory and empirical evidence clearly indicate that phy- logenies (trees) of different genes (loci) should not display Abstract not available at time of publication. precisely matched topologies. However, some genes should display topologically related phylogenies. Based on that, Shmuel Friedland we could group them into one or more (for genetic hy- Department of Mathematics ,University of Illinois, brids) clusters in the ”tree space”. In addition, unusual Chicago evolutionary histories or effects of selection may result in [email protected] ”outlier” genes with phylogenies that fall outside the main distribution(s) of trees in tree space. To identify phyloge- netic outliers in a given set of ortholog groups from mul- MS35 tiple genomes, we present a new phylogenomic method, On a Proof of the Set-theoretic Version of the CURatio, which uses likelihood ratios of gene trees. It suc- Salmon Conjecture - Part I cessfully detects phylogenetic outliers in a simulation data set and a set of annoteated genomes of the fungal family,

Let Vr(m, n, l) be the irreducible variety of tensors of Clavicipitaceae. m×n×l tensors of border rank at most r, i.e. the closure of the set of tensors of rank at most r. In 2007, Elizabeth All- Qiwen Kang, Christopher Schardl, Neil Moore University of Kentucky man posed the problem of determining the ideal I4(4, 4, 4) [email protected], [email protected], generated by all polynomials vanishing on V4(4, 4, 4). In [email protected] 2005 Sturmfels stated the salmon conjecture that I4(4, 4, 4) is generated by polynomials of degree 5, (the Strassen com- mutative conditions), and 9, (originating from Strassen’s Ruriko Yoshida result that V4(3, 3, 3) is a hypersurface of degree 9). A first Naval Postgraduate School nontrivial step in characterizing V4(4, 4, 4) is to character- [email protected] ize V4(3, 3, 4). In 2004 Landsberg and Manivel show that V4(3, 3, 4) satisfies a set of polynomial equations of degree 6 which are not in the ideal generated by the equations of MS36 degree 5 from the original conjecture. The revised version Toric Ideals of Neural Codes of the salmon conjecture stated by Sturmfels in 2008 claims that I4(4, 4, 4) is generated by polynomials of degree 5, 6 A rat has special neurons that encode its geographic loca- and 9. This in particular implies the set-theoretic version tion. These neurons are called place cells and each place of the salmon conjecture: V4(4, 4, 4) is the zero set of ho- cell points to a region in the space, called a place field. mogeneous polynomials of degree 5, 6 and 9. In this talk I Neural codes are collections of the firing patterns of place will outline the complete solution of the set-theoretic ver- cells. In this talk, we investigate how to algorithmically sion of the salmon conjecture. It is based on my solo paper draw a place field diagram of a neural code, building on and the joint paper with Elizabeth Gross which appeared existing work studying neural codes, ideas developed in in journals in 2013 and 2012 respectively. the field of information visualization, and the toric ideal of a neural code. This talk is based on joint work with Eliz- Shmuel Friedland abeth Gross (San Jose State University) and Nora Youngs [email protected] (Colby College). [email protected] Nida K. Obatake Texas A&M University MS35 [email protected] Tensor Network Ranks Elizabeth Gross At the beginning of this talk, we will introduce the back- San Jose State University ground of tensor network states (TNS) in various areas [email protected] such as quantum physics, quantum chemistry and numer- ical partial differential equations. Famous TNS includes Nora Youngs tensor trains (TT), matrix product states (MPS) and pro- Colby College jected entangled pair states (PEPS). Then we will explain Department of Mathematics and Statistics 96 AG17 Abstracts

[email protected] unknown.

Eddie Aamari MS36 INRIA [email protected] Introduction to Biological Algebraic Statistics Jisu Kim This talk will provide an introduction to biological alge- Carnegie Mellon University braic statistics. [email protected] Megan Owen Department of Mathematics and Computer Science Fr´ed´eric Chazal City University of New York, Lehman College INRIA Saclay Ile-de-France [email protected] [email protected]

Bertrand Michel MS36 Ecole´ Centrale Nantes [email protected] Dimensionality Reduction via Tropical Geometry

A dimensionality reduction is applied to high-dimensional Alessandro Rinaldo, Larry Wasserman data sets in order to solve the problem called the curse of Carnegie Mellon University dimensionality. This term refers to the prob- lems associ- [email protected], [email protected] ated with multivariate data analysis as the dimensionality increases. This problem of multidimensionality is acute in MS37 the rapidly growing area of phylogenomics, which can pro- vide insight into relationships and evolutionary patterns Topological Summaries of Tumor Images Improve of a diversity of organisms, from humans, plants and an- Prediction of Disease Free Survival in Glioblastoma imals, to microbes and viruses. In this project, we are Multiforme interested in applying the tropical metric in the max-plus In this work we propose a novel statistic, the smooth Euler algebra to a dimen- sionality reduction over the space of characteristic transform (SECT). The SECT is designed to rooted equidistant phylogenetic trees on m leaves, that is integrate shape information into standard statistical mod- realized as the set of all ultrametrics. In this project, the els. More specifically, the SECT allows us to represent proposed process of reducing the dimension of the multidi- shapes as a collection of vectors with little to no loss in in- mensional data sets on the treespace is to take data points formation. As a result, detailed shape information can be in the space into a lower dimensional plane which mini- combined with biomarkers such as gene expression in stan- mizes the sum of distance between each point in the data dard statistical frameworks, such as linear mixed models. set and their orthogonal projection onto the plane, that is, We illustrate the utility of the SECT in radiogenomics by an optimization problem such that minimizing projection predicting disease free survival in glioblas- toma multiforme residuals between data points and their projections on the patients based on the shape of their tumor as assayed by plane via the tropical metric in the max-plus algebra. magnetic resonance imaging. We show that the SECT fea- tures outperform gene expression, volumetric features, and Ruriko Yoshida morphometric features in predicting disease free survival. Statistics University of Kentucky Lorin Crawford [email protected] Duke University [email protected] Xu Zhang University of Kentucky Anthea Monod, Andrew Chen [email protected] Columbia University Departments of Systems Biology and Biomedical Informatics MS37 [email protected], [email protected] Approximation and Geometry of the Reach Sayan Mukherjee Various problems within computational geometry and Duke University manifold learning encode geometric regularity through the [email protected] so-called reach, a generalized convexity parameter. The D reach τM of a submanifold M ⊂ R is the maximal offset Raul Rabadan radius on which the projection onto M is well defined. The Columbia University quantity τM renders a certain minimal scale of M, giving [email protected] bounds on both maximum curvature and possible bottle- neck structures. In this talk, we will study the geometry of the reach through an approximation theory perspective. MS37 We present new geometric results on the reach of subman- The Geometry of Synchronization Problems and ifolds without boundary. An estimatorτ ˆn of τM will be Learning Group Actions described, in an idealized i.i.d. framework where tangent spaces are known. The estimatorτ ˆn is showed to achieve We develop a geometric framework, based on the classical uniform expected loss bounds over a C3-like model. Mini- theory of fibre bundles, to characterize the cohomologi- max upper and lower bounds are derived. We will conclude cal nature of a large class of synchronization-type problems with an extension to a model in which tangent spaces are in the context of graph inference and combinatorial op- AG17 Abstracts 97

timization. We identify each synchronization problem in [email protected] topological group G on connected graph Γ with a flat prin- cipal G-bundle over Γ, thus establishing a classification re- sult for synchronization problems using the representation MS38 variety of the fundamental group of Γ into G.Wethen Body-and-cad Rigidity develop a twisted Hodge theory on flat vector bundles as- sociated with these flat principal G-bundles, and provide a Computer-aided-design (CAD) software allows a user to af- geometric realization of the graph connection Laplacian as fix affine linear spaces (points, lines, planes) to rigid bodies the lowest-degree Hodge Laplacian in the twisted de Rham- and impose geometric constraints among them. For exam- Hodge cochain complex. Motivated by these geometric in- ple, maintaining a fixed angle between two lines, or requir- tuitions, we propose to study the problem of learning group ing a point to coincide with a line. Such a system of rigid actions — partitioning a collection of objects based on the bodies and geometric constraints is called a body-and-cad local synchronizability of pairwise correspondence relations framework. If all of the constraints are point-point distance — and provide a heuristic synchronization-based algorithm constraints, we have the familiar body-and-bar framework for solving this type of problems. We demonstrate the ef- studied by Tay, and White-Whiteley. In this talk I will ficacy of this algorithm on simulated and real datasets. give an overview of body-and-cad rigidity theory laid out in Haller et al, and will present a combinatorial character- ization of body-and-cad rigidity theory. Tingran Gao Department of Mathematics Jessica Sidman Duke University Mount Holyoke College [email protected] [email protected]

Jacek Brodzki University of Southampton MS38 [email protected] Generic Global and Universal Rigidity

Sayan Mukherjee A (bar-joint) framework (G, p) is a graph G,alongwitha Duke University placement p ofitsverticesintoEd. A framework is said [email protected] to be universally rigid if any other (G, q)inany dimension D ≥ d that has the same edge lengths as (G, p) is related to (G, p) (embedded in a d-dimensional affine subspace) by a rigid body motion. I’ll describe an algebraic charac- MS37 terisation of which graphs G have generic universally rigid The Least Squares Klein Bottle For Image Patches frameworks (G, p) and a close connection to a widely used semidefinite programming algorithm for the graph realisa- tion or distance geometry problem. Joint work with Bob It has been observed that an important subspace of high- Connelly and Shlomo Gortler. contrast image patches has the topology of the Klein bot- tle. In this work, we offer a simple explanation for why Louis Theran the Klein bottle appears when modeling edges in images, Freie Universitat Berlin derived from properties of odd and even functions, and in [email protected] doing so show that there are many Klein bottles that one may consider. We consider the task of inferring the best Klein bottle to fit a given set of image patches from a fam- MS38 ily of possibilities, and compare the ability of our proposed Discriminants and Exceptional Configurations of method to represent image patches to other dimension re- Bar-and-joint Frameworks duction techniques. A graph is minimally rigid in the plane if a generic as- Brad Nelson, Gunnar Carlsson signment of edge lengths results in a rigid bar-and-joint Stanford University framework. In particular, there are finitely many options [email protected], [email protected] for distances between unconnected vertices. I will discuss a discriminant associated to a minimally rigid graph and exceptional configurations of edge lengths that result in a moving framework. This is ongoing work with Zvi Rosen, MS38 Jessica Sidman, and Louis Theran.

Tropical Geometry for Rigidity Theory and Matrix ZviH.Rosen Completion Department of Mathematics University of Pennsylvania Tropical geometry offers us a set of tools for illuminating [email protected] the combinatorics of an algebraic matroid. I will discuss how one might use these tools for problems in rigidity the- Jessica Sidman ory and matrix completion. In particular, I will show how Mount Holyoke College to obtain a combinatorial description of the rank-2 com- [email protected] pletion matroid by using tropical geometry to reduce this to a problem about phylogenetic trees which can then be Louis Theran solved. Freie Universitat Berlin [email protected] Daniel I. Bernstein North Carolina State University Cynthia Vinzant 98 AG17 Abstracts

North Carolina State University wavelets by using tools from Algebraic Geometry, such as Dept. of Mathematics Sums of Squares and Quillen-Suslin theorem. [email protected] Youngmi Hur Yonsei University MS39 [email protected] Identifiability: The Uniqueness of the Reconstruc- tion MS39 I will show how methods from Complex Algebraic Geom- Identifiability of An X-rank Decomposition of Poly- etry can be used to determine when the decomposition of nomial Maps a tensor in terms of elementary objects is unique, even in cases where the celebrated Kruskal criterion does not apply. In this talk, we study a polynomial decomposition model For symmetric tensors, the ranks for which generic unique- that arises in problems of system identification, signal pro- ness holds were recently completely classified . For specific cessing and machine learning. We show that this de- (non necessarily symmetric) tensors, we have only partial composition is a special case of the X-rank decomposi- methods to detect the uniqueness, basically for tensors of tion — a concept in algebraic geometry that generalizes submaximal rank or for decompositions which satisfy some the tensor CP decomposition. We prove new results on extra condition (real decompositions, simultaneous decom- generic/maximal rank and on identifiability of the polyno- positions, ...) mial decomposition model. Luca Chiantini Universita di Siena Pierre Comon [email protected] GIPSA-Lab CNRS Grenoble, France [email protected] MS39 Modelling User Feedback in Recommender Sys- Yang Qi tems: Tensor-based Approach University of Chicago [email protected] The main task for conventional recommender systems is to predict what user may like based on known user pref- Konstantin Usevich erences and past collective user behavior. Typically this GIPSA-Lab behavior is gathered in the form of an explicit (e.g. num- [email protected] ber of stars) or implicit (e.g. number of views, clicks) user feedback. The main question then is how to translate these ”signals” from users into a predictive model. One of the MS40 standard ways is to use numeric feedback values directly Mogami Constructions of Manifolds from Trees of as a measure of utility and solve some sort of a matrix Tetrahedra completion task. However, such an approach neglects the fact that user feedback by its nature is subjective and does A 3-ball is a simplicial complex homeomorphic to the unit not align well with the concept of cardinal values. In or- ball in R3. A “tree of tetrahedra” is a 3-ball whose dual der to remove this inconsistency we propose to treat user graph is a tree. It is easy to see that every (connected) 3- feedback as an ordinal variable and model it along with manifold can be obtained from some tree of tetrahedra by users and items in a ternary way. We employ a third-order recursively gluing together two boundary triangles. The tensor factorization technique and use efficient computa- quantum physicist Tsugui Mogami has studied “Mogami tional methods to make our model applicable in real world manifolds”, that is, those manifolds that can be obtained settings, allowing for instant interactions even with new from a tree of tetrahedra by recursively gluing together two users. One of the key features of the model is that it be- *incident* boundary triangles. In 1995 he conjectured that comes equally sensitive to both positive (i.e. high rating) all 3-balls are Mogami. Mogami’s conjecture would imply a and negative (i.e. low rating) signals and allows to ac- much desired exponential bound for the number of 3-balls curately predict relevant items even from a negative only with N tetrahedra. We show that Mogami’s conjecture feedback. Additionally we show that polarity of feedbacks does not hold. plays an important role not only in the model construction but also in the prediction quality assessment, leading to Bruno Benedetti unobvious conclusions on algorithms performance. University of Miami Evgeny Frolov,IvanOseledets [email protected] Skolkovo Institute of Science and Technology [email protected], [email protected] MS40 Equivariant Matroid Theory MS39 Algebraic Geometry in Wavelet Construction I will introduce a theory of group actions on semimatroids, motivated by the study of toric arrangements and encom- Wavelet is one of standard tools in Signal and Image Pro- passing classical matroid theory as well as more recent ob- cessings and its theory is based on Harmonic Analysis. In jects such as arithmetic Tutte polynomials and intersection this talk, I will provide an introduction to wavelets and posets of abelian arrangements. discuss challenges of wavelets, especially from the point of view of constructing them in multidimensional setting. I Emanuele Delucchi will then present some of recent methods for constructing University of Fribourg AG17 Abstracts 99

[email protected] Inference via Groebner Bases

Recently, tools from algebraic geometry have been em- MS40 ployed to infer the network structure of biological systems. Low Rank Vector Bundles and Simplicial Com- In particular for a given set of data points over a finite plexes with Symmetry field, the ideal of points can be used to describe the space of polynomial functions that fit the data. Each reduced This talk will describe several examples where geometric Groebner basis of the ideal represents a distinct choice of objects with symmetry have played a role in the construc- minimal polynomial model for the underlying network. Be- tion of varieties and vector bundles with special properties. cause these models may give rise to vastly different pre- Time Permitting, there will also be a discussion of the role dictions about the network, identifying which ideals have of symmetry in probabilistic computations in the setting unique reduced Groebner bases is of importance. In this of enumerative algebraic geometry. talk, we identify geometric properties of the data and alge- braic properties of the ideal that result in a unique reduced Chris Peterson Groebner basis. Colorado State University [email protected] Brandilyn Stigler Southern Methodist University [email protected] MS40 The Convex Hull of Two Circles in R3 MS41 We describe convex hulls of the simplest compact space On the Perfect Reconstruction of the Topology of curves, reducible quartics consisting of two circles. When Dynamic Networks the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of The network inference problem consists in reconstructing degree eight whose ruling forms a genus one curve. We the structure or wiring diagram of a dynamic network from classify which curves arise, classify the face lattices of the time-series data. Even though this problem has been stud- convex hulls and determine which are spectrahedra. We ied in the past, there is no algorithm that guarantees per- also discuss an approach to these convex hulls using pro- fect reconstruction of the structure of a dynamic network. jective duality. In this talk I will present a framework and algorithm to solve the network inference problem that, given enough Evan D. Nash data, is guaranteed to reconstruct the structure with zero The Ohio State University errors. The framework uses tools from algebraic geometry. [email protected] Alan Veliz-Cuba Ata F. Pir, Frank Sottile, Li Ying Department of Mathematics Texas A&M University University of Dayton atafi[email protected], [email protected], [email protected] [email protected]

MS41 MS41 Model-dependent and Model-independent Control Turing, a Software Package with Crowd-sourcing of Biological Network Models Capabilities for the Analysis of Finite Polynomial Dynamical Systems Network models of intracellular signaling and regulation are ubiquitous in systems biology research because of their Polynomial dynamical systems over finite fields are rich ability to integrate the current knowledge of a biological and interesting mathematical objects that bring together process and test new findings and hypotheses. An often dynamical systems theory, combinatorics, and computer asked question is how to control a network model and drive algebra. A special instance are Boolean networks. They it towards its dynamical attractors (which have been found have found applications in a range of fields, including sys- to be identifiable with phenotypes or stable patterns of ac- tems biology, computer science, engineering, and physics. tivity of the modeled system), and which nodes and in- The software package Turing provides a collection of al- terventions are required to do so. In this talk, we will gorithms to construct such systems and analyze their dy- introduce two recently developed network control meth- namics, among other capabilities. An important feature ods -feedback vertex set control and stable motif control- of the software package is that it enables algorithm devel- that use the graph structure of a network model to identify opers to add an implemented algorithm to the package on nodes that drive the system towards an attractor of inter- their own, and connect it with other existing algorithms est (i.e., nodes sufficient for attractor control). Feedback into workflows. This uses recent new software engineering vertex set control makes predictions that apply to all net- tools. In this way, Turing is a fully crowd-sourced tool for work models with a given graph structure and stable motif finite polynomial dynamical systems. control makes predictions for a specific model instance, and this allows us to compare the results of model-independent Reinhard Laubenbacher and model-dependent network control. We illustrate these Center for Quantitative Medicine, UConn Health methods with various examples and discuss the aspects of Jackson Laboratory for Genomic Medicine each method that makes its predictions dependent or inde- [email protected] pendent of the model. Jorge Zanudo MS41 Pennsylvania State University Model Selection Strategies in Biological Network [email protected] 100 AG17 Abstracts

Reka Albert recently constructed LRCs with one and two recovery sets Department of Physics from algebraic curves. This talk presents a generalization Penn State University of this construction to t = 3 recovery sets, using iterated [email protected] fiber products of curves. Codes with arbitrarily many recovery sets are constructed, employing maximal curves Gang Yang from fiber products devised by Van der Geer and Van der The Pennsylvania State University Vlugt. This is joint work with Kathryn Haymaker and [email protected] Gretchen Matthews. Beth Malmskog MS42 Villanova University Algebraic Problems in Error Correcting Codes Mo- [email protected] tivated by Distributed Storage Applications We introduce several problems in coding theory that arose MS43 recently from the demands of distributed storage. A com- Relative Entropy Relaxations for Signomial Opti- mon feature of these problems is local correction of errors mization or erasures, by which we mean either accessing a small part of the codeword or transmitting a small amount of infor- Due to its favorable analytical properties, the relative en- mation to the decoder. We emphasize algebraic techniques tropy function plays a prominent role in a variety of con- in the construction of codes. In particular, a natural appli- texts in information theory and in statistics. In this talk, cation of the basic algebraic geometric construction gives we discuss some of the beneficial computational properties rise to good families of codes with locality. of this function by describing a class of relative-entropy- based convex relaxations for obtaining bounds on signomi- Alexander Barg als programs (SPs), which arise commonly in many prob- University of Maryland lems domains. SPs are non-convex in general, and families [email protected] of NP-hard problems can be reduced to SPs. By appealing to representation theorems from real algebraic geometry, MS42 we show that sequences of bounds obtained by solving in- creasingly larger relative entropy programs converge to the Multi-cyclic Locally Repairable Code global optima for broad classes of SPs. The central idea Locally repairable code is class of error-correcting codes in underlying our approach is a connection between the rela- which each code symbol can be recovered from a set of r tive entropy function and efficient proofs of nonnegativity other symbols, for some integer r less than the code length. via the arithmetic-geometric-mean inequality. The code parameter r is called the locality. Codes with small locality have efficient repair for each code symbol, Venkat Chandrasekaran and thereby find applications in distributed storage sys- California Institute of Technology tems. In this talk, we present some locally repairable codes [email protected] whose code symbols can be arranged in a multi-dimensional array, and are closed under cyclic shift in each dimension. Parikshit Shah The multi-cyclic structure enables fast encoding and de- University of Wisconsin/Philips Research coding algorithms. The code constructions are based on [email protected] multi-dimensional discrete Fourier transform and algebraic plane curves. MS43 Kenneth Chum Factorizations of PSD Matrix Polynomials and The Chinese University of Hong Kong their Smith Normal Forms [email protected] It is well-known, that any univariate polynomial matrix A MS42 over the complex numbers that takes only positive semidef- inite values on the real line, can be factored as A = B∗B Examples of Locally Recoverable Codes Arising for a polynomial square matrix B. This is a simultaneous from Galois Covers of Curves generalization of both the fact that a univariate real poly- In my masters thesis project explicit new examples of lo- nomial that is psd factors as a complex polynomial times its conjugate and that a constant psd matrix A can be fac- cally recoverable codes are proposed, in the spirit of earlier ∗ examples by Barg, Tamo and Vladuts. The talk describes tored as B B For real A, in general, one cannot choose B the geometry behind these examples and discusses their to be also a real square matrix. However, if A is of size × t parameters and their implementation in MAGMA. n n, then a factorization A = B B exists, where B is a real rectangular matrix of size (n +1)× n. It turns out Dani Hulzebos that these correspond to the factorizations of the Smith University of Groningen normal form of A, an invariant not usually associated with [email protected] symmetric matrices in their role as quadratic forms. A consequence is, that the factorizations can usually be eas- ily counted, which in turn has an interesting application to MS42 minimal length sums of squares of linear forms on varieties Locally Recoverable Codes with Multiple Recovery of minimal degree. Sets: A General Fiber Product Construction Christoph Hanselka Locally recoverable codes (LRCs) have benefits for dis- University of Auckland tributed storage applications. Barg, Tamo, and Vladut [email protected] AG17 Abstracts 101

Rainer Sinn [email protected] Georgia Tech [email protected] MS44 Numerical Challenges to Successful Decomposition MS43 of Real Algebraic Surfaces Reciprocal Linear Spaces Computation of real algebraic objects remains a challenge in modern software implementation. Symbolic decompo- A reciprocal linear space is the image of a linear space un- sitions, such as the Cylindrical Algebraic Decomposition der coordinate-wise inversion. These fundamental varieties implemented in Maple, e.g., face doubly-exponential com- describe the analytic centers of hyperplane arrangements plexity bounds. Numerical cellular decompositions, such and appear as part of the defining equations of the central as that implemented in Bertini real, face both complexity path of a linear program. Their structure is controlled by and numerical issues. Conditioning and scale of problems, an underlying matroid. This provides a large family of hy- naive implementations of necessary symbolic algorithms, perbolic varieties, recently introduced by Shamovich and and tracking repeatedly toward singular points on already Vinnikov. Here we give a definite determinantal represen- singular objects all cause algorithms to take a long time, tation to the Chow form of a reciprocal linear space. One or to commonly fail at certain points. This talk will dis- consequence is the existence of symmetric rank-one Ulrich cuss some of these ongoing and persistent issues, and ideas sheaves on reciprocal linear spaces. Another is a represen- about what can be done both in theory and in software to tation of the entropic discriminant as a sum of squares. For remedy them. generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph Dani Brake and generalizations to simplicial matroids. This raises in- University of Notre Dame teresting questions about the combinatorics of hyperbolic Dept. of App. Comp. Math. & Stats. varieties and connections with the positive Grassmannian. [email protected]

Mario Kummer MPI Leipzig MS44 [email protected] A Linear Homotopy Method for Finding General- ized Tensor Eigenpairs Cynthia Vinzant North Carolina State University The tensor eigenvalue problems were first introduced in Dept. of Mathematics 2015 by Qi and Lim independently. Since then, tensor [email protected] eigenvalues have found applications in various areas, such as automatic control, statistical data analysis, diffusion tensor imaging, image authenticity verification, high order MS43 Markov chains, spectral hypergraph theory, and quantum entanglement. In this talk, we consider a generalized tensor Infeasible Subsystems of Spectrahedral Systems eigenvalue problem, which covers various types of tensor eigenvalues proposed in the literature, including eigenval- Given real symmetric n × n-matrices A0,...,Am,letA(y) m ues, E-eigenvalues, H-eigenvalues, Z-eigenvalues, and D- 0 − denote the linear matrix pencil A(y)=A i=1 yiAi. eigenvalues. We propose a linear homotopy method for Farkas’ lemma for semidefinite programming then provides solving this tensor eigenproblem. This method follows the a characterization of the feasibility of the spectrahedron optimal number of paths and finds all isolated generalized SA in terms of an alternative spectrahedron. In the well- eigenpairs. It can also find some generalized eigenpairs studied special case of linear programming, the index sets contained in the positive dimensional components (if there of irreducible inconsistent subsystems are exactly the ver- are any). We have implemented the method in a MATLAB tices of the alternative polyhedron (Gleeson, Ryan, 1990). software package, TenEig. Numerical results are provided Here, we study the nonlinear situation of general spectrahe- to show the efficiency of the proposed method. This talk is dra. We show that one direction of the Theorem of Gleeson based on joint work with Liping Chen and Liangmin Zhou. and Ryan can be generalized. The reverse direction, how- ever, is not true in general, which we show based on study- ing the semialgebraic inequalities underlying the positive Lixing Han semidefiniteness condition of a certain counterexample. University of Michigan at Flint lxhan@umflint.edu Kai Kellner J.W. Goethe-Universit¨at Frankfurt am Main, Germany MS44 [email protected] Homotopies for Overdetermined Systems with Ap- plications in Computer Vision Marc Pfetsch Research Group Optimization, Department of Many problems in computer vision are represented using a Mathematics parametrized overdetermined system of polynomials which TU Darmstadt must be solved quickly and efficiently. We propose new [email protected] numerical algebraic geometric methods to efficiently solve parameterized overdetermined systems in projective space Thorsten Theobald with the new approaches using locally adaptive methods J.W. Goethe-Universit¨at and sparse matrix calculations. Examples will be provided Frankfurt am Main, Germany to compare the new methods with traditional approaches 102 AG17 Abstracts

in numerical algebraic geometry. Georg Grasegger RICAM - Linz Margaret Regan [email protected] University of Notre Dame NotreDameINUSA Christoph Koutschan [email protected] RICAM Linz Austrian Academy of Sciences Jonathan Hauenstein [email protected] University of Notre Dame Dept. of App. Comp. Math. & Stats. Niels Lubbes [email protected] RICAM - Linz [email protected] Sameer Agarwal Google Inc Josef Schicho [email protected] RISC - Linz [email protected]

MS44 A Dynamic Enumeration Approach for Computing MS45 Tropical Prevarieties Detecting Projective Symmetries and Equivalences of Rational Curves by Solving Polynomial Systems The computation of the tropical prevariety P of a polyno- of Equations mial system f requires the intersection of normal cones to the Newton polytopes of f. It is a generalization of the It has been observed that many application-oriented prob- mixed volume computation, and a necessary first step in lems in geometric modelling are solved by using techniques the generalization of polyhedral homotopies from isolated from symbolic computation, e.g. by solving polynomial solutions to positive dimensional solution sets. We apply systems of equations. We investigate the problem of de- the idea of dynamic enumeration to computing prevarieties, tecting equivalences and symmetries, which is an essential and compare it to other algorithms. With our methods, we problem in Pattern Recognition, Computer Graphics and have computed the tropical prevariety of cyclic-16; we will Computer Vision. In particular, we consider two input mention further experimental results as well. curves in Rd, given by proper rational parameterizations in reduced form. We decide whether these curves can be Anders Jensen transformed into each other using a projective transforma- Department of Mathematical Sciences tion, i.e., we are looking for projective equivalences between Aarhus University the two curves. In order to do so, we create a polynomial [email protected] system of equations in 4 unknowns which define a linear rational reparameterization of the curves. For creating the Jeff Sommars system we use homogeneous coordinates to obtain a poly- University of Illinois at Chicago nomial representation from the rational input and projec- Chicago, Illinois USA tive coordinates to identify the projective transformation. [email protected] Furthermore, we investigate the special cases of detecting affine equivalences and symmetries as well as polynomial input curves. Jan Verschelde Department of Mathematics, Statistics and Computer Michael Hauer Science Johannes Kepler University of Linz University of Illinois at Chicago [email protected] [email protected] Bert Juettler Johannes Kepler University of Linz MS45 Institute of Applied Geometry The Number of Embeddings of a Laman Graph [email protected]

Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are MS45 finitely many ways, up to isometries, to realize a Laman On the Connection Between Lexicographic Groeb- graph in the plane. Such realizations can be seen as so- ner Bases and Triangular Sets lutions of systems of quadratic equations prescribing the distances between pairs of points. Here we provide a re- Lexicographic Groebner bases and triangular sets are stan- cursion formula for the number of complex solutions of such dard tools in polynomial elimination theory. In this talk, systems. This is a joint work with J.Capco, G. Grasegger, we present new results on the intrinsic structures of lex- C. Koutschan, N. Lubbes and J. Schicho. icographic Greobner bases and relations between lexico- graphic Groenber bases and the minimal triangular sets Matteo Gallet contained in them called W-characteristic sets. It is shown RICAM, Austrian Academy of Sciences that either this W-characteristic set is normal or there are [email protected] explicit (pseudo-)divisibility relations between the polyno- mials in it. Based on these properties of W-characteristic Jose Capco sets, we introduce the concept of characteristic pair con- RISC - Linz sisting of a reduced lexicographic Groebner basis and its [email protected] normal W-characteristic set, and design an algorithm for AG17 Abstracts 103

decomposing any polynomial set into finitely many char- is an expression of M as Hadamard product, i.e., entry- acteristic pairs with associated zero relations, which pro- wise product, of several rank r matrices of the same size. vide representations of the zero of the polynomial set in The smallest length of such a decomposition is called the terms of those of Groebner bases and in terms of those of r-th Hadamard rank of M. The study of these decompo- triangular sets simultaneously. Several nice properties of sitions reduces to studying particular algebraic varieties: the decomposition and the resulting characteristic pairs, Hadamard powers of secant varieties of Segre and Veronese in particular relationships between the Groebner basis and varieties. These algebraic varieties have been recently in- the W-characterisitic set in each pair, are established. This troduced and they give also geometric models for particular talk is based on joint work with Dongming Wang and Rina statistical models called Restricted Boltzmann Machines. Dong. In this talk, I will introduce these definitions and I will explain some algebrogeometric toll to study these objects. Chenqi Mou In particular, I will focus on Hadamard decompositions of Beihang University, China generic matrices. I will report results from joint projects [email protected] with N.Friedenberg, R.Williams and J.Kileel.

Alessandro Oneto MS45 INRIA Nice (France) Radical Varieties [email protected] We introduce some algebraic constructions that are useful for studying the variety defined by a radical parametriza- MS46 tion (a tuple of functions involving complex numbers, n Decomposing Tensors into Frames variables, the four field operations and radical extrac- tions). We also present algorithms to implicitize rad- A symmetric tensor of small rank decomposes into a config- ical parametrizations and to check whether a radical uration of only few vectors. We study the variety of tensors parametrization can be reparametrized into a rational for which this configuration is a unit norm tight frame. parametrization. Elina Robeva David Sevilla Massachusetts Institute of Technology University of Extremadura [email protected] [email protected] Elina Robeva MS46 UC Berkeley [email protected] Monomials: Complex vs Real Rank and Waring Loci MS47 In this talk we will see some recent results about the War- ing rank for homogenous polinomials, such as results on the Properties of Tree-valued Dimensionality Reduc- real rank of monomials, the simultaneous rank of monomi- tion als, and Waring loci. Waring loci are introduced to study all linear forms which could appear in some minimal sum The metric geometry of the moduli space of rooted leaf- of powers decompsoition of a given form. We will see how labeled trees was shown by Billera, Holmes, and Vogtmann Waring loci relate to Strassen’s Conjecture. to support unique geodesics, providing the foundation for statistics on sets of phylogenies. Results of Owen and Enrico Carlini Provan show how to compute the BHV metric in poly- Politecnico di Torino nomial time. However, for large trees that arise in appli- [email protected] cations (e.g., large scale viral evolution), working directly in tree space is often still intractable. To handle this case, Zairis et al. introduced an intrinsic method of tree dimen- MS46 sionality reduction which maps a tree to a distribution of On Critical Rank-k Approximations to Tensors trees in low-dimensional tree space with bounded metric distortion. In this talk we explain how to use these point By the Eckart-Young theorem, the critical rank-k approx- clouds in tree space for geometric and combinatorial infer- imations to a matrix A all lie in the span of critical rank-1 ence. approximations to A. I will discuss analogues of this result for higher-order tensors. Gill Grindstaff,Th´er`ese Wu The University of Texas at Austin Jan Draisma gillian.grindstaff@math.utexas.edu, [email protected] Eindhoven University of Technology, The Netherlands Department of Mathematics and Computer Science [email protected] MS47 Tropical Fermat-Weber Points Giorgio Ottaviani, Alicia Tocino S´anchez University of Firenze We investigate the computation of Fermat-Weber points [email protected]fi.it, [email protected] under the tropical metric, motivated by its application to the space of equidistant phylogenetic trees with N leaves realized as the tropical linear space of all ultrametrics. MS46 While the Fr´echet mean with the CAT(0)-metric of Billera- Hadamard Decompositions of Matrices Holmes-Vogtman has been studied by many authors, the Fermat-Weber point under tropical metric in tree spaces Given a matrix M,ar-th Hadamard decomposition of M is not well understood. In this paper we investigate the 104 AG17 Abstracts

Fermat-Weber points under the tropical metric and we [email protected] show that the set of tropical Fermat-Weber points is a clas- sical convex polytope. We identify conditions under which this set is a singleton. MS48 Latent Tree Models and the Em Algorithm Bo Lin UC Berkeley Latent tree models are tree structured graphical models USA where some random variables are observable while others [email protected] are hidden or latent. In this talk we will show how we estimate the volume of a latent tree model known as the Ruriko Yoshida 3-leaf model, where one node is a hidden variable with 2 Naval Postgraduate School states and is the parent of three observable variables with [email protected] 2 states, in the 7-dimensional simplex, and the volume of another 3-leaf model, where the root is a 3-state hidden variable, and is the parent of three observable variables MS47 with 3, 3 and 2 states, in the 17-dimensional simplex. We will also discuss the algebraic boundary of these models and Tensors and Algebra Give Interpretable Groups for we observe the behavior of the estimates of the Expecta- Crosstalk Mechanisms in Breast Cancer tion Maximization algorithm (EM algorithm), an iterative method typically used to try to find a maximum likelihood We introduce a tensor-based clustering method to extract estimator. sparse structure from high-dimensional, multi-indexed datasets. The framework is designed to enable clustering Hector D. Banos Cervantes data in the presence of structural requirements which can Department of Mathematics and Statistics be encoded as algebraic constraints in a linear program. University of Alaska We showcase our method on a collection of experiments [email protected] measuring the response of genetically diverse breast cancer cell lines to an array of ligands. Each experiment consists of a cell line-ligand combination, and contains time-course MS48 measurements of the early-signaling kinases MAPK and Parameter Identification in Directed Graphical AKT at two different ligand dose levels. By imposing ap- Models propriate structural constraints and respecting the multi- indexed structure of the data, our clustering analysis can We treat statistical models that relate random variables of be optimized for biological interpretation and therapeutic interest via a linear equation system that features stochas- understanding. tic noise. These models, also known as structural equa- tion models, are naturally represented by a mixed graph, Anna Seigal with directed edges indicating non-zero coefficients in the UC Berkeley linear equations and bidirected edges indicating possible [email protected] correlations among noise terms. In this talk we report on progress on combinatorial conditions for parameter identi- Mariano Beguerisse-D´ıaz fiability. Identifiability holds if coefficients associated with University of Oxford the edges of the graph can be uniquely recovered from the [email protected] covariance matrix they define.

Birgit Schoeberl Mathias Drton Merrimack Pharmaceuticals Department of Statistics [email protected] University of Washington [email protected] Mario Niepel Harvard Medical School MS48 [email protected] Combinatorics of Bayesian Networks

Heather Harrington Graphical models based on directed acyclic graphs (DAGs), University of Oxford also known as Bayesian networks, are used to model com- [email protected] plex cause-effect systems across a variety of research areas including computational biology, sociology, and environ- mental management. A fundamental problem in causal- MS47 ity is to learn an unknown DAG G based only on a set Asymptotic Distributions of Sample Means on of observed conditional independence (CI) relations. The Piece-wise Linear Stratified Spaces typical approach to this learning problem searches over the space of all Markov equivalent DAGs, i.e. classes of DAGs We discuss the asymptotic behavior of intrinsic means on encoding identical CI relations inherited via the Markov piece-wise Euclidean stratified spaces, and on stratified assumption. However, since both the space of DAGs and spaces endowed with a chord distance associated with an the space of Markov equivalence classes are notably larger embedding into a numerical space. The results are then ap- than the space of permutations, recent permutation-based plied to data analysis on spaces of phylogenetic trees with algorithms have been proposed. In this talk we examine a given number of leafs. a pair of simplex-type permutation-based algorithms that can be thought of as a search over the nodes of a new type Chen Shen of generalized permutohedron called a DAG associahedron. Florida State University We will discuss consistency guarantees for these algorithms AG17 Abstracts 105

as well as address their efficiency in relation to the problem Hal Schenck of Markov equivalence. Mathematics Department University of Illinois Liam Solus,LiamSolus [email protected] KTH Royal Institute of Technology [email protected], [email protected] MS49 Adityanarayanan Radhakrishnan Semialgebraic Splines Massachusetts Institute of Technology [email protected] Semialgebraic splines are functions that are piecewise poly- nomial with respect to a cell decomposition of their domain Lenka Matejovicova into sets defined by polynomial inequalities. We study bi- IST Austria variate semialgebraic splines, computing the the dimen- [email protected] sion of the space of splines with large degree in two ex- treme cases when the cell decomposition has a single in- ternal vertex. First, when the forms defining the edges Caroline Uhler, Yuhao Wang span a two-dimensional space of forms of degree n—then Massachusetts Institute of Technology the curves they define meet in n2 points in the complex [email protected], [email protected] projective plane. In the other extreme, the curves have distinct slopes at the vertex and do not simultaneously vanish at any other point. This is joint work with Michael MS48 DiPasquale and Lanyin Sun On Sampling Graphical Markov Models Frank Sottile Abstract not available at time of publication. Texas A&M University [email protected] Prasad Tetali Georgia Institute of Technology Sun Lanyin [email protected] Dalian Univerity of Science and Technology [email protected] MS49 Michael Di Pasquale Algebraic Methods in Spline Theory Oklahoma State Univeristyr [email protected] Multivariate splines are a cornerstone to approximation theory and numerical analysis. A fundamental question in the theory of splines is to determine the dimension of MS49 (and a basis for) the space of splines of degree at most d n Geometric Realizations of the Space of Splines on on a partition in R . In 1988, Billera introduced methods a Simplicial Complex from homological and commutative algebra to address this question; in particular he proved a conjecture of Strang re- r 1 We consider the space of C -continuous splines (or piece- garding the dimension of the C spline space on a planar wise polynomial functions) defined on a simplicial complex. triangulation. A main theme of Billera’s argument is a hy- Besides the practical applications of splines, including the brid of topological and algebraic techniques, later refined solution of partial differential equations by the finite el- by Schenck and Stillman. In this talk we explain Billera’s ement method, and the approximation of shapes in geo- insight and some of the results that have been obtained metric modeling, the space of Cr-continuous splines forms using these techniques. Time permitting, we will highlight a ring, and one can study its algebraic structure. More connections to difficult conjectures in algebraic geometry precisely, the space of C0-continuous splines is a quotient and to the theory of hyperplane arrangements. of the Stanley-Reisner ring of the corresponding simpli- cial complex, and the geometric realization of the Stanley- Michael DiPasquale, Michael DiPasquale Reisner ring reflects the structure of the simplicial com- Oklahoma State University plex. In the talk, we shall consider the generalized Stanley- [email protected], [email protected] Reisner rings associated to a simplicial complex, namely the ring of spline functions with higher order of global con- tinuity on the simplicial complex, and give a description of MS49 their geometric realizations for particular instances of the Subdivision and Spline Spaces dual graph of the complex. This is a joint work with R. Piene. A standard construction in approximation theory is mesh refinement. For a simplicial or polyhedral mesh Δ ⊆ Rk, Nelly Villamizar we study the subdivision Δ obtained by subdividing a Johann Radon Institute for Computational and Applied maximal cell of Δ. We give sufficient conditions for the Mathema module of splines on Δ to split as the direct sum of splines [email protected] on Δ and splines on the subdivided cell. As a consequence, we obtain dimension formulas and explicit bases for several commonly used subdivisions and their multivariate gener- MS50 alizations. Robust Permanence of Polynomial Dynamical Sys- tems Tatyana Sorokina Towson University A “permanent” dynamical system is one whose positive [email protected] solutions stay bounded away from zero and infinity. The 106 AG17 Abstracts

permanence property has important applications in bio- MAK and BST (Biochemical Systems Theory) models of chemistry, cell biology, and ecology. Inspired by reaction biological systems with these kinetics. network theory, we define a class of polynomial dynamical systems called N -tropically endotactic, using a given poly- Dylan Antonio Talabis hedral fan N . We show that for appropriate fans N these Institute of Mathematical Sciences and Physics polynomial dynamical systems are permanent, irrespective University of the Philippines Los Ba˜nos to the values of (possibly time-dependent) parameters in [email protected] these systems. These results generalize the permanence of 2D reversible and weakly reversible mass-action systems. Carlene Arceo Institute of Mathematics James Brunner University of the Philippines Diliman University of Wisconsin [email protected] [email protected] Editha Jose Gheorghe Craciun Institute of Mathematical Sciences and Physics Department of Mathematics, University of University of the Philippines Los Ba˜nos Wisconsin-Madison [email protected] [email protected] Angelyn R. Lao Mathematics Department MS50 De La Salle University Algebraic Approaches to Biological Interaction [email protected] Networks Eduardo Mendoza Algebraic analysis of biological interaction networks Faculty of Physics and Center for Nanoscience promises (biological) insights that are hard to obtain by Ludwig Maximilian University other mathematical approaches. In this introductory talk [email protected] I will • discuss the benefits of algebraic approaches (especially MS50 in comparison to purely numerical approaches), Joining and Decomposing Reaction Networks • present areas of active research and their relation to the mini-symposium and Much research in systems biology focuses on how coupling two or more specific pathways affects systems-level behav- • comment on the usefulness and abundance of alge- ior; however, theory for tackling this problem in general braic parameterizations of steady states. has lagged behind. In this talk, we discuss how combin- ing or decomposing networks affects three properties that Carsten Conradi reaction networks may possess - identifiability, multista- HTW Berlin tionarity, and steady-state invariants having certain struc- [email protected] ture. In particular, we will show some sufficient conditions for obtaining an identifiable model from two identifiable submodels, how to obtain the steady-state invariants of a MS50 larger model from the steady-state invariants of its sub- Reactant Subspaces and Positive Equilibria of models, and discuss when multistationarity is preserved Power Law Kinetic Systems when combining networks. Nicolette Meshkat The linear subspace generated by a chemical reaction net- Santa Clara University work’s (CRN) reactant complexes in species (composition) [email protected] space, which we call the reactant subspace R,anditsim- pact on a kinetic systems dynamics, have so far received lit- tle attention in chemical reaction network theory (CRNT). Elizabeth Gross To our knowledge, the reactant subspace has been studied San Jose State University only in Injectivity Theory for mass action kinetics (MAK) [email protected] and was initiated by Craciun and Feinberg (2005) . In this talk, we will introduce the concepts of reactant rank Heather Harrington and reactant deficiency, and compare them with their ana- Mathematical Institute logues currently used in CRNT. We will discuss networks University of Oxford with the stoichiometric subspace S contained in R,which [email protected] we call reactant-determined stoichiometric subspace (RSS) networks that are important in MAK and BST embedded Anne Shiu systems. We introduce the concept of kinetic flux subspace Texas A&M University for such networks, as an extension of the kinetic order sub- [email protected] space of M¨uller and Regensburger (2014) for GMAK sys- tems on cycle terminal networks, i.e. those in which every complex is a reactant complex. In conclusion, we apply MS51 these concepts and results to study positive equilibria of the Obstructions to Graph Homomorphisms set PL-ILK (Power Law Inflow excluded Linkage class lin- early independent Kinetics). We show that the Deficiency In his seminal proof of the Kneser conjecture, Lovasz in- Zero and Deficiency One Theorems of mass action kinet- troduced the neighborhood complex of a graph and showed ics can be extended to these kinetics and give examples of that the Z2-equivariant topology of these spaces provided AG17 Abstracts 107

lower bounds on chromatic number. Similar homomor- also implies that all simplicial manifolds with g2 =2are phism spaces of graphs were studied by Babson, Kozlov, polytopal spheres. and others. We discuss some further extensions of these ideas, including connections to statistical physics, categor- Hailun Zheng ical homotopy theory, and group actions involving larger University of Washington symmetry groups. Parts of this are joint with Ragnar Freij. [email protected]

Anton Dochtermann MS52 Texas State University Persistence-based Summaries for Metric Graphs [email protected] In this talk, we will focus on giving a qualitative description of information that one can capture from metric graphs MS51 using topological summaries. In particular, we will give a Convex Geometry and Compactified Jacobians complete characterization of the 1-dimensional persistence diagrams for metric graphs under a particular intrinsic set- A nodal curve may be thought of as a “limit” of smooth ting. We will also look at two persistence-based distances curves. I will discuss (generalized) Jacobians associated to that one may define for metric graphs and discuss progress such curves. In this setting, the combinatorics of the “dual toward establishing an inequality relating these two dis- graph” play an essential role. I will describe some beauti- tances. ful combinatorics and convex geometry that is related to compactifications of these varieties. (This talk is based on Ellen Gasparovic joint projects with Alberto Bellardini.) Union College [email protected] Farbod Shokrieh Cornell University [email protected] MS52 Induced Dynamics in the Space of Persistence Di- agrams MS51 Hyperplane Equipartitions Plus Constraints Persistent homology can be used to summarize topological characteristics of scalar fields into compact representations Although equivariant methods have yielded a number of called persistence diagrams. The PD stability theorem for fruitful applications in combinatorial geometry, their in- Lipschitz functions ensures that smooth variations in the ability to settle the long-standing but now-settled Tverberg underlying function will give rise to continuous variations conjecture has made clear the need to move “beyond” the in the metric space of PDs, Per. For instance, mapping use of Borsuk-Ulam type theorems alone. Such concerns functions along a solution to a dissipative PDE to their hold equally well for hyperplane mass equipartition prob- diagrams will give rise to continuous dynamics in Per. In lems dating back to Gr¨unbaum, for which the best known this talk we discuss recent numerical investigations of these topological upper bounds nearly always exceed the conjec- induced dynamics, with a focus on dynamic and structural tured values arising from simple dimension counting. By properties of underlying attractors of systems exhibiting analogy with the “constraint” method of Blagojevi´c, Frick, complex spatial and temporal behaviors. and Ziegler, we show how this gap can be removed by the imposition of further conditions – on the hyperplanes them- Francis Motta selves (e.g., orthogonality and/or affine containment), as Duke University well as by adding further masses with specified partition- Department of Mathematics types (e.g., cascades and/or those of a “Makeev” variety) – [email protected] thereby yielding a variety of optimal results still obtainable via classical group cohomological techniques. MS52 Steven Simon Persistence-based Summaries for Metric Graphs – Bard College Revisited [email protected] We use persistence to obtain qualitative-quantitative sum- maries of metric graphs. We analyze the information con- MS51 tained in these persistence-based summaries of graphs, and A Characterization of Simplicial Manifolds with compare their discriminative powers. This is joint work g2 ≤ 2 with Ellen Gasparovic, Maria Gommel, Emilie Purvine, Bei Wang, Yusu Wang, and Lori Ziegelmeier. The celebrated low bound theorem states that any sim- plicial manifold of dimension ≥ 3satisfiesg2 ≥ 0, and Radmila Sazdanovic equality holds if and only if it is a stacked sphere. Further- North Carolina State University more, more recently, the class of all simplicial spheres with [email protected] g2 = 1 was characterized by Nevo and Novinsky, by an ar- gument based on rigidity theory for graphs. In this talk, I will first define three different retriangulations of simplicial MS52 complexes that preserve the homeomorphism type. Then Cliques and Cavities in Neuroscience I will show that all simplicial manifolds with g2 ≤ 2can be obtained by retriangulating a polytopal sphere with a Encoding the axon bundles between brain regions as a com- smaller g2. This implies Nevo and Novinskys result for plex network has provided novel insights into brain func- simplicial spheres of dimension ≥ 4. More surprisingly, it tion and disease. Standard network tools describe local or 108 AG17 Abstracts

aggregate global phenomena, however many neural func- best previously known examples in Grassl’s tables. tions occur at the mesoscale, involving complex patterns of interactions between multiple brain regions. Here we John B. Little employ persistent homology to illuminate essential cycles College of the Holy Cross within the structural brain network which may allow for [email protected] distributed computations. We present long-lived cycles in multiple dimensions, and demonstrate the non-random na- ture of these persistent features across eight healthy indi- MS53 viduals. Finally we propose implications of these persistent Optimal Locally Recoverable Codes from Algebraic cycles for cognitive function and highlight the potential of Surfaces this topological view of the brain network. In this talk we will explain a construction of locally recov- Ann Sizemore,ChadGiusti erable codes (LRCs) using algebraic surfaces that yields University of Pennsylvania optimal codes, i.e., codes that meet the Singleton-type [email protected], [email protected] bound for codes with locality constraints. The length of the codes produced is roughly the square of the size of the alphabet. This suggests that an analogue for optimal MS53 LRC codes of the main conjecture for maximal distance separable codes merits investigation. This is joint work Hierarchical Locally Recoverable Codes on Alge- with Sasha Barg, Kathryn Haymaker, Everett Howe, and braic Curves Gretchen Matthews.

Hierarchical codes, introduced in [SAK15], are a particular Anthony Varilly Alvarado type of locally recoverable code in which the code restricted Rice University to a recovering set is itself locally recoverable. The bene- [email protected] fits are that very simple erasure patterns can be recovered at the small locality level while still handling larger era- sure patterns using the larger recovering sets. I will show MS54 that the natural locally recoverable codes constructed in Gram Spectrahedra [BTV15] have a built-in, but more flexible, hierarchical structure. I will also introduce a new construction for codes Representations of nonnegative polynomials as sums of with n levels of hierarchy from infinite towers of algebraic squares are central to real algebraic geometry and the sub- curves. Finally, I will examine the asymptotic bounds on ject of active research. The sum-of-squares representations such codes and apply an example obtained from extracting of a given polynomial are parametrized by the convex body sub-towers of curves from the Garcia-Stichtenoth tower. of positive semidefinite Gram matrices, called the Gram spectrahedron. This is a fundamental object in polynomial Sean F. Ballentine optimization and convex algebraic geometry. We summa- University of Maryland rize results on sums of squares that fit naturally into the [email protected] context of Gram spectrahedra, present some new results, and highlight related open questions. We discuss sum-of- squares representations of minimal length and relate them MS53 to Hermitian Gram spectrahedra and point evaluations on Recent Results on Codes for Distributed Storage toric varieties.

An overview of our recent results on codes for distributed Lynn Chua storage will be presented with a focus on codes for locality University of California at Berkeley and multiple erasure correction. [email protected]

Vijay Kumar Daniel Plaumann Indian Institute of Science TU Dortmund [email protected] [email protected]

Rainer Sinn MS53 Georgia Tech Evaluation Codes from Algebraic Surfaces over a [email protected] Finite Field Cynthia Vinzant Following Goppa’s recipe, algebraic varieties of dimension North Carolina State University greater than 1 over a finite field can also be used to con- Dept. of Mathematics struct evaluation codes. In this talk we will study the [email protected] potential for finding good codes starting from algebraic surfaces. Limiting the Picard number of the surface (the rank of the Neron-Severi group over the finite field) puts MS54 restrictions on the irreducible curves on the surface that Constrained Optimization via SONC Polynomials can appear as components of divisors in the hyperplane section divisor class (and this is important since reducible Determining lower bounds for real polynomials is a cen- divisors often yield codewords of small weight in the associ- tral problem in polynomial optimization being NP-hard in ated evaluation codes). The sectional genus of the surface general. Finding efficiently computable relaxations for this also plays a key role. We will present theoretical results problem often leads to approximating the cone of nonneg- bounding the minimum distance and experimental results ative polynomials by sums of squares (SOS) using semidef- giving such codes with minimum distance better than the inite programming (Lasserre relaxations). A well known AG17 Abstracts 109

issue of this approach is that the size of these semidefinite [email protected], [email protected] programs grows rapidly with the number of variables or degree of the polynomials. In this talk I will present some Akshay Gupte recent developments in polynomial optimization based on Department of Mathematical Sciences a new approximation of nonnegative polynomials, namely Clemson University via the cone of sums of nonnegative circuit polynomials [email protected] (SONC). These approximations often behave differently than SOS approximations. Moreover I will present a Pos- itivstellensatz for SONC polynomials, which yields a con- MS55 verging hierarchy of lower bounds for solving arbitrary con- Model Selection for Gaussian Mixtures with Nu- strained optimization problems on compact sets. The cor- merical Algebraic Geometry responding lower bounds can be computed efficiently via relative entropy programming using interior point meth- Gaussian Mixture Models (GMMs) are among the most ods. The talk is based on joint work with Sadik Iliman statistically maturemethods for clustering and density esti- and Timo de Wolff. mation with many applications in science and engineering. GMM parameters are typically estimated from training Mareike Dressler data using the iterative Expectation-Maximization (EM) Goethe University, Frankfurt am Main algorithm, which requires knowing the number of Gaussian [email protected] components a priori. In this study we propose an approach using numerical algebraic geometry to identify the optimal number of Gaussian components in a GMM. The proposed MS54 approach transforms a GGM into equivalent polynomial re- Complexity in Polynomial Optimization with Inte- gression splines and uses homotopy continuation methods ger Variables to find the model, or, equivalently, the number of compo- nents that is most compatible with the training data. As Lenstra showed, a linear objective function can be opti- mized over the integer points in a polyhedron in polynomial Adel Alaeddini time in any fixed dimension. Surprisingly, it is unknown if University of Texas, San Antonio a quadratic objective function can be minimized over the [email protected] integer points in a points in a polyhedron in polynomial time. We will present recent results on this question of the Elizabeth Gross complexity of polynomials integer optimization. San Jose State University [email protected] Robert Hildebrand IBM Watson, Yorktown Heights Sara Shirinkam [email protected] University of Texas, San Antonio [email protected] MS54 Deriving Convex Hull Forms of Special Symmetric MS55 Multilinear Polynomials Reduced Basis Homotopy Method for Computing Multiple Solutions of Nonlinear PDEs This talk derives the convex hull forms of various symmet- ric multilinear polynomials (SMPs) over box constraints. In this talk, I will present a new homotopy continuation The associated regions are shown to be polyhedral via method based on spectral method to compute multiple so- a reformulation-linearization-technique (RLT). The RLT lutions of nonlinear differential equations. Motivated by gives the convex hull forms, in an extended variable space, the reduced basis method for solving parametrized partial for all SMPs over any collection of box constraints. To ob- differential equations, we construct the spectral approxi- tain the convex hulls in the original variable space, a pro- mation space adaptively using a greedy algorithm. Nu- jection operation can therefore be used. Such a projection, merical homotopy continuation method based on this low- however, is combinatorial in nature because it requires the dimensional approximation space is computationally very identification of all the extreme directions of a suitably- efficient for finding multiple solutions of differential equa- defined cone. In order to circumvent this combinatorial tions. Various numerical examples are given to illustrate challenge, we generate and verify all facet-defining inequal- the efficiency of the new approach. ities (facets) directly in the original variable space using a two-step procedure. The first step exploits the problem Wenrui Hao structure for the purpose of devising necessary conditions Penn State for establishing a valid linear inequality as being a facet. State College, PA USA The second step again uses the problem structure, but this [email protected] time to motivate and verify families of facets, and to invoke the necessary conditions to conclude that all facets have been obtained. Notably, and unlike the higher-dimensional MS55 counterpart, the convex hull forms in the original variable Certification via Liaison Pruning space depend upon the chosen boxes. Different such forms are provided for various symmetric functions and for var- Certification via Smales α-theory requires a square system. ious sets of box constraints. As a byproduct, given any Schubert problems in type-A Schubert calculus have square facet, the set of all points which satisfy the facet at equal- formulations arising from duality or by lifting to a more ity is identified. refined flag manifold, but generalizing these techniques to geometries of other types may result in non-square sys- Yibo Xu, Warren Adams tems. We propose that one certify numerical output of Clemson University a non-square system indirectly: we use a liaison between 110 AG17 Abstracts

the non-square system and another system which is square. sizes of certain immigrant populations in the Netherlands When the union of the varieties corresponding to the linked (in 2007). systems is also given by a square system, one may use inclusion-exclusion with α-theory to verify numerical so- Ann Johnston lutions to the original system. Harvey Mudd College [email protected] Nickolas Hein University of Nebraska at Kearney Sonja Petrovic [email protected] Illinois Institute of Technology, Department of Applied Mathematics, Chicago IL 60616 [email protected] MS55 Computational Algebraic Geometry Methods in Aleksandra Slavkovic Machine Learning Penn State Univeristy [email protected] The talk will present novel interpretations of some of the machine learning algorithms in terms of algebraic geom- etry. Then, a review on recent results will be presented. MS56 Reconstructing Complex Cellular Mixtures from Dhagash Mehta Heterogeneous Tissue Samples University of Notre Dame Ever-improving technologies have revealed that even seem- [email protected] ingly homogeneous tissues can exhibit extensive genetic and genomic variability from cell-to-cell. This variability MS56 can be of great value in reconstructing developmental pro- cesses in healthy and diseased tissues. Our ability to mea- Valid Plane Trees and RNA sure this genetic variability is still limited, however, cre- ating a need for computational methods to assist genomic There is a well known correspondence between the sec- profiling in reconstructing the composition of complex cel- ondary structure of a folded strand of RNA with n nu- lular mixtures from partial measurements. This talk will cleotides and a (non-perfect, possibly crossing) matching of consider variations on the problem of reconstructing ge- the set [n]. By restricting this correspondence to perfect, nomic mixtures derived from differentiating cell popula- non-crossing (psuedoknot-free) matchings, we can look at tions. It will first examine geometric models of mixture secondary structures as plane trees. The plane trees that composition and approaches for applying them to resolv- are “valid” for a particular word in a given alphabet, e.g. ing complex mixture structure. It will further explore how A,C,G,U, are those that pair the ith and jth letters only this mixture deconvolution problem can be productively if they are complementary, like C and G. One valid plane combined with combinatorial models of phylogenetic tree tree can be transformed into another through “unzipping” inference to better leverage shared ancestry in reconstruct- and “rezipping” bonds in particular ways. The graph of ing more complex cell mixtures. Variants of these strate- these movements for any given word has many interesting gies will be demonstrated with application to reconstruct- properties which will be discussed in this talk. ing models of progression of cancer genomic samples and Elizabeth Drellich models of differentiation of cellular lineages in tissue devel- University of North Texas opment. [email protected] Russell Schwartz Biological Sciences and Computational Biology Julianna Tymoczko, Francis Black Carnegie Mellon University Smith College [email protected] [email protected], [email protected]

MS56 MS56 A Convex Realization of Neural Codes on Grids An Algebraic Approach to Categorical Data Fusion for Population Size Estimation The brain encodes spatial structure through a combina- torial code of neural activity. Experiments suggest such Information from two or more administrative registries codes correspond to convex areas of the subject’s environ- can be used to estimate a population’s size (via capture- ment. We present an intrinsic condition that implies a recapture methodology). In estimating total population neural code may correspond to a collection of convex sets size, it is necessary to estimate the number of individuals and give a bound on the minimal dimension underlying in the population that are missed by all registries. This such a realization. requires a collection of situation-driven log-linear model assumptions to be taken. It is common to assume either Robert L. Williams independent inclusion probabilities or a fixed odds ratio Texas A&M across the different registries. We provide new popula- [email protected] tion size estimation results (using algebraic tools) assuming only a given a odds ratio bound. Further, we extend previ- ously proposed algebraic approaches to handling capture- MS57 recapture population size estimation to the setting of two Connectedness of Tensor Ranks or more registries that include unshared covariates, where the problem becomes the more general problem of data To avoid non-smooth points in optimizing low-rank tensor fusion. As a running example, we look at estimating the approximation problems, it is often desirable to restrict AG17 Abstracts 111

the feasible set to tensors of a specific rank, as opposed method for solving join approximation problems. For some to tensors not more than a specific rank. This can poten- special join sets, we characterize the behavior of sequences tially create another issue as the set of tensors of a specific in the join set converging to the latter’s boundary points. rank may well be a disconnected set, and path-following Finally, the discussion is specialized to the tensor rank de- algorithms starting in one component would fail to find an composition and we provide several numerical experiments optimizer located in another. This is especially a problem confirming the main theoretical results. over the reals — for example, it is well-known that the set of real full-rank square matrices is not a connected set. Paul Breiding We will discuss path-connectedness of tensor rank, border TU Berlin rank, and multilinear rank. We show that the set of real [email protected] tensors of a fixed tensor rank or border rank r is always path connected if r is less than the complex generic rank Nick Vannieuwenhoven (subgeneric). The set of real d-tensors of multilinear rank Department of Computer Science (r1, ..., rd) has more complicated connectedness properties; KU Leuven we will show that in the subgeneric case it is connected if [email protected] r1 = ... = rd or if each ri is less than the product of all other rj ’s for i =1, ..., d, but not in general. This is joint work with Pierre Comon, Yang Qi, and Ke Ye. MS58 An Invariants-Based Method for Efficient Identi- Lek-Heng Lim,KeYe,YangQi fication of Hybrid Species under the Coalescent University of Chicago Model [email protected], [email protected], [email protected] Coalescent-based species tree inference has become widely used in the analysis of genome-scale multilocus and SNP Pierre Comon datasets when the goal is inference of a species-level phy- GIPSA-Lab CNRS logeny. However, numerous evolutionary processes are Grenoble, France known to violate the assumptions of a coalescence-only [email protected] model and complicate inference of the species tree. One such process is hybrid speciation, in which a species shares its ancestry with two distinct species. Although many MS57 methods have been proposed to detect hybrid speciation, Symmetric Tensor Nuclear Norms only a few have considered both hybridization and coales- cence in a unified framework, and these are generally lim- This talk discusses nuclear norms of symmetric tensors. As ited to the setting in which putative hybrid species must be recently shown by Friedland and Lim, the nuclear norm of identified in advance. This talk will provide an overview to a symmetric tensor can be achieved at a symmetric decom- a method that is based on a model that considers both coa- position. We discuss how to compute symmetric tensor nu- lescence and hybridization together, and uses phylogenetic clear norms, depending on the tensor order and the ground linear invariants to construct a hypothesis test for detecting field. Lasserre relaxations are proposed for the computa- and quantifying the extent of hybridization. This is joint tion. The theoretical properties of the relaxations are stud- work with Laura Kubatko, The Ohio State University. ied. For symmetric tensors, we can compute their nuclear norms, as well as the nuclear decompositions. The pro- Julia Chifman posed methods can be extended to nonsymmetric tensors. American University [email protected]

Jiawang Nie University of California, San Diego MS58 [email protected] Bounds on the Expected Size of the Maximum Agreement Subtree

MS57 We prove polynomial upper and lower bounds on the ex- The Condition Number of Join Decompositions pected size of the maximum agreement subtree of two ran- dom binary phylogenetic trees under both the uniform dis- Join decompositions generalize some ubiquitous decompo- tribution and Yule-Harding distribution. This positively sitions in multilinear algebra, namely tensor rank, Waring, answers a question posed in earlier work. Determining and block term decompositions. A join decomposition is a tight upper and lower bounds remains an open problem. fixed-length sum of points from several smooth submani- This is joint work with Daniel I. Bernstein, Lam Si Tung folds embedded in a mutual vector space. The set of points Ho, Colby Long, Mike Steel, and Katherine St. John. admitting a join decomposition is called a join set. Given a point in the ambient vector space, the join approxima- Seth Sullivant tion problem consists of finding the closest point in the join North Carolina State University set. We examine the geometric condition number of join [email protected] decompositions. It is characterized as a distance to a set of ill-posed points in a supplementary product of Grassmanni- ans. We prove that this condition number can be computed MS58 efficiently as the smallest singular value of an auxiliary ma- Finite Phylogenetic Complexity and Combinatorics trix. We show how the join approximation problem may of Tables be formulated as a Riemannian optimization problem, de- spite the fact that join sets are not habitually Riemannian In algebraic statistics, Jukes-Cantor and Kimura models manifolds. We show that the geometric condition number are of great importance. Sturmfels and Sullivant general- governs the convergence of the Riemannian GaussNewton ized these models associating to any finite abelian group 112 AG17 Abstracts

G a family of toric varieties X(G, K1,n). We investigate fying rank conditions on submatrices. I will discuss how to the generators of their ideals. We show that for any fi- use matrix Schubert varieties (and their analogs for sym- nite abelian group G there exists a constant φ, depending metric and upper triangular matrices) to study two prob- only on G, such that the ideals of X(G, K1,n) are gener- lems from algebraic statistics: first, I’ll explain how spe- ated in degree at most φ. This is joint work with Mateusz cial conditional independence models for Gaussian random Michalek. variables are intersections of symmetric matrix Schubert varieties, and use this obtain a combinatorial primary de- Emanuele Ventura composition algorithm for some conditional independence Aalto University ideals. Then, I’ll characterize the vanishing ideals of Gaus- [email protected] sian graphical models for generalized Markov chains. This is joint work with Alex Fink and Seth Sullivant.

MS58 Alex Fink Stochastic Safety Radius on NJ and BME Methods Queen Mary University of London for Small Trees UK a.fi[email protected] A distance-based method to reconstruct a phylogenetic tree × with n leaves takes a distance matrix, n n symmetric ma- Jenna Rajchgot trix with 0s in the diagonal as its input and it reconstructs Department of Mathematics and Statistics atreewithn leaves using tools in combinatorics. A safety University of Saskatchewan radius is a radius from a tree metric, a distance matrix re- [email protected] alizing a true tree where the input distance matrices all lie within, in order to satisfy a precise combinatorial condition under which the distance-based method is guaranteed to Seth Sullivant return a correct tree. A stochastic safety radius is a safety North Carolina State University radius under which the distance-based method is guaran- [email protected] teed to return a correct tree within a certain probability. In this talk we will investigate stochastic safety radii for MS59 the neighbor-joining (NJ) method and balanced minimal evolution (BME) method for n =5. Totally Positive Graphical Models

Jing Xi We study maximum likelihood estimation for Gaussian North Carolina State University models that are multivariate totally positive of order two [email protected] (MTP2), i.e. where the covariance matrix is an inverse M-matrix. Using conic duality theory we show that the Jin Xie maximum likelihood estimator (MLE) for MTP2 Gaussian University of Kentucky models exists already for 2 observations, which is in high [email protected] contrast with the existence of the MLE in Gaussian graph- ical models without the MTP2 constraint. In addition, we discuss how the MTP2 constraint induces sparsity in the Ruriko Yoshida underlying graphical model, and we determine a lower and Naval Postgraduate School upper bound on the graphical model. We end by describing [email protected] an application of this theory to factor analysis.

Stefan Forcey Caroline Uhler University of Akron Massachusetts Institute of Technology [email protected] [email protected]

Piotr Zwiernik MS59 Mittag-Leffler Institute Computing the ML Degree of Hierarchical Log- Stockholm, Sweden linear Models [email protected] The maximum likelihood degree is the number of complex critical points of the likelihood function on a projective va- Steffen Lauritzen riety. A wide class of such varieties is provided by hierar- University of Copenhagen chical log-linear models and graphical models, a subclass of [email protected] toric varieties. We will show how to compute the maximum likelihood degree of these models and exhibit examples. MS59 Serkan Hosten Bayesian Networks and Generalized Permutohedra Department of Mathematics San Francisco State University A graphical model encodes conditional independence rela- [email protected] tions via the Markov properties. For an undirected graph these conditional independence relations are represented by a simple polytope known as the graph associahedron, MS59 which can be constructed as a Minkowski sum of standard Matrix Schubert Varieties and Gaussian Condi- simplices. There is an analogous polytope for conditional tional Independence Models independence relations coming from any regular Gaussian model, and it can be defined using relative entropy. For Matrix Schubert varieties are certain subvarieties of the directed acyclic graphical models we give a construction affine space of square matrices which are defined by speci- of this polytope as a Minkowski sum of matroid poly- AG17 Abstracts 113

topes. The motivation came from the problem of learning joint with Holly Mandel and Claudia Yun. Bayesian networks from observational data. Julianna Tymoczko Fatemeh Mohammadi Smith College Technical University of Berlin [email protected] [email protected]

Caroline Uhler MS61 Massachusetts Institute of Technology Graphical Equilibria for Deterministic and [email protected] Stochastic Reaction Networks Chemical reaction networks can be modelled either deter- Charles Wang ministically, by means of a system of ordinary differential Georgia Tech equations, or stochastically, by means of a continuous time [email protected] Markov chain. In both cases, a graph can be associated to the model. In the deterministic setting, the relationship Josephine Yu between dynamical features of the model and structural Georgia Institute of Technology properties of the graph have been intensively studied since [email protected] the Seventies, with a special focus on complex balanced equilibria. We explore further the concept of complex bal- anced equilibria, by considering equilibria of subgraphs of MS60 the network. Moreover, we explore the connection between Smooth Splines on Surfaces with General Topology graphical equilibria in the deterministic and in the stochas- tic models, with a focus on the stationary distributions of We analyze the space of piecewise polynomial functions the Markov chain in the stochastic setting. globally differentiable on a mesh of general topology. This linear space of spline functions is characterized by glueing Daniele Cappelletti data across the shared edges. Using algebraic techniques, University of Copenhagen which involve the analysis of the module of syzygies of the [email protected] glueing data, we give a dimension formula for the space of geometrically smooth splines of degree k, on surfaces of arbitrary topology when k is big enough. We provide ex- MS61 plicit constructions of basis functions attached respectively Regions of Multistationarity in Chemical Reaction to vertices, edges and faces. Applications to fitting and re- Networks construction problems illustrate the approach. This is a joint work with A. Blidia and N. Villamizar. Given a real sparse polynomial system, we present a general framework to find explicit coefficients for which the system Bernard Mourrain has more than one positive solution, based on a recent arti- INRIA Sophia Antipolis cle by Bihan and Spaenlehauer. We apply this approach to [email protected] find explicit reaction rate constants and total conservation constants in biochemical reaction networks for which the associated dynamical system is multistationary. MS60 Spaces of Splines, Vector Bundles, and Reflexive Alicia Dickenstein Sheaves Universidad de Buenos Aires [email protected] In this talk we discuss a number of connections between certain local and global problems in approximation the- Fr´ed´eric Bihan ory, related to spaces of splines of various degrees and or- Universit´e Savoie Mont Blanc ders of smoothness, and certain vector bundles/reflexive France sheaves on complex projective spaces. We also give some [email protected] results that link theorems on the vanishing of the higher cohomology of vector bundles to formulas for the dimen- Magal´ı Giaroli sion of spline spaces and discuss a possible connection with Universidad de Buenos Aires semi-stability as it applies to vector bundles and reflexive [email protected] sheaves on projective spaces.

Peter F. Stiller MS61 Texas A&M University Towards Quasi-stationary Distributions for a Class [email protected] of Reaction Networks

The long-term behavior of a given reaction network may MS60 depend crucially on whether it is modeled deterministically Splines on Lattices and Equivariant Cohomology of or stochastically. In particular, the possibility of extinc- Certain Affine Springer Fibers tion, which is a widely occurring phenomenon in nature, is only captured by the latter. Indeed, the deterministic We describe an algebraic generalization of splines and give model solution is an approximation of the solution for the a combinatorial basis for the splines on infinite graphs aris- stochastic model only on finite time intervals. A conse- ing from various lattices. As an application, we show how quence of this is that the counterpart to a stable station- to construct the equivariant cohomology of an infinite ge- ary solution in the deterministically modeled system is not ometric object called an affine Springer fiber. This work is a stationary distribution of the stochastic model, but in- 114 AG17 Abstracts

stead a so-called quasi-stationary distribution, which is the Technische Universit¨at Berlin stationary measure when we condition on the process not [email protected] going extinct. Quasi-stationary distributions are notori- ously hard to calculate explicitly. Here, we look at a class of reaction networks exhibiting absolute concentration ro- MS62 bustness and possessing a conservation law which allows us Some Relatives of Matroid Polytopes to take the limit under which the stationary distribution of a reduced network is precisely the quasi-stationary dis- The goal of the talk is to introduce certain classes of poly- tribution of the desired species in the original full network. topes that behave pretty similarly to matroid basis poly- topes and serve as a new technique to understand geomet- Mads C. Hansen rically what happens if we construct a matroid polytope by University of Copenhagen adding one vertex at a time in a suitably organized way. [email protected] Jose Samper University of Miami MS61 [email protected] Identifying Parameter Regions for Multistationar- ity MS62 Biological systems have complex dynamical behavior. The Non-acyclicity of Coset Lattices and Generation of behavior might depend on specific parameter values or Finite Groups might be sensitive to small variations in parameter values. Bistability, and generally multistationarity, is a property In a 2016 paper, John Shareshian and I showed that the that often is associated with cellular decision-making, sig- coset lattice of any finite group has non-trivial homology in nal propagation in cellular cascades and on/off responses characteristic 2, hence is not contractible. The general idea in cells; however it is a property that is surprisingly dif- is to apply Smith Theory to each chief factor of a group, ficult to demonstrate by mathematically means. We in- but the proof requires the classification, and has several troduce a method to partition the parameter space of a difficult details. I’ll discuss recent improvements and sim- parameterized system of ODEs into regions for which the plifications of Bob Guralnick, Shareshian, and myself on system has a unique or multiple equilibria. The method is coset lattices and related group-generation results. based on the computation of the Brouwer degree of a par- ticular function, and it creates a multivariate polynomial Russ Woodroofe with parameter depending coefficients. Using algebraic Mississippi State University techniques, the signs of the coefficients reveal parameter [email protected] regions with and without multistationarity. A particular strength of the method is the avoidance of numerical anal- ysis and parameter sampling. We demonstrate the method MS63 on biological models, and show that we often obtain a com- Gromov-Hausdorff Limit of Wasserstein Spaces on plete partitioning of the parameter space with respect to Point Clouds multistationarity. We obtain conditions, which often are interpretable in biochemical terms involving for example With the objective of studying the consistency of a large Michaelis-Menten constants or catalytic activity constants. variety of machine learning procedures of evolution type, Therefore, the method not only decides on a mathematical we study discrete Wasserstein spaces on point clouds and in question but also offers a meaningful biochemical expla- particular their limit in the Gromov-Hausdorff sense. More nation of why multistationarity arises in a given system. precisely, we consider a point cloud Xn := {x1, ..., xn} uni- formly distributed on a compact manifold M , and con- struct a geometric graph on the cloud by connecting points Carsten Wiuf, Elisenda Feliu that are within distance  of each other. We let P (Xn)be University of Copenhagen the space of probability measures on Xn and endow it with [email protected], [email protected] a discrete Wasserstein distance Wn as introduced by Maas for general finite Markov chains. We show that as long Carsten Conradi as  decays towards zero slower than an explicit rate de- HTW Berlin pending on the level of uniformity of Xn, then the space [email protected] (P (Xn),Wn) converges in the Gromov-Hausdorff sense to- wards the space of probability measures on M endowed Maya Mincheva with the Wasserstein distance. The motivation and ap- Northern Illinois University plications of this result will be discussed during the talk. [email protected] Nicolas Garcia Trillos MS62 Carnegie Mellon University Chromatic Numbers of Simplicial Manifolds nicolas garcia [email protected]

Higher chromatic numbers of simplicial complexes natu- rally generalize the chromatic number of a graph. In this MS63 talk, we focus on explicit examples of triangulated sur- Sheaves and Numerical Analysis faces and triangulated 3-manifolds with a non-trivial 2- chromatic number of at least 5, based on symmetric Steiner Sheaves are mathematical objects that combine bits of lo- triple systems and iterated moment curve constructions. cal information into a consistent whole. Sheaf theory has been entwined with the study of differential equations since Frank H. Lutz it was noticed by Ehrenpreis that differential equations AG17 Abstracts 115

give rise to sheaves of solutions. These sheaves admit var- ular data, including solvation free energies, partition coef- ious analytical techniques, and consistency relationships ficients, protein-drug binding affinities, and protein muta- between discretization and non-discretized version of the tion impacts. equation are exposed, which means that different approx- imation methods can be rated based on their impact on Guowei Wei their solutions. Shadows of the sheaf theoretic perspec- Department of Mathematics tive are apparent in the construction of volume meshers Michigan State University (which construct pullbacks and pushforwards of sheaves [email protected] of functions), and finite element solvers (which construct the space of global sections of a sheaf). Unlike traditional methods for analyzing numerical approximations, the con- MS64 sistency relationships are local to each individual variable, Locally Decodable Codes and Practical Applica- so approximation methods of different accuracy can be eas- tions ily mixed across the model. Originating from theoretical computer science, the notion Michael Robinson of locally decodable codes was formalized by Katz and Tre- Department of Mathematics and Statistics visan in 2000. For a locally decodable code, recovering one American University symbol of the original message is done by looking at very [email protected] few codeword symbols. Katz and Trevisan showed that the notion of locally decodable codes is equivalent to multi- server information theoretically secure private information MS63 (PIR) schemes. A PIR scheme enables a user to cryp- Topological Rhythm Hierarchy Quantification in tographically query a remote database, while the query Musical Audio remains unknown to the server holding the database. In the beginning of the 2010’s, a lot of progress have been Music is full of repetition at many scales, including the done, with the construction of multiplicity codes (Kop- rhythm, or the ”pulse” of the music. Automatically seg- party, Saraf and Yekhnainin), and new affine-invariant menting musical audio into beat intervals is an important codes, lifted codes, (Guo, Koppary and Sudan). Both con- preprocessing step for many algorithms in music informa- structions provides codes with rate close to 1, and sub- tion retrieval, including music section segmentation and linear, very small, locality. In this talk, we present mul- cover song identification. Most algorithms for automati- tiplicity codes, and discuss parameters for practical, non cally finding ”beat onsets,” or rhythm interval delimiters, asymptotic values. Using the underlying geometry, these rely on Fourier methods and autocorrelation, and they out- codes enable PIR schemes, with small number of servers put beats at what is deemed to be a single, dominant tempo and small storage overhead. We also report on our imple- level. However, in most genres, rhythmic indicators oc- mentation with timings. Also, we briefly present how these cur in hierarchies (sometimes referred to as ”microbeats” locally decodable codes may be used for Proofs of Retriev- and ”macrobeats”) in complex patterns, often with missed ability, which allows users to check that uploaded data is beats and syncopation. This commonly leads to doubling available on a remote server, while storing a small fraction and halving of the estimated beat onsets in 4/4 music, and of data. Joint work with Nicholas Coxon, Julien Lavauzelle similar problems in odd meters. In this work, we argue for and Francoise Levy-dit-Vehel. a multiscale understanding of rhythm, and an alternative, Daniel Augot geometric approach to recurrence analysis that is better INRIA suited to rhythm hierarchies under this complexity. Using [email protected] tools from topological data analysis and spectral graph the- ory, we uncover and quantify rhythm hierarchies using slid- ing window embeddings of audio novelty functions derived MS64 from audio spectrograms. We use circular coordinates to AG Codes as Products of Reed-Solomon Codes and indicate beat phase at different tempo scales, and changes Applications in orientability as an indicator for the relative strength of the microbeats/macrobeats in different sections. In this talk, we consider families of algebraic geometry (AG) codes which can be expressed as products of Reed- Chris Tralie Solomon codes. This representation allows for applications Duke University to decoding and distributed storage. [email protected] Gretchen L. Matthews Clemson University MS63 [email protected] Topological Deep Learning of Biomolecular Data

The exponential growth of biological data has offered a rev- MS64 olutionary opportunity for mathematically driven advances Distributed Coding for Evolving Content in Evolv- in biological sciences. Conventional geometric analysis is ing Networks frequently inundated with too much structural detail to be computationally tractable for massive biomolecular data, We consider the use of coding where error correction is while traditional topological tools often incur too much re- used to convey updates and erasure correction used to re- duction of the original data to be practically useful. Persis- spond to changes in topologies. In such settings, coding tent homology, a new branch of algebraic topology, is able is not used to handle exceptions but, rather, as a means to bridge the gap between geometry and topology. I will of continuously curating content that undergoes updates discuss how to combine persistent topology with cutting over changing networks. We consider distributed systems edge machine learning and deep learning to arrive at the in which nodes generally do not maintain files but rather state of the art predictions of a vast variety of biomolec- just coded portions of them. Updating such coded versions 116 AG17 Abstracts

without recourse to creating new coded portions from an talk is based on joint work with Greg Blekherman, Daniel updated version of the updated file poses interesting new Plaumann, and Cynthia Vinzant and Christoph Hanselka. problems, for which we show some new approaches. Rainer Sinn,GregBlekherman Muriel Medard, N Prakash, Vitaly Abdrashitov Georgia Institute of Technology MIT [email protected], [email protected] [email protected], [email protected], [email protected] Christoph Hanselka University of Auckland MS65 [email protected] On Representing the Positive Semidefinite Cone using the Second-order Cone Daniel Plaumann We show that it is not possible to represent the 3 × 3pos- TU Dortmund itive semidefinite cone using any finite number of second- [email protected] order cones. This answers a question of Lewis and Glineur, Parrilo, Saunderson. In fact we show that the slice consist- Cynthia Vinzant ing of Hankel matrices does not admit a second-order cone North Carolina State University representation. Our proof relies on exhibiting a sequence Dept. of Mathematics of submatrices of the slack matrix of the 3 × 3 positive [email protected] semidefinite cone whose second-order cone rank grows to infinity. MS65 Hamza Fawzi Do Sums-of-squares Dream of Free Resolutions? University of Cambridge [email protected] Let X ⊆ P n be a reduced scheme over the reals defined by an ideal I. We show that the number of steps for which the minimal free resolution of I is linear is a lower bound for the MS65 next-to-minimal rank of extreme rays of the cone dual to Bad Semidefinite Programs: They All Look the the sums of squares in X. As a consequence, we obtain: (1) Same A complete classification of totally real reduced schemes for which nonnegative quadratic forms are sums of squares. (2) Semidefinite programs (SDPs) often behave pathologically: New certificates of exactness for semidefinite relaxations of the optimal values of the primal and dual programs may polynomial optimization problems on projective varieties. differ, and may not be attained. This research was moti- vated by the curious similarity of the pathological SDPs in Greg Blekherman, Rainer Sinn the literature: in our main result we characterize patho- Georgia Institute of Technology logical semidefinite systems by certain excluded matrices, [email protected], [email protected] which are easy to spot in all published instances. We show how to transform semidefinite systems into a canon- Mauricio Velasco ical form, so their pathological nature becomes trivial to Universidad de los Andes verify. The transformation is surprisingly simple, as it re- [email protected] lies mostly on elementary row operations inherited from Gaussian elimination. As a byproduct, we prove an in- teresting corollary in convex geometry: we show how to MS66 transform linear maps acting on symmetric matrices into a Numerical Algebraic Geometry for Geolocation of canonical form, so it is trivial to verify whether the image RF Emitters of the semidefinite cone under the map is closed. Being able to accurately locate RF emitters is vital to lo- Gabor Pataki cating satellites or objects on the ground. This problem is University of North Carolina, at Chapel Hill challenging because it is passive in nature- we often do not [email protected] know when signals are initially sent from an emitter. But, with more than one receiver, we can calculate the differ- ence in time or frequency that signals are received by the MS65 receivers. The equations that result from this approach Hermitian Factorizations of Univariate Matrix to geolocation are polynomial in nature. In this talk I will Polynomials discuss some of my work with this problem using numerical algebraic geometry. We consider univariate real symmetric matrix polynomi- als A, i.e. a symmetric n × n matrix whose entries are Karleigh Cameron polynomials in one variable t with real coefficients. Such a Colorado State University polynomial A is positive semidefinite if the real symmetric Fort Collins, CO 80523 matrix A(t) is positive semidefinite for every real number [email protected] t. It is known that a positive semidefinite univariate ma- trix polynomial is a sum of squares, i.e. it can be written as A(t)=BtB,whereB is a symmetric d × n real uni- MS66 variate matrix polynomial. We give an optimal bound on Topological Data Analysis for Real Algebraic Vari- the number of squares d = n + 1 by relating the problem eties to a sums of squares problem on a toric variety, namely a rational normal scroll. Using the theory of quadratic forms We study real algebraic varieties using topological data over rings, we can count the number of representations of analysis. Topological data analysis (TDA) provides a grow- smallest length for a generic matrix polynomial A.This ing body of tools for computing geometric and topolog- AG17 Abstracts 117

ical information about spaces from a finite sampling of University of Notre Dame points. We present a new adaptive algorithm for finding [email protected] provably dense samples of points on real algebraic varieties given a set of defining polynomials. The algorithm utilizes Charles Wampler methods from numerical algebraic geometry to give formal General Motors Research Laboratories, Enterprise guarantees about the density of the sampling and it also Systems Lab employs geometric heuristics to minimize the sampling. 30500 Mound Road, Warren, MI 48090-9055, USA Since TDA methods consume significant computational re- [email protected] sources that scale poorly in the number of sample points, our sample minimization makes applying TDA methods more feasible. We demonstrate our algorithm with exam- MS67 ples and present our findings. Computing GIT-fans with Symmetry Parker Edwards University of Florida The GIT-fan is a combinatorial structure which describes Gainesville, FL USA all reasonable quotients obtainable from the action of an pedwards@ufl.edu algebraic group on an algebraic variety. In the case of an algebraic torus acting on an affine variety, based on work Emilie Dufresne of Berchtold and Hausen, Keicher has developed an algo- University of Oxford rithm for computing the GIT-fan. The algorithm relies on [email protected] Groebner basis and polyhedral computations. Due to the complexity, in important examples from algebraic geom- Heather Harrington etry a direct application of the algorithm is not feasible. Mathematical Institute In this talk, I will describe how to make these examples University of Oxford accessible by the use of efficient algorithms for computing [email protected] saturations and by taking into account actions of symmetry groups. Jonathan Hauenstein University of Notre Dame Janko Boehm Dept. of App. Comp. Math. & Stats. TU Kaiserslautern [email protected] [email protected]

MS66 MS67 Computing the Canonical Decomposition of Unbal- anced Tensors by Homotopy Method The Computation of Discriminants in Mass-action Networks The canonical decomposition of the tensor whose maximal dimension is greater than its rank is considered. We de- The steady states of a mass-action network are the non- rive the upper bound of rank under which computing the negative real solutions to a parametric system of poly- canonical decomposition is equivalent to solving a struc- nomial equations. It is a central problem to classify the tured polynomial system that is determined by the full number of isolated real solutions of such a system in cer- rank factorization of the matricization of the tensor. Un- tain linear subspaces. One approach to this problem is the der the generic uniqueness conditions, the CPD solutions computation of discriminants. We exemplify this method of the system are isolated so that these solutions can be on the dual-site phosphorylation network. achieved by homotopy method. Tsung-Lin Lee Alexandru Iosif National Sun Yat-set University Otto-von-Guericke Universit¨at Magdeburg [email protected] [email protected]

MS66 MS67 Exceptional Stewart-Gough Platforms and Segre When is a Polynomial Ideal Binomial After Chang- Embeddings ing Coordinates? Stewart-Gough platforms are robots made up of a base and platform connected by six legs of fixed length. A It is an important problem in computational algebraic ge- generic Stewart-Gough platform is rigid whereas excep- ometry to find good representations of algebraic varieties. tional Stewart-Gough platforms have self-motion. This Ideals generated by binomials define schemes with much talk will focus on a family of exceptional Stewart-Gough simpler geometry and combinatorics than arbitrary poly- platforms, called Segre-dependent Stewart-Gough plat- nomial ideals. The property of being generated by binomi- forms, which arise from a linear dependence among point als is obviously not invariant under changes of coordinates, pairs under the Segre embedding. so one might ask whether it is possible for a given ideal I to find new coordinates in which I is binomial. Similarly, one Samantha Sherman might ask for new coordinates in which I is homogeneous Notre Dame with respect to some (multi-) grading. In this talk I will Notre Dame, IN USA present algorithms for these and related tasks. [email protected] Lukas Katth¨an Jon Hauenstein Goethe Universit¨at Frankfurt 118 AG17 Abstracts

[email protected] problem is based on a generalization of trace norm regular- ization, which has been used extensively for learning low rank matrices, to the tensor setting. We highlight some MS67 limitations of this approach and discuss alternative convex Computing Chow Forms, Hurwitz Forms, and Be- relaxations. We present a general algorithm for solving the yond associated regularization problems and address statistical and computational issues behind these methods. Finally The Chow form of a projective variety is a single poly- we present two extensions of the above methodology within nomial defining this variety, although its vanishing ideal the setting of kernel methods and multitask learning. is generated by more than one equation. We obtain it as follows: For a k-dimensional variety in projective n-space, Massimiliano Pontil the set of (n − k − 1)-dimensional planes that intersect the University College London variety is a hypersurface in the Grassmannians of these [email protected] planes. The Chow form is the polynomial in Plcker coor- dinates defining this hypersurface. In this talk we discuss how to compute Chow forms and how to recover the van- MS69 ishing ideal of the underlying projective variety, as well An Algebraic Framework for Determining as generalizations of Chow forms and their computational Weighted Rearrangement Distance aspects. Variations in genome arrangements are an important Kathl´en Kohn source of phylogenetic information and have been used Technische Universit¨at Berlin to inform phylogenetic studies since the 1930s. The re- Germany arrangement distance between a pair of genomes is usu- [email protected] ally defined as the minimal number of events from a set of allowed operations required to transform one genome into another. An implicit assumption underlying most of MS68 the current methods is that all rearrangement operators Tensor Decomposition for Learning Latent Vari- included in the model are equally probable. Empirical bio- able Models: Algorithms and Challenges logical evidence, however, suggests that this is not the case i.e., some rearrangement events occur more frequently than Tensor decomposition has become an important tool in others. In this talk, I will present a flexible group-theoretic learning latent variable models. In this talk we will briefly framework that can de adapted to determine a minimal talk about how tensor decomposition can be applied to weighted distance between a pairs of genomes. This work learn some simple models. We will also discuss how new makes use of the well-developed theory of rewriting sys- tensor algorithms that are more noise tolerant and can han- tems and to the best of our knowledge, is the first known dle higher rank could benefit the learning applications. use of rewriting systems to a problem in comparative phy- logenetics. Rong Ge Duke University Sangeeta Bhatia [email protected] University of Western Sydney Deans Unit School of Computing, Engineering, and Mathematics MS68 [email protected] Adaptive Sparsity in Machine Learning and Auton- omy MS69 We discuss a framework based on Stone spaces to under- stand connections between artificial neural networks (sub- An Overview of Novel Applications of Algebraic symbolic) and reasoning along a lattice (symbolic). We and Topological Biology then discuss the competing constraints on information pro- cessing present in natural systems of intelligence, which In this talk we will give an overview of this mini-symposium we call ”adaptive sparsity,” and, time permitting, consider and how techniques in algebraic objects such as multi- some design principles of an adaptively sparse autonomous graded modules, ideals, and algebraic varieties as well as agent. topological data analysis techniques and their associated statistics can be used to organize, quantify, and create Sara Jamshidi novel understanding of biological data. Applications stud- Pennsylvania State University ied in this mini-symposium include genomics for phylo- [email protected] genetics and cancer, neuroscience, the structure of RNA, DNA, and chromosomes, blood clotting, and morphological objects such as fly wings. We will explain how the math- MS68 ematical techniques for these topics naturally complement On Learning Low Rank Tensors each other and identify promising directions in this rapidly developing area of mathematical biology. During the past few years, there has been a growing in- terest on the problem of learning a tensor from a set of Ruth E. Davidson linear measurements. This methodology has been applied University of Illinois Urbana Champaign to various fields, ranging from collaborative filtering, to [email protected] computer vision, to medical imaging, among others. We review research efforts in this area, with particular focus on machine learning methods based on regularized empiri- MS69 cal error minimization. A prominent methodology for this Inferring Rooted Species Trees from Unrooted AG17 Abstracts 119

Gene Trees novices but flexible enough to accomodate advanced usage by Macaulay2 experts. We will also preview some exciting Methods for inferring species trees (parameters) from gene new features currently under development. trees (data) have traditionally either used rooted gene trees to infer the rooted species tree or unrooted gene trees to Christopher Oneill infer the unrooted species tree, where the root had to be University of California Davis determined by other means, such as an outgroup. In 2011, [email protected] invariants were used to show that distributions of topolo- gies of unrooted gene trees identify the rooted species tree David Kahle when there are five or more taxa. Here we use maximum Baylor University likelihood and Approximate Bayesian Computation (ABC) david [email protected] to infer the rooted species tree topology from unrooted gene tree topologies. Jeff Sommars James H. Degnan,AyedAlanzi University of Illinois at Chicago University of New Mexico Chicago, Illinois USA [email protected], [email protected] [email protected]

MS69 MS70 Dimensional Reduction for Phylogenetic Tree SEMID: An R Package for Parameter Identifiabil- Models ity in Linear Structural Equation Models

I will present a general method of dimensional reduction Linear structural equation models relate a collection of ran- for phylogenetic tree models. The method reduces the di- dom variables using linear interdependencies and additive mension of the model space on a phylogenetic tree from noise. Each such model can be naturally associated with a exponential in the number of extant taxa, to quadratic in mixed graph whose nodes correspond to the random vari- the number of taxa. A key feature is the identification of ables. In the graph, directed edges imply a direct linear an invariant subspace which depends only bilinearly on the causal effect of one random variable on the other, while model parameters; in contrast to the usual multi-linear de- bidirected edges signify the existence of unobserved con- pendence in the full model space. The talk will concentrate founding. A key question of interest in these models is on the algebraic foundations of the dimensional reduction, that of generic identifiability, that is, whether or not ran- particularly discussing the identification of a novel repre- domly chosen edge weights can be recovered from the joint sentation of the “Markov’ group embedded within the usual covariance matrix of the random variables with probability array of tensor products. Practical applications, including 1. While a number of sufficient combinatorial conditions the computation of split (edge) weights on phylogenetic for generic identifiability have been discovered, little soft- trees from observed sequence data, will also be discussed. ware has been released publicly to make these tools easily applicable. To this end, we have developed the R package Jeremy Sumner SEMID which provides a simple API to a number of the University of Tasmania state-of-the-art generic identifiability algorithms. More- School of Physical Sciences over, our package allows for said algorithms to be used [email protected] together to enhance their applicability and eases general experimentation with mixed graphs.

MS70 Luca Weihs Algebraic Statistics in R A State of the Union University of Washington [email protected] The aim of the Software and Computation in Algebraic Statistics sessions is to provide a forum to discuss recent advances in applied algebraic statistics, with a particular MS70 emphasis on algorithms and implementations. To that end, Semigroups – A Computational Approach in this talk I provide an introduction to and overview of applied algebraic statistics in R, the packages for algebraic The question whether there exists an integral solution statistical computations that are available in the R ecosys- to the system of linear equations with non-negativity m×n tem, and the external software that support them. I will constraints, Ax = b, x ≥ 0, where A ∈ Z and m also highlight directions where improvements are needed b ∈ Z , finds its applications in many areas such as and plans are underway. This talk assumes little back- operations research, number theory, combinatorics, and ground knowledge of algebraic statistics. statistics. In order to solve this problem, we have to understand the semigroup generated by the columns of David Kahle the matrix A and the structure of the “holes” which Baylor University are the difference between the semigroup and its satu- david [email protected] ration. In this talk, we discuss the implementation of an algorithm by Hemmecke, Takemura, and Yoshida that computes the set of holes of a semigroup and we dis- MS70 cuss applications to problems in combinatorics. More- Using Macaulay2 from within R: The M2R Package over, we compute the set of holes for the common diag- onal effect model and we show that the nth linear order- The R package m2r provides a persistent interface to the ing polytope has the integer-decomposition property for algebra system Macaulay2, making it an ideal utility for n ≤ 7. The software is available at http://ehrhart.math.fu- algebraic statistics computations. In this talk, we will ex- berlin.de/People/fkohl/HASE/. plore a variety of m2r functionality, and discuss several design decisions that make m2r easy to use for Macaulay2 Ruriko Yoshida 120 AG17 Abstracts

Statistics ture of algebraic polynomials cannot be used in this general University of Kentucky setting. The results strengthens the structural similarity [email protected] between algebraic polynomial and general Tchebycheffian spline spaces. Florian Kohl Freie Universit¨at Berlin Cesare Bracco [email protected] Department of Mathematics - University of Florence cesare.bracco@unifi.it Yanxi Li University of Kentucky Tom Lyche [email protected] University of Oslo Department of Mathematics [email protected] Johannes Rauh Max Planck Institute for Mathematics in the Sciences [email protected] Carla Manni Department of Mathematics University of Rome MS71 [email protected] Basis and Dimension of Trivariate Geometrically Continuous Isogeometric Functions on Two-Patch Fabio Roman Domains University of Turin [email protected] The notion of geometric continuity, which has originated in geometric design, experiences intense interest due to Hendrik Speleers its applications in isogeometric analysis,especially when University of Rome ”Tor Vergata” constructing smooth discretizations on multi-patch geome- [email protected] tries. In this context it is important to study the space of geometrically continuous functions on such domains. More precisely, it is of interest to investigate the dimen- MS71 sion and to construct local bases for this space. Results Algebraic Geometry and Interpolation Problems for the bivariate case were published recently [Kapl, Vit- rih, Juettler, Birner, Isogeometric Analysis with Geomet- I will discuss several problems in multivariate interpolation rically Continuous Functions on Two-Patch Geometries, that can be approached using tools of algebraic geometry. CAMWA 2015] and it was possible to show that locally Here is one of them: Given an integer k what is the mini- supported basis functions and spaces with good approxi- mal number of k-dimensional subspaces of polynomials so 2 mation properties can be constructed for bilinear geome- that for any k distinct points in C the interpolation prob- try mappings as well as for more general analysis-suitable lem is well-posed in one of these subspaces. We will show G1 multi-patch parametrizations [Collin, Sangalli, Takacs, the connection between this problem and the geometry of Analysis-suitable G1 multi-patch parametrizations for C1 subspace arrangements. isogeometric spaces, CAGD 2016]. In this talk we extend these results to the trivariate case. In particular, we focus Boris Shekhtman on two hexahedral volumetric domains given as general B- Department of Mathematics and Statistics Spline maps with an analysis-suitable parametrization of University of South Florida the interface. [email protected]

Katharina Birner Johannes Kepler University MS71 [email protected] Interpolating with Hyperplane Arrangements via Generalized Star Configurations Varieties Bert Juettler Generalized star configuration varieties are projective vari- Johannes Kepler University of Linz eties that are the common zero locus of all the fold products Institute of Applied Geometry of the same size of a given set of linear forms. It turns out [email protected] that every subspace arrangement is a star configuration va- riety, and this helps view the subspace arrangement as the Angelos Mantzaflaris singular locus of a certain “multiplicity’ of some hyperplane RICAM arrangement (hence the use of the term ”interpolation”). Austrian Academy of Sciences One application of this point of view is that one can ob- angelos.mantzafl[email protected] tain an upper bound on the number of equations needed to define the subspace arrangement.

MS71 Stefan Tohaneanu Dimension of Tchebycheffian Spline Spaces on T- University of Idaho meshes [email protected]

The dimension of polynomial spline spaces on a prescribed T-mesh for a given componentwise degree and smoothness MS72 has been addressed by several authors using different tech- Deficiency-based Approaches to Steady State niques. In this talk we consider the dimension problem Parametrizations: New and Old for general Tchebycheffian splines using a homological ap- proach. The extension is nontrivial because the ring struc- The now-classical Deficiency Zero Theorem implies that, AG17 Abstracts 121

for networks which are weakly reversible (WR) and have a Copenhage zero deficiency (ZD), the steady state variety of the cor- [email protected] responding mass action system has a simple monomial parametrization. Recent work has furthermore shown that Elisenda Feliu, Carsten Wiuf monomial parametrizations can be obtained for networks University of Copenhagen which are not WR/ZD but may be corresponded, through [email protected], [email protected] a process known as network translations, to WR/ZD gen- eralized networks. In this talk, we go one step further to consider networks which are weakly reversible but have a MS72 more complicated deficiencies. Preliminary work suggests Polynomial Dynamical Systems Derived from Eu- a rich pathway to rational parametrizations for many such clidean Embedded Graphs networks. Some of the most common mathematical models of reac- Matthew D. Johnston tion networks are based on mass-action kinetics and give San Jose State University n rise to polynomial dynamical systems on R 0.Anysuch [email protected] > dynamical system can be derived from an Euclidean em- bedded graph, which is a directed graph whose vertices are Rn MS72 in . A toric dynamical system is an example of such a system, and is known to have a unique stable equilibrium Intermediates, Enzymes, Binomiality and Multi- within each affine invariant subspace. M¨uller and Regens- stationarity burger have studied a generalization of mass-action kinet- ics, where essentially the polynomial dynamical system is In this work we consider reaction networks with intermedi- given by a pair of Euclidean embedded graphs. We will ates (as described in [Feliu and Wiuf 2013]), with enzymes discuss some properties of the set of equilibria of these sys- (as described in [P´erezMill´an, Dickenstein, Shiu, and Con- tems, which can be regarded as generalizations of certain radi 2012]) and networks with toric steady states (as con- properties of toric dynamical systems. sidered in [Marcondes de Freitas, Feliu, and Wiuf 2016]). Specifically, given a core reaction network and an exten- Polly Yu sion of it obtained by adding intermediates or enzymes, we University of Wisconsin Madison study how the Gr¨obner bases of the steady state ideal of [email protected] the core reaction network relate to the Gr¨obner bases of the steady state ideal of extended networks. We use this to (1) find Gr¨obner bases of large networks from Grbner bases of MS73 simplified versions of them, thereby reducing substantially the computational cost; (2) infer when extensions of net- The Tropical Nullstellensatz works with toric steady states also have toric steady states; (3) pinpoint what/whether intermediates can be target as Tropical versions of the Nullstellensatz have been intro- responsible for multistationarity. duced since the start of tropical geometry. I will survey these versions, and then discuss a new version, joint with Elisenda Feliu, Amirhossein Sadeghi Manesh Felipe Rinc´on, that uses the theory of valuated matroids University of Copenhagen to get a statement more directly analogous to the classical [email protected], [email protected] Nullstellensatz. Diane Maclagan MS72 University of Warwick [email protected] Parametrising the Steady State Variety by Linear Elimination MS73 The steady states of a chemical reaction network with mass-action kinetics are solutions to a system of polyno- Tropical Hyperelliptic Curves in the Plane mial equations. Even for small systems, finding the steady states of the system is a very demanding task and there- Classically, any hyperelliptic curve admits a non- degenerate planar representation with an equation of the fore methods that reduce the number of variables are desir- 2 able. In [Feliu, Wiuf: Variable elimination in chemical re- form y = f(x). I will show that tropically, things are far action networks with mass-action kinetics. SIAM J. APPL. more restrictive: tropical hyperelliptic curves in the plane MATH.] the authors give one such method, in which so- are limited to a basic type of graphs called chains, with called non-interacting species are eliminated from the sys- nontrivial conditions on the achievable metrics. tem of steady state equations. We extend this method for the elimination of what we call reactant non-interacting Ralph Morrison species and give some conditions that ensure the positiv- Williams College ity of the elimination obtained, that is, for positive val- [email protected] ues of the reaction rate constants and the non-eliminated species, we ensure that the eliminated species are positive as well. Further, we show that iterative use of this elimi- MS73 nation procedure leads to the elimination of more general Gonality Sequences of Complete Graphs and Com- sets of species. In particular, if enough species can be elim- plete Bipartite Graphs inated, then this method provides a parametrisation of the positive part of the steady state variety. Abstract not available at time of publication.

Meritxell Saez Marta Panizzut Department of Mathematical Sciences. University of TU Berlin 122 AG17 Abstracts

[email protected] is a natural way to define morphisms between these cat- egories that generalizes the stability results of TDA to a broad class of objects by showing that the morphisms are MS73 1-Lipschitz. If time allows it, I will present a new example Computing Tropical Variaties using Triangular De- of such a morphism, known as the hom-tree functor, which composition provides a new bound on the Reeb graph interleaving dis- tance. Explicit computation of tropical varieties sounds like one of the most fundamental tasks in computational tropical Anastasios Stefanou geometry, but it is far from trivial. In this talk, I will give SUNY Albany a survey the existing techniques and explain how triangular [email protected] decomposition can be used to speed up the bottleneck of the computation. MS74 Yue Ren The Topology of Biological Aggregations: Experi- TU Kaiserslautern ments and Simulations [email protected] We apply tools from topological data analysis to experi- mental and simulation data of models inspired by biologi- MS74 cal aggregations such as bird flocks, fish schools, and insect A Morse-theoretic Algorithm to Compute Persis- swarms. Our data consist of numerical simulation output tent Homology, with Generators from the models of Vicsek and D’Orsogna as well as exper- imental data of pea aphids and associated models. These We introduce two Morse-theoretic methods for memory- models are dynamical systems describing the movement of efficient computation of persistent homology on a filtered agents who interact via alignment, attraction, and/or re- clique complex. The first determines a discrete Morse vec- pulsion. Each simulation time frame is a point cloud in tor field that enables the deletion of n-dimensional sim- position-velocity space. We analyze the topological struc- plices dynamically as the filtered complex is constructed, ture of these point clouds. To interpret the persistent ho- while preserving the first n-1 homology groups. The sec- mology of our results, we introduce a visualization that ond appeals to an algebraic interpretation of the Morse displays Betti numbers over simulation time and topolog- complex to recover, in a memory-efficient fashion, barcode ical persistence scale. We compare our topological results representatives for the input space. These methods have to order parameters typically used to quantify the global been combined in the open-source library Eirene. Experi- behavior of aggregations, such as polarization and angular ments with random, geometric, and empirical data suggest momentum. The topological calculations reveal events and that for complexes with large cliques, the first approach structure not captured by the order parameters. may decrease the size of the input to a standard homology solver by several orders of magnitude, while the memory Lori Ziegelmeier requirement of the second remains approximately linear in Dept. of Mathematics, Statistics, and Computer Science the size of the input. Our approach is broadly informed Macalester College by methods in discrete optimization and matroid theory. [email protected] Future directions in computational topology, discrete con- vexity, and optimization will be discussed. MS75 Greg Henselman Princeton University Well-rounded Lattices and Applications to Physical [email protected] Layer Security In the classical wireless communication scenario a sender, MS74 Alice, desires to transmit a message to a receiver, Bob, Local Cohomology and Stratification through a possibly noisy channel. Due to the increasing demand for wireless security, it is natural to consider the In this talk, we examine an efficient method to construct more complicated, yet realistic, situation in which an eaves- the coarsest stratification of a regular CW complex into dropper, Eve, tries to obtain Bob’s information trough a (homology) manifolds. degraded channel. In this scenario, known as the wire- tap channel, one wishes to achieve the following three con- Vidit Nanda tradictory objectives: high information rate between Al- Alan Turing Institute ice and Bob, high reliability at the legitimate receiver, Oxford University and minimal mutual information between the transmitted [email protected] message and Eve’s decoded message. Several design cri- teria based on lattice codes have been proposed with this aim. In this talk we introduce some design criteria based MS74 on well-rounded lattice coset codes. Well-rounded lattices Interleavings on Categories with Lax [0, ∞)-Action naturally arise in the study of arithmetic both in num- ber fields and in quaternion algebras. We show how to The interleaving distance is a powerful tool in TDA which obtain families of well-rounded lattices from different alge- has been shown to provide a metric for such topological sig- braic constructions and how to choose the most convenient natures as persistence diagrams and Reeb graphs. In this one for the desired secure communication purpose. The talk we generalize the idea of interleavings to a broader presentation is based on collaboration with A. Barreal, M. class of objects, namely categories with lax [0, ∞)-action. T. Damir, L. Fukshansky, O. W. Gnilke, C. Hollanti, D. This allows us to show that many commonly used dis- Karpuk, A. Karrila, and H. T. N. Tran. tances, such as the L∞ and Hausdorff metrics, are in fact special cases of interleaving distances. In addition, there Piermarco Milione AG17 Abstracts 123

Aalto University University of Campinas piermarco.milione@aalto.fi [email protected]

MS75 MS76 Rank-metric Codes of Zero Defect Polyhedral Approximations to Nonnegative Poly- nomials We propose a definition of rank defect for codes endowed with the rank metric, and study those whose defect is zero. Consider a collection of real homogeneous degree 2d poly- These codes generalize optimal (i.e., MRD) codes, and exist nomials A = {r1,r2,...,rm}. We denote the vector space for all choices of the parameters. We show combinatorial spanned by {ri} with PA, and the nonnegative polynomials properties of their rank distributions and duality theory. in PA formaconethatwedenotebyPosA.Inthisnote,we study polyhedral approximations and approximation lim- Alberto Ravagnani its to PosA. In the case of full support (i.e PA = Pn,2d) University of Neuchatel we show that any constant ratio polyhedral approxima- [email protected] tion to PosA has to have exponentially many facets. In the case of a sparse subspace PA with dim(PA)=m,we develop polyhedral approximations to PosA basedonspec- MS75 tral sparsification. A corollary of our work is a polyhedral −2 Weight Two Masking in the McEliece Public Key constant ratio approximation with O(nm ) many facets. System Alperen Ergur The National Institute of Standards and Technology North Carolina State University (NIST) encouraged a year ago research in public key cryp- Department of Mathematics tosystems which would resist the computing capability of a [email protected] quantum computer. One of the most promising candidates for post-quantum cryptography are code-based cryptosys- tems. The idea goes back to a proposal by McEliece who MS76 proposed the use of classical Goppa codes, disguised by Tropical Spectrahedra a monomial transformation. The main drawback of the original proposal was the large key size. For this rea- We introduce tropical spectrahedra, defined as the images son many researchers proposed alternative systems hav- by the nonarchimedean valuation of spectrahedra over the ing smaller key size. In this talk we present a variant field of real Puiseux series. We provide an explicit charac- of the McEliece system, proposed by Bolkema, Gluesing- terization of generic tropical spectrahedra, involving prin- Luerssen, Kelley, Lauter, Malmskog, Rosenthal using Gen- cipal tropical minors of size at most 2. To do so, we study eralized Reed Solomon (GRS) codes and a matrix with con- the images by the nonarchimedean valuation of semialge- stant row weight two as scrambling transformation. This braic sets. We prove in particular that, under a regularity variant is a special case of the BBCRS scheme, proposed by assumption, the image by the valuation of a basic semi- Baldi, Bianchi, Chiaraluce, Rosenthal, Schipani, for which algebraic set is obtained by tropicalizing the inequalities Couvreur, Gaborit, Gauthier-Uma˜na, Otmani, Tillich pro- which define it. We finally show that the projections of vided a distinguisher attack, in case that the square code tropical spectrahedra are precisely the sets of winning cer- has not maximal dimension. In this talk we provide evi- tificates of stochastic mean-payoff games, and discuss the dence that the weight two masking leads to maximal di- applications of these results to semidefinite programming mension of the square code avoiding in this way the distin- over nonarchimedean fields. guisher attack. Some of the presented results were devel- oped in the Master thesis of Weger. Xavier Allamigeon INRIA and CMAP, Ecole Polytechnique, CNRS Joachim Rosenthal [email protected] Institut f¨ur Mathematik Universit¨at Z¨urich Stephane Gaubert [email protected] INRIA and CMAP, Ecole Polytechnique [email protected] Violetta Weger University of Zurich Mateusz Skomra Mathematics Institute CMAP, Ecole Polytechnique, CNRS and INRIA [email protected] [email protected]

MS75 MS76 Quantum Codes from AG Codes of Castle Type Polynomial Norms

Our talk is concerning Algebraic Geometry codes produc- We establish when the d-th root of a multivariate degree-d ing quantum error-correcting codes by the CSS construc- polynomial produces a norm. In the quadratic case, it is tion; in fact, we pay particular attention to the family of well-known that this is the case if and only if the matrix Castle codes and show that many of the examples known associated with the quadratic form is positive definite. We so far in the literature belong to this family of codes. We present a generalization of this result to higher order poly- systematize these constructions by showing the common nomials and show that there is a hierarchy of semidefinite theory that underlies all of them. It is based in a joint programs that can detect all polynomial norms. work with C. Munuera and W. Tenorio. Amir Ali Ahmadi Fernando Torres Princeton University 124 AG17 Abstracts

a a [email protected] [email protected]

Etienne Klerk, de Faculty of Economic Sciences MS77 Tilburg University [email protected] Singular Value Homotopy for Finding Critical Pa- rameters in Differential Equations Georgina Hall Princeton University Many nonlinear differential equations are parameterized in [email protected] some form. Often the value of this parameter can affect the number of solutions to the equation. In this talk I will de- scribe a numerical algebraic geometry technique for finding MS76 the critical parameter value where the number of solutions of a differential equation changes. This technique involves Semidefinite Approximations of Reachable Sets for forming a homotopy that drives the smallest singular value Discrete-time Polynomial Systems of the Jacobian of a system to zero. We consider the problem of approximating the reachable set of a discrete-time polynomial system from a constrained James B. Collins semialgebraic set of initial conditions under general semi- West Texas A&M University algebraic set constraints. Assuming inclusion in a simple [email protected] set like a box or an ellipsoid given a priori, we provide a method to compute certified outer approximations of the Jon Hauenstein reachable set. The proposed method consists of building a University of Notre Dame hierarchy of relaxations for an infinite-dimensional moment [email protected] problem. Under certain assumptions, the optimal value of this problem is the volume of the reachable set and the op- timum solution is the restriction of the Lebesgue measure MS77 on this set. Then, one can outer approximate the reach- able set as closely as desired with a hierarchy of super level On Isolation of Simple Multiple Zeros of Polyno- sets of increasing degree polynomials. For each fixed de- mial Systems gree, finding the coefficients of the polynomial boils down to computing the optimal solution of a convex semidefinite Solving polynomial systems is a fundamental problem program. When the degree of the polynomial approxima- in numerical algebraic geometry, where computing iso- tion tends to infinity, we provide strong convergence guar- lated zeros is of particular interest to many researchers. antees of the super level sets towards the reachable set. Smale and others developed alpha-theory that certifies the We also present some application examples together with quadratic convergence of Newton’s method and provides a numerical results. lower bound for isolating a simple zero from other zeros. Dedieu and Shub proposed an explicit isolation bound of a Victor Magron simple double zero and a numerical criteria to certify a clus- CNRS VERIMAG ter of two zeros (counting the multiplicity). Giusti, Lecerf, [email protected] Salvy and Yakoubsohn investigated the location and the approximation of clusters of zeros of analytic maps of em- Didier Henrion bedding dimension one via implicit theorem and deflation. LAAS-CNRS, University of Toulouse, France In this talk, inspired by Giusti et al.’s technique of reduc- [email protected] tion to one variable, and based on our previous works on computing the multiplicity structure of simple multiple ze- Pierre-Loic Garoche ros, we generalized Dedieu and Shub’s results for simple ONERA multiple zeros with higher multiplicities. [email protected] Nan Li Xavier Thirioux Tianjin University ENSEEIHT Tianjin, China [email protected] [email protected]

MS77 MS77 Computing Newton Polytopes via Numerical Alge- On Sparse Homotopy and Toric Varieties braic Geometry

Computing an oracle representation for the Newton poly- The recent advances obtained on Smale’s 17th problem tope of a hypersurface can be done using an algorithm by provide a rigorous mathematical theory for homotopy Hauenstein and Sottile. In this talk, we discuss how the methods in polynomial solving. They are quantitative in numerical nature of this algorithm differentiates it from the sense that the expected number of homotopy steps is other oracle representations and how these differences can bounded above. Yet, it is usually assumed that the input be harnessed to speed up computations. We also exhibit polynomials are provided in dense representation. More- results from its most recent implementation. over, a certain unitarily invariant probability distribution is assumed, which only makes sense for dense polynomial Taylor Brysiewicz systems. Texas A&M I shall report on the generalization of this theory to a large College Station, TX USA class of polynomial systems of practical interest, namely AG17 Abstracts 125

sparse systems of the form the resulting algorithms to higher dimensions. a1 a2 an Lars Kastner F1(Z)= f aZ Z ···Z a∈A1 i, 1 2 n FU Berlin . . [email protected] . a1 a2 an a ··· Fn(Z)= a∈An fi, Z1 Z2 Zn MS78 where each Ai is a finite set. The natural input size Homological Algebra in Macaulay2 for those systems is i #Ai which can be exponentially smaller than the dense input size. Another important in- We will discuss the design and implementation of a new variant is the generic number of roots in Cn \{0} of such collection of homological algebra data types and methods systems of equations, known as the mixed volume of the in the Macaulay2 software system. Examples from alge- supporting polytopes. It can be much smaller than the braic geometry will be used to illustrate these algorithms. dense B´ezout bound. Performing homotopy on the proper This talk is based on joint work with Mike Stillman. compactification of Cn \{0}(a toric variety) allows to ob- tain sharper time bounds and practical improvements of Gregory G. Smith the homotopy algorithms. Queen’s University [email protected] Gregorio Malajovich Universidade Federal do Rio de Janeiro [email protected] MS78 Asymptotic Syzygies via Numerical Linear Algebra and High Throughput Computing MS78 A Combinatorial Smoothness Criterion for Spher- Ein and Lazerfield pioneered the concept of asymptotic ical Varieties syzygies. Later Ein, Erman, and Lazerfield among oth- ers expanded this concept further with several conjectures. One core algorithm in commutative algebra is given by the However there is very little in the way of data to support Jacobian criterion to check smoothness of algebraic vari- or refute these conjectures, owing largely to the difficulty eties. This criterion can be impractical due to the number of computing betti numbers for all but the smallest exam- and size of the minors involved. In this talk we want to ples. We introduce a technique of using numerical linear present alternative algorithms to check smoothness for the algebra combined with high throughput to compute these class of spherical varieties which contains those of toric va- betti numbers. We then use this technique to formulate and provide support for several conjectures about the syzy- rieties, flag varieties and symmetric varieties, and which 2 forms a remarkable class of algebraic varieties with an ac- gies of the Veronese of P . tion by an algebraic group having an open dense orbit. In Jay Yang the toric case there is a well-known simple combinatorial University of Wisconsin-Madison smoothness criterion whereas in the spherical case Brion, U.S.A. Camus and Gagliardi have shown smoothness criteria for [email protected] spherical varieties which either are rather involved or rely on certain classification results. We suggest a purely com- binatorial smoothness criterion by introducing a rational MS79 invariant which only depends on the combinatorics of the spherical variety. We have a conjectural inequality this No Occurrence Obstructions in Geometric Com- invariant should satisfy where the equality case would im- plexity Theory ply a combinatorial criterion for the spherical variety to be The permanent versus determinant conjecture is a major isomorphic to a toric one. Our conjecture would also im- problem in complexity theory that is equivalent to the sep- ply the generalized Mukai conjecture for spherical varieties. aration of the complexity classes VPs and VNP. Mulmuley Batyrev and Moreau have also given a conjectural smooth- and Sohoni suggested to study a strengthened version of ness criterion for spherical varieties which uses stringy Eu- this conjecture over the complex numbers that amounts ler numbers. We complete our talk by summarizing in to separating the orbit closures of the determinant and which cases the above mentioned approaches are known to padded permanent polynomials. In that paper it was also be true. This is joint work with Giuliano Gagliardi. proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representa- Johannes Hofscheier tions of GL 2 (C), which occur in one coordinate ring of Otto-von-Guericke Universit¨at Magdeburg n the orbit closure, but not in the other. We prove that Germany this approach is impossible. However, we do not rule out [email protected] the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed by Mul- MS78 muley and Sohoni. Injective Resolutions in Toric Geometry Peter Buergisser TU Berlin The description of graded modules over semigroup algebras [email protected] allows one to resolve modules injectively in a combinatorial fashion. Even though these modules are infinitely gener- ated algebraically, one can use combinatorial software to MS79 deal with derived functors using injective resolutions. We The Geometry of Rank-one Tensor Completion will demonstrate this for the class of two-dimensional cyclic quotient singularities and elaborate on how to generalize We show how to apply tools from algebraic geometry to de- 126 AG17 Abstracts

cide if a partial tensor is completable to a rank-one tensor, [email protected] and if such a completion exists, how to find it. Both, com- pletions to complex and real rank-one tensors are studied. Conditions for finite and unique completability to rank-one MS80 tensors are given. Finally, we study real rank-one com- Topological Methods for Cancer Genomics pletability inside the standard simplex and characterize the algebraic boundary of the completable region. Developing suitable targeted therapies for cancer requires the stratification of cancer patients according to their Thomas Kahle molecular alterations. Commonly used methods for iden- OvGU Magdeburg tifying driver alterations seek signatures of positive selec- [email protected] tion based on recurrence across large cohorts of patients. Because of the large complexity involved in modelling the Kaie Kubjas background mutation rate, these strategies have limited Aalto University power to identify functional variants that occur at low fre- kaie.kubjas@aalto.fi quencies or in hyper-mutated tumors. Nonetheless, most clonal mutations in patients of a cancer type have a preva- Mario Kummer lence below 5 percent, and many of these mutations are MPI Leipzig expected to be clinically relevant. Building upon Topo- [email protected] logical Data Analysis, we present an unsupervised statis- tical framework for the identification of genetic alterations Zvi H. Rosen (point mutations, small insertions and deletions, and gene Department of Mathematics fusions) that consistently drive global expression patterns University of Pennsylvania in tumors. The use of topological methods is particularly [email protected] tailored to this problem, as the phenotypic space of tumors of a cancer type is relatively continuous, with many tumors having characteristics of multiple sub-types according to traditional classification schemes. Application of this ap- MS79 proach to large cancer databases, such as those generated in Border Ranks of Monomials the context of ongoing precision medicine initiatives, leads to the identification of new potential therapeutic targets that, because of their low prevalence, have remained elu- Young flattenings, introduced by Landsberg and Ottaviani, sive to current methods of detection. give determinantal equations for secant varieties and pro- vide lower bounds for border ranks of tensors. We find Pablo Camara special monomial-optimal Young flattenings that provide Columbia University the best possible lower bound for all monomials up to de- Department of Systems Biology, Irving Cancer Research gree 6. For degree 7 and higher these flattenings no longer Center suffice for all monomials. To overcome this problem we in- [email protected] troduce partial Young flattenings and use them to give a lower bound on the border rank of monomials which agrees with Landsberg and Teitler’s upper bound. Udi Rubin Columbia University [email protected] Luke Oeding Auburn University [email protected] Arnold Levine Institute for Advanced Study [email protected] MS80 Raul Rabadan Mathematical Methods to Quantify the Topological Columbia University Complexity of Chromosomes [email protected]

Uncovering the basic principles that govern the three di- mensional (3D) organization of genomes poses one of the MS80 main challenges in mathematical biology of the postge- Genome Rearrangement Distance under Inversions nomic era. Certain viruses and some organisms, such as and Deletions: A Semigroup Model trypanosomes, accommodate knotted or linked genomes. Others, such as bacteria, are known to have unknotted In this talk I will describe an algebraic model of genome genomes. It remains to be determined if the genomes of rearrangement that allows inversion and deletion of seg- higher organisms admit topologically complex forms. In ments of DNA. This results in a model that can be trans- this talk I will present a new method to quantify the topo- lated into semigroup theory, working within the symmetric logical complexity of chromosome organization. We test inverse monoid. I will describe an algorithm for approxi- our methods with hi-C data from budding yeast. Our re- mating the minimal distance between two genomes under sults suggest that the Rabl conformation of chromosomes this model. Joint work with James Mitchell and Julius provides a mechanism for topology simplification. Jonuas from St Andrews University, Scotland, and Chad Clark from Western Sydney University, Australia. Javier Asuaga University of California-Davis Andrew Francis Departments of Mathematics and Molecular and Cell University of Western Sydney Biology Centre for Research in Mathematics AG17 Abstracts 127

[email protected] connect the data-specific fiber. We apply this method to fit neuronal and protein data to some degree-based ERGMs.

MS80 Despina Stasi Topological Features in Cancer Gene Expression Illinois Institute of Technology Data [email protected]

We present a new method for exploring cancer gene ex- pression data based on tools from algebraic topology. Our MS81 method selects a small relevant subset from tens of thou- Phylogenetic Trees, a M2 Package sands of genes while simultaneously identifying nontrivial higher order topological features, i.e., holes, in the data. We introduce the package PhylogeneticTrees for Macaulay2 We first circumvent the problem of high dimensionality by which allows users to compute phylogenetic invariants for dualizing the data, i.e., by studying genes as points in the group-based tree models. We provide some background patient space. Then we select a small subset of the genes information on phylogenetic algebraic geometry and show as landmarks to construct topological structures that cap- how the package PhylogeneticTrees can be used to calculate ture persistent, i.e., topologically significant, features of a generating set for a phylogenetic ideal as well as a lower the data set in its first homology group. Furthermore, we bound for its dimension. Finally, we show how methods demonstrate that many members of these loops have been within the package can be used to compute a generating implicated for cancer biogenesis in scientific literature. We set for the join of any two ideals. illustrate our method on five different data sets belonging Allen Stewart to brain, breast, leukemia, and ovarian cancers. Seattle University Svetlana Lookwood [email protected] Washington State Universitry [email protected] Hector Banos University of Alaska, Fairbanks Bala Krishnamoorthy [email protected] Department of Mathematics Washington State University-Vancouver Nathaniel Bushek [email protected] University of Alaska, Anchorage [email protected]

MS81 Ruth E. Davidson Fast Exact Tests for Stochastic Block Models using University of Illinois Urbana Champaign Algebraic Statistics [email protected]

Stochastic Block models (SBM) with unknown block struc- Elizabeth Gross ture are widely used to detect communities in real world San Jose State University networks. Testing the goodness of fit of such models is a [email protected] challenging task due to the fact that the parameters of an SBM are usually estimated from a single observed network. Pamela Harris Usual asymptotic tests are not valid. We develop a finite Williams College sample goodness-of-fit tests for three different variants of [email protected] Stochastic Block models with unknown blocks. The finite sample test is based on the posterior predictive distribu- tion of the SBM with unknown blocks. A key building Robert Krone block for sampling from this distribution is sampler from Queen’s University fibers of models with known block assignments. Sampling Department of Mathematics and Statistics from these fibers is carried out using Markov bases. As in- [email protected] termediate results, we describe the Markov bases and the marginal polytope of Stochastic Block models with known Colby Long block assignments. Ohio State University [email protected] Vishesh Karwa Harvard University Robert Walker Carnegie Mellon University University of Michigan [email protected] [email protected]

MS81 MS82 Parameter Hypergraphs, Fiber Walks and Nearest Points on Toric Varieties Goodness-of-fit Testing for Biological Network Data We determine a combinitorial expression for the Euclidean distance degree of a projective toric variety. This extends A log-linear model gives rise to a parameter hypergraph the formula of Matsui and Takeuchi for the degree of the A- and is associated with the toric ideal of the monomial al- discriminant in terms of local Euler obstructions. Our pri- gebra parametrized by the edges of the hypergraph. Under- mary goal is the development of reliable algorithmic tools standing the generators of this ideal gives insight into the for computing the points on a real toric variety that are model itself. We utilize the combinatorics of the generators closest to a given data point. This also leads to combinato- to dynamically produce fiber walks that are guaranteed to rial expressions for the polar degrees, the degree and codi- 128 AG17 Abstracts

mension of the dual variety, and the Chern-Mather class the network structure. In experimental set-ups we often of a projective toric variety. This is joint work with Bernd encounter the problem of noisy data points for which we Sturmfels. want to find the corresponding steady state predicted by the model. Depending on the network there may be many Martin Helmer such points and the number of which is given by the eu- Department of Mathematics clidean distance degree (EDdegree). In this talk I show University of California, Berkeley how certain properties of networks relate to the EDdegree [email protected] and how the runtime of numerical algebraic geomtery com- putations scales with the EDdegree. I will illustrate our Bernd Sturmfels findings by the examples of multisite phosphorylation and MPI Leipzig and UC Berkeley kinetic proofreading. [email protected] MichaelF.Adamer University of Oxford MS82 [email protected] The ED Degree of the Camera Variety Martin Helmer We construct a resolution of the projective camera varieties Department of Mathematics that satisfies two strong geometric conditions. This allows University of California, Berkeley us to compute the Chern-Mather class of the projective [email protected] camera variety for an arbitrary number of cameras. In particular, we obtain a tight bound on the ED degree of theaffinecameravariety. MS83 A Convergent and Efficient Algorithm for Calcu- Daniel Lowengrub lating Equilibrium for Chemical Networks of Re- Department of Mathematics versible Binding Reactions University of California, Berkeley [email protected] We have a particular interest in chemical networks of re- versible binding reactions that are complete, a qualifier MS82 that essentially affirms the conservation of the chemical species that are the building blocks of the network. These Local Euler Obstructions of Toric Varieties networks are pervasive in pharmacostatic, the subdiscipline We use Matsui and Takeuchi’s formula for toric A- of pharmacology concerned with the characterization of the discriminants to give algorithms for computing local Euler equilibrium parameters and states of core interactions of obstructions of toric surfaces and 3-folds. As an application physiologic and therapeutic interest. We have been on a we compute polar degrees and the EDdegree of weighted quest for a method to calculate the equilibrium states of projective surfaces and 3-folds. these networks that is generic, guaranteed to succeed, and fast; or what we called worry-free. We will report on the Bernt-Ivar U. Nødland tangible progress we recently made in that direction. Department of Mathematics University of Oslo Gilles Gnacadja [email protected] Amgen [email protected]

MS82 MS83 Polar Varieties and Euclidean Distance Degree Multistationarity in Interaction Networks Reciprocal polar varietes of a given, possibly singular, pro- jective variety are defined with respect to a non-degenerate Interaction networks are used to model many biological quadric hypersurface. The quadric induces a notion of or- systems. Multistationarity (existence of multiple, compat- thogonality in the projective space, hence also a notion of ible positive equilibria) underlies switching behavior in the ‘Euclidean geometry.’ Thus one can define the Euclidean interaction network. For networks considered with mass- distance degree and the Euclidean normal bundle, as well action kinetics, the number of equilibria corresponds to the as the classical concepts of focal loci and caustics of reflec- number of positive zeros of a multivariate polynomial sys- tions. The Euclidean distance degree and the degree of the tem. Such a system of polynomials is parametrized by a focal locus can be expressed in terms of the degrees of the large number of positive rate constants, whose values are classical polar varieties. I will give some examples in the often unknown. We will state some recent results regard- case of curves, surfaces, and toric varieties. ing the maximum number of equilibria that a system can have over all possible values of rate constants. Ragni Piene Department of Mathematics Badal Joshi University of Oslo California State University, San Marcos [email protected] [email protected]

MS83 MS83 Algebraic Complexity of Chemical Reaction Net- Inheritance of Bistability in Mass Action Reaction works Through EDdegrees Networks

Steady state chemical reaction models can be thought of This talk focuses on bistability (the existence of multiple as algebraic varieties whose properties are determined by stable positive equilibria), a dynamical property that un- AG17 Abstracts 129

derlies important cellular processes, and a recurring theme Mustapha BOUHTOU in recent work on reaction networks. Namely, we consider Orange Labs the question: ”when can we conclude that a network ad- [email protected] mits multiple stable positive equilibria based onanalysis of its subnetworks? We identify a number of operations on Jean Bernard Eytard reaction networks that preserve bistability as we build up INRIA - CMAP, Ecole Polytechnique the network, and we illustrate the power of this approach [email protected] on the much-studied Huang-Ferrell MAPK cascade. This work is related to Joshi and Shius “atoms of multistation- Stephane Gaubert arity”, and falls broadly into the theory of motifs, a central INRIA and CMAP, Ecole Polytechnique theme in systems biology. [email protected] Casian Pantea West Virginia University MS84 [email protected] Abstract Tropical Linear Programming Murad Banaji Middlesex University London Analogously to classical oriented matroid programming, we [email protected] present a feasibility algorithm for signed tropical oriented matroids. They encode the structure of tropical linear in- equality systems and generalize them. The feasibility prob- MS84 lem for these inequality systems is of special interest as it is in NP ∩ co-NP but no polynomial time algorithm is known. Construction of Lindstrom Valuations of Algebraic We formulate pivoting between basic covectors which only Extensions depends on the combinatorial structure and not on the co- efficients of the inequalities. For this, we use the structure An algebraic matroid is a combinatorial structure associ- of (not necessarily regular) subdivisions of products of sim- ated to a collection of elements in a field extension, or plices. equivalently, to chosen set of coordinates for an algebraic variety. Algebraic matroids in positive characteristic are Georg Loho notoriously subtle and many basic questions are still open. TU Berlin Recently, Bollen, Draisma, and Pendavingh have intro- [email protected] duced a more refined structure called a matroid flock on top of an algebraic matroid, which they prove to be equivalent to a valuated matroid. I will give a direct construction of MS84 this Lindstrm valuation, and its connections with tropical geometry. Computing Tropical Linear Spaces

Dustin Cartwright Tropical linear spaces are a core object in tropical geom- University of Tennessee, Knoxville etry. They are polyhedral complexes which are dual to [email protected] matroid subdivisions of hypersimplices. This gives a hint why matroid theory is importend in tropical geometry. Re- search on matroid subdivisions and related objects goes MS84 back to Dress and Wenzel and to Kapranov. Speyer in- stigated a systematic study in the context of tropical ge- A Tropical Approach to Bilevel Programming ometry, while algorithms for trivial valuation have been developed and implemented by Rincon. In this talk I will In recent work, Baldwin, Klemperer, Tran and Yu have present a new method for computing tropical linear spaces applied tropical geometry methods to a class of problems and more general duals of polyhedral subdivisions. It is in mathematical economy (existence of competitive equi- based on Ganter’s algorithm (1984) for finite closure sys- libria for indivisible goods). A key idea here is to represent tems. The presented method is implemented in polymake. the agent’s response by an arrangement of tropical hyper- This is joint work with Simon Hampe and Michael Joswig. surfaces. Here, we apply this idea to bilevel programming problems. The latter are a class of (generally hard) opti- mization problems, omnipresent in optimal pricing: a high Benjamin Schr¨oter level agent, like a company, wishing to optimize a total in- Georgia Tech come or a measure of total satisfaction of the customers, [email protected] announces a price for a shared resource. Then, every cus- tomer determines his consumption by optimizing an in- dividual utility function depending on this price. These MS85 consumptions finally determine the objective function of the high level agent. By exploiting tropical geometry and Resultants Modulo P discrete convexity methods (M-convexity in the sense of Murota), we show that a class of bilevel programming prob- Several problems in elimination theory involving arithmetic lems can be solved in polynomial time. We illustrate these over the integers (like resultants, the Nullstellensatz, etc.) results by an application provided by Orange, in which we have as an outcome an integer number which if it is not zero use price incentives to balance the load in a mobile net- modulo a prime p, then several results over the complex work. number (dimension, number of zeroes, etc.) descend to the residual field. But what happens when p does divide Marianne Akian this number? We present some bounds on the the p-th INRIA and CMAP, Ecole Polytechnique valuation of the resultant which are tight in generic cases, [email protected] and applications to computations of Milnor Numbers and 130 AG17 Abstracts

degrees of gcd’s. Formulae and Applications

Laurent Buse We show how resultants and subresultants can be described INRIA, Sophia Antipolis in terms of the evaluation of the considered two polynomi- [email protected]¿ als in any set of points not necessarily the roots of one of them and not necessarily different. We present the applica- Carlos D’Andrea, Martin Sombra tion of this formulae in the context of the so called ”Poly- Universitat de Barcelona nomial Algebra by Values” and will show how to deal with [email protected], [email protected] intersection problems involving algebraic curves when their equations are presented “by values’ (i.e., in the Lagrange, the Hermite or the Birkhoff basis). This work has been par- MS85 tially supported by the Spanish Ministerio de Econom´ıa y Offsets and Subresultants Competitividad and by the European Regional Develop- ment Fund (ERDF), under the project MTM2014-54141-P The aim of this talk is to explain how Subresultants can be useful to describe the geometry of offsets to planar ra- tional curves. Intuitively, the offset curve toadistanceδ Laureano Gonzalez-Vega toacurveC is the “parallel” curve to C at the distance Departamento de Matematicas, Estadistica y δ. Offsets have many important applications in Computer Computacion Aided Design, such as tool path generation, NC milling, Universidad de Cantabria design of thick curved surfaces and tolerance analysis. If C [email protected] is a planar rational curve, its offset is an algebraic curve, but it is generally not rational. Moreover, the offset im- plicit equation typically has much higher degree than C, MS85 many terms and big coefficients. Therefore, deriving the topology of the offset from its implicit equation is usually Introductory Talk time-consuming and quite often impossible in practice. In this talk, on the one hand, we will present a new algorithm This talk will introduce the topic and the goals of the min- to find a finite set containing all the affine parameter values isymposium. In particular, we highlight the different defi- giving rise to real, non-isolated singularities of the offset. nitions of resultants and subresultants and their historical This is essential to trim the offset. On the other hand, we developments. Then we give an overview of some of the will present another different method for computing the application areas where these notions are used. topology of the offset which uses the parametrizations of the exterior and interior offsets, which are given in terms of Agnes Szanto the unitary normal vector to the curve, involving a radical. North Carolina State University Both algorithms do not require to compute or make use [email protected] of the implicit equation of the offset and use Subresultants as an essential tool. Besides, they have been implemented andtestedinthesoftwareMaple. MS86 Gema M. Diaz-Toca Improved Constructions of Nested Code Pairs Universidad de Murcia, Spain [email protected] Two new constructions of linear code pairs C2 ⊂ C1 are given for which the codimension and the relative minimum ⊥ ⊥ Laureano Gonzalez-Vega distances M1(C1,C2)andM1(C2 ,C1 )aregood.Bythis Departamento de Matematicas, Estadistica y we mean that for any two out of the three parameters the Computacion third parameter of the constructed code pair is large. Such Universidad de Cantabria pairs of nested codes are indispensable for the determina- [email protected] tion of good linear ramp secret sharing schemes. They can also be used to ensure reliable communication over asym- Mario Fioravanti metric quantum channels. The new constructions result Universidad de Cantabria, Spain from carefully applying the Feng-Rao bounds to a family mario.fi[email protected] of codes defined from multivariate polynomials and Carte- sian product point sets. Jorge Caravantes Universidad Complutense de Madrid, Spain Carlos Galindo [email protected] University Jaume I [email protected] Ioana Necula Universidad de Sevilla, Spain. Olav Geil [email protected] Aalborg University [email protected] Juan Gerardo Alc´azar Universidad de Alcal´a de Henares, Spain. Fernando Hernando [email protected] San Diego State University [email protected]

MS85 Diego Ruano Resultants and Subresultants through Evaluation: Aalborg University AG17 Abstracts 131

[email protected] of Nan/Boston, and Mao/Thill/Hassibi.

John A. Mackenzie MS86 Strathclyde University Quasi-cyclic Subcodes of Cyclic Codes [email protected]

In an earlier work with McGuire, we obtained weight di- Michael E. O’Sullivan visibility results on certain trace codes via the relation of San Diego State University codewords to supersingular curves over finite fields. These [email protected] codes turned out to be quasi-cyclic codes inside cyclic codes. This led us to study quasi-cyclic subcodes of cyclic Sarah Roy codes in a general context. Together with Belfiore and Unaffiliated Ozkaya, we completely characterized possible indices of [email protected] quasi-cyclic subcodes in a cyclic code for a very broad class of cyclic codes. Moreover, we obtained enumeration results for quasi-cyclic subcodes of a fixed index and showed that MS87 the problem of enumeration is equivalent to enumeration Algebraic Certificates of Disconnectedness of certain vector subspaces in finite fields. The speaker will present these results in the talk. Given two disjoint semialgebraic subsets of a given compact semialgebraic set, we describe an algorithm that certifies Cem G¨uneri that the two subsets are disconnected, i.e. that there is no Sabanci University continuous path connecting them. The certificate of dis- [email protected] connectedness is a polynomial satisfying specific positivity conditions. It is computed by solving a hierarchy of convex moment-sum-of-squares semidefinite programs. MS86 Communication Efficient and Strongly Secure Se- Didier Henrion cret Sharing Schemes Based on Algebraic Geome- LAAS-CNRS, University of Toulouse, France try Codes [email protected]

Secret sharing schemes with optimal and universal commu- Mohab Safey El Din nication overheads have been obtained independently by UPMC, Univ Paris 6 Bitar et al. and Huang et al. However, their constructions [email protected] require a finite field of size q>n,wheren is the number of shares, and do not provide strong security. In this work, we give a general framework to construct communication effi- MS87 cient secret sharing schemes based on sequences of nested Semidefinitely Representable Convex Sets linear codes, which allows to use in particular algebraic geometry codes and allows to obtain strongly secure and We show that there are many (compact) convex semi- communication efficient schemes. Using this framework, algebraic sets in euclidean space that do not have a semidef- we obtain: 1) schemes with universal and close to optimal inite representation. This gives a negative answer to a communication overheads for arbitrarily large lengths n question of Nemirovski, resp. it shows that the Helton-Nie and a fixed finite field, 2) the first construction of schemes conjecture is false. with universal and optimal communication overheads and optimal strong security (for restricted lengths), having in Claus Scheiderer particular the security advantages of perfect schemes and Univ Konstanz (Germany) the storage efficiency of ramp schemes, and 3) schemes with [email protected] universal and close to optimal communication overheads and close to optimal strong security defined for arbitrarily large lengths n and a fixed finite field. MS87 Tranfer Function Realizations, Sums of Hermitian Umberto Martinez-Penas Squares, and Determinantal Representations Aalborg University [email protected] I will show that any polynomial p ∈ C[z1,...,zd]thathas no zeroes on the closed unit polydisc in Cd admits a mul- tiple pq so that pq =det(I − ZK)whereZ is a diago- MS86 nal matrix with the variables z1,...,zd on the diagonal: ⊕···⊕ ∈ n×n ··· The Ingleton Ratio, the Inclusion-exclusion Ratio Z = z1In1 zdInd ,andK C (n = n1 + +nd) ∗ and the Entropy Region is a strict contraction: K K

that appear in the generalized Lax conjecture. This is a hom4ps joint work with A. Grinshpan, D. Kaliuzhnyi-Verbovetskyi, [email protected] and H. Woerdeman.

Victor Vinnikov MS88 Ben Gurion University of the Negev Solving Polynomial Systems via Homotopy Contin- Beer Sheva, ISRAEL uation and Monodromy [email protected] We develop an algorithm to find all solutions of a generic system in a family of polynomial systems with paramet- MS87 ric coefficients using numerical homotopy continuation and The Relation Between SOS and SONC the action of the monodromy group. We argue that the expected number of homotopy paths that this algorithm In 2014, Iliman and I introduced a new nonnegativity cer- needs to follow is roughly linear in the number of solu- tificate based on sums of nonnegative circuit polynomials tions. We demonstrate that our software implementation (SONC),whichareindependent of sums of squares. More is competitive with the existing state-of-the art methods precisely, the SONC cone and the SOS cone intersect, but implemented in other software packages. they are not contained in each other. If a single circuit polynomials f is nonnegative, then it is a sum of squares Tim Duff if and only if a particular one of its exponents is contained Georgia Tech in the maximal mediated set corresponding to the Newton Atlanta, GA USA polytope of f. Maximal mediated sets, which are purely tduff[email protected] combinatorial objects, were first introduced by Reznick in ’89, but are not well understood yet. In this talk I will give Cvetelina Hill an overview about the relation between SOS and SONC Georgia Institute of Technology and outline my current research on this topic. [email protected] Timo de Wolff Texas A&M University Anders Jensen dewolff@math.tamu.edu Department of Mathematical Sciences Aarhus University [email protected] MS88 Geometry and Topology of the Solution Variety Kisun Lee Georgia Institute of Technology In this talk I will describe a geometrical framework for the [email protected] study of homotopy methods for solving numerical prob- lems. Some details about the geometry and the topology Anton Leykin of one of the fundamental objects in this geometrical frame- School of Mathematics work, the so–called solution variety, are known and will be Georgia Institute of Technology given in different cases including solving systems of poly- [email protected] nomial equations. Results concerning the applications for practical computing will be presented either as facts or as Jeff Sommars conjectures. University of Illinois at Chicago Carlos Beltr´an Chicago, Illinois USA Departamento de Matematicas [email protected] Universidad de Cantabria [email protected] MS88 Computing the Homology of Real Projective Sets MS88 We describe and analyze a numerical algorithm for com- Conic Elimination Technique for Mixed Cells Com- puting the homology of real projective varieties. Its cost putation depends on the condition of the input as well as on its Mixed volume computation is an important problem in nu- size: it is singly exponential in the number of variables merical algebraic geometry. Via the theory of BKK bound, (the dimension of the ambient space) and polynomial in the generic root count of a square system of polynomial the condition and the degrees of the defining polynomials. equations in the algebraic torus is given by the mixed vol- The algorithm relies on a new connection between Smale’s ume of the Newton polytopes of the polynomials. More- γ quantity and the injectivity radius of the normal bundle over, one practical approach in computing mixed volume of the variety restricted to the sphere. produces the by-product known as mixed cells which are Felipe Cucker the critical ingredient in constructing a polyhedral homo- City University, Hong Kong topy method for solving system of polynomial equations. [email protected] Among many different ways in computing mixed volume or mixed cells, the scheme based on systematic extensions of subfaces has been proved to be efficient and scalable. This Teresa Krick talk presents certain generalizations and improvements of Universidad de Buenos Aires and IMAS, CONICET this scheme. [email protected]

Tianran Chen Michael Shub Michigan State University IMAS, CONICET, Argentina and Graduate School of AG17 Abstracts 133

CUNY [email protected] [email protected]

MS89 MS89 Random Fields and the Enumerative Geometry of The Number of a Eigenvectors of a Random Tensor Lines on Real and Complex Hypersurfaces

Generalizing matrices to tables of dimension higher than We derive a formula expressing the average number of real two one obtains the notion of a tensor. The concept of lines on a random hypersurface of degree 2n − 3inreal eigenvalues of matrices can be transferred to tensors ac- projective space in terms of the expected modulus of the cordingly. Remarkably, for real symmetric tensors the determinant of a special random matrix. We determine number of real eigenvalues is not constant - different to the asymptotics at the logarithmic scale, and in the case n=3 matrix case. In this talk I will explain how to transfer the we prove that the average√ number of real lines on a random definition of the real Ginibre Ensemble and the Gaussian cubic surface equals: 6 2 − 3. Our technique can also be Orthogonal Ensemble from matrices to tensors and show used to express the number of complex lines on a generic you a formula for the expected number of real eigenvalues hypersurface of degree 2n − 3 in complex projective space for each of the two tensor random variables. in terms of the determinant of a random Hermitian ma- trix. As a special case we obtain a new proof of the clas- Paul Breiding sical statement that there are 27 complex lines on a cubic. TU Berlin This is joint work with Saugata Basu (Purdue Univeristy, [email protected] United States), Antonio Lerario (SISSA, Italy) and Chris Peterson (Colorado State University, United States).

MS89 Erik Lundberg Probabilistic Schubert Calculus Florida Atlantic University [email protected] Classical Schubert calculus deals with the intersection of Schubert varieties in general position. We present an at- tempt at developing such a theory over the reals. By the MS90 title we understand the investigation of the expected num- Core Variety of a Linear Functional and Truncated ber of points of intersection of real Schubert varieties in Moment Problem random position. We define a notion of expected degree of real Grassmannians that turns out to be the key quantity We show how to associate to a linear functional L on the governing questions of random incidence geometry. Us- vector space of polynomials of bounded degree (truncated ing integral geometry, we prove a result that decouples a moment sequence) its core variety V (L), which records the random incidence geometry problem into a volume com- possible supports of representing measures. In particular putation in real projective space and the determination L can be represented by a measure if and only if the core of the expected degree. Over the complex numbers, the variety is non-empty. We show several applications of this same decoupling result is a consequence of the ring struc- including a simple proof of Bayer-Techmann theorem. ture of the cohomology of the Grassmannian. We prove an asymptotically sharp upper bound on the expected de- Greg Blekherman gree of the real Grassmannians G(k,n). Moreover, if both k Georgia Institute of Technology and n go to infinity, the expected degree turns out to have [email protected] the same asymptotic growth (in the logarithmic scale) as the square root of the degree of the corresponding com- Lawrence A. Fialkow plex Grassmannian. This finding is in the spirit of the real SUNY New Paltz average Bezout’s theorem due to Shub and Smale. In the fi[email protected] case of the Grassmannian of lines, we can provide a finer asymptotic. This is joint work with Antonio Lerario, see arXiv:1612.06893 MS90 Moment Problems and Inequalities for Power Sums Peter B¨urgisser n r TU Berlin The power sum Mr(x)= k=1 xk can be thought of as [email protected] the r-th moment of a measure with unit point masses at the points t = xk. In this sense, the Cauchy-Schwartz inequality follows from the solution to the simplest case of MS89 the Hamburger Moment Problem. We will explore various Percolation of Random Nodal Lines non-obvious power sum inequalities which follow in this way; in some cases the extreme value comes when n =2. n 1 · 3 − If we fix a rectangle in the affine real space and if we choose Special attention will be given to M M > M4 , unless a real polynomial with degree d at random, the probabil- the speaker finds better examples in the next six months. ity P (d) that a component of its vanishing locus crosses the rectangle in its length is clearly positive. But is P (d) Bruce Reznick uniformly bounded from below when d increases? I will University of Illinois at Urbana-Champaign explain a positive answer to a very close question involving [email protected] real analytic functions. This is a joint work with Vincent Beffara. MS90 Damien Gayet Optimization Approaches to Quadrature Institut Fourier Grenoble Let d and k be positive integers. Let μ be a positive Borel 134 AG17 Abstracts

measure on R2 possessing finite moments up to degree d. exhibits both positive and negative gene regulation within Using methods from optimization, we study the question a single operon. In this talk, we will describe our proposed of the minimal m ∈ N such that there exists a quadrature Boolean network model for the ara operon, which consists rule for μ with m nodes which is exact for all polynomials of both a physical wiring diagram, and the logical functions of degree at most d. We show that if d =2d − 1andthe that govern each node. Additionally, we will describe the support of μ is contained in an algebraic curve of degree k, results of model validation and current and future research. then there exists a quadrature rule for μ with at most d ·k many nodes all placed on the curve (and positive weights). Lee Andrew Jenkins This generalizes Gauss quadrature where the curve is a line University of Georgia and (the odd case of) Szeg˝o quadrature where the curve is [email protected] a circle to arbitrary plane algebraic curves. In the even case, i.e., d =2d this result generalizes to compact curves. MS91 We use this result to show that, any plane measure μ has a quadrature rule with at most 3/2d(d −1)+1 many nodes, Detecting Binomiality of Chemical Reaction Net- which is exact up to degree 2d − 1. All our results are works obtaines by minimizing a certain linear functional on the polynomials of degree d. And our proof uses both results We discuss the problem of deciding if a given polynomial from convex optimisation and from real algebraic geometry. ideal is binomial. While the methods are general, our main motivation and source of examples is the simplification of steady state equations of chemical reaction networks. For Cordian Riener homogeneous ideals we give an efficient, Groebner-free al- Aalto University gorithm for binomiality detection, based on linear algebra [email protected] only. On inhomogeneous input the algorithm can only give a sufficient condition for binomiality. As a remedy we con- Markus Schweighofer struct a heuristic toolbox that can lead to simplifications University of Konstanz even if the given ideal is not binomial. [email protected] Thomas Kahle OvGU Magdeburg MS90 [email protected] Hankel and Quasi-Hankel Matrix Completion: Per- formance of the Nuclear Norm Relaxation MS91 Minimal rank completion of matrices with missing values Statistics of Topological RNA Structures is a difficult nonconvex optimization problem. A popular convex relaxation is based on minimization of the nuclear We study properties of topological RNA structures, norm (sum of singular values) of the matrix. For this re- i.e. RNA contact structures with cross-serial interactions laxation, an important question is when the two optimiza- that are filtered by their topological genus. RNA secondary tion problems lead to the same solution. Our focus lies on structures within this framework are topological structures the completion of structured matrices with fixed pattern of having genus zero. We derive a new bivariate generating missing values, in particular, on Hankel and quasi-Hankel function whose singular expansion allows us to analyze the matrix completion. The latter appears as a subproblem distributions of arcs, stacks, hairpin- , interior- and multi- in the computation of symmetric tensor canonical polyadic loops. We then extend this analysis to H-type pseudoknots, decomposition. In the talk, we report recent results on the kissing hairpins as well as 3-knots and compute their re- performance of the nuclear norm minimization for comple- spective expectation values. Finally we discuss our results tion of rank-r complex Hankel and quasi-Hankel matrices. and put them into context with data obtained by uniform sampling structures of fixed genus.

Konstantin Usevich Thomas Li GIPSA-Lab Virginia Tech [email protected] Biocomplexity Institute [email protected] Pierre Comon GIPSA-Lab CNRS Christian Reidys Grenoble, France Biocomplexity Institute [email protected] Virginia Tech [email protected]

MS91 A Boolean Network Model of the L-arabinose MS91 Operon Finding Steady-states Solution of Fibrin Network Dynamics with Numerical Algebraic Geometry In genetics, an operon is a segment of DNA that contains several co-transcribed genes, which together form a func- Mechanical properties of fiber network are an essential fac- tional regulatory unit. Operons have primarily been stud- tor in determining the stability of blood clots. In the first ied in prokaryotes, with both the lactose (lac) and trypto- part of this talk, I will present a simple ODEs kinetic model phan (trp) operons in E. coli having been classically mod- for a network of Fibrin fibers, will summarize the results eled with differential equations and more recently, with obtained with the model. The second part of the talk will Boolean networks. The L-arabinose (ara) operon in E. coli instead focus on how this type of problems can benefit from encodes proteins that function in the catabolism of ara- Numerical Algebraic Geometry. In particular, after giving binose. It differs from the lac and trp operons in that it a brief review on some of the methods and the software AG17 Abstracts 135

BertiniTM, I will illustrate how we can reduce the com- Design. Ray tracing, collision detection, and solid model- putational time need for simulations with the previously ing all benefit from implicitization procedures for rational described network model. surfaces. The univariate resultant of two moving lines gen- erated by a μ-basis for a rational curve represents the im- Francesco Pancaldi plicit equation of the rational curve. But although the mul- University of Notre Dame tivariate resultant of three moving planes corresponding to Department of Applied and Computational Mathematics a μ-basis for a rational surface is guaranteed to contain the and Stat implicit equation of the surface as a factor, μ-bases for ra- [email protected] tional surfaces are difficult to compute. Moreover, μ-bases for a rational surface often have high degrees, so these resul- tants generally contain many extraneous factors. Here we MS92 develop fast algorithms to implicitize rational tensor prod- Rational Plane Curves, Projections, and Degree = uct surfaces by computing the resultant of three moving 6andMu=2 planes corresponding to three syzygies with low degrees. These syzygies are easy to compute, and the resultants of I will begin with the relation between the μ-invariant of a the corresponding moving planes generally contain fewer rational plane curve and projections from rational normal extraneous factors than the resultants of the moving planes scrolls, based on the work of Bernardi, Gimigliano and Id`a. corresponding to μ-bases. We predict and compute all the I will then focus on the case of degree = 6, where μ =2 possible extraneous factors that may appear in these resul- or 3. For each of these μ-values, the generic curve has 10 tants. Examples are provided to clarify and illuminate the rational double points. Is there some way the geometry of theory. these 10 points tells us whether μ is 2 or 3? I will present one answer based on 2-cycles of a rational map of degree Liyong Shen 5 from the projective line to itself. This is joint work with Chinese Academy of Science Carlos D’Andrea and Teresa Cortadellas. [email protected]

David Cox Ron Goldman Amherst College Rice University [email protected] Computer Science [email protected] MS92 Implicitizing Rational Tensor Product Surfaces by MS93 the Resultant of Three Moving Planes ED Degrees of Orthogonally Invariant Varieties

Implicitizing rational surfaces is a fundamental computa- In a 2016 paper, Drusvyatskiy, Lee, Ottaviani, and Thomas tional task in Computer Graphics and Computer Aided established a ”transfer principle” for orthogonally invariant Design. Ray tracing, collision detection, and solid model- matrix varieties. After extracting the essence of their proof, ing all benefit from implicitization procedures for rational we will report on ongoing work classifying the orthogonal surfaces. The univariate resultant of two moving lines gen- representations amenable to this approach. erated by a mu-basis for a rational curve represents the im- plicit equation of the rational curve. But although the mul- Arthur Bik tivariate resultant of three moving planes corresponding to Mathematisches Institut a mu-basis for a rational surface is guaranteed to contain Universit¨at Bern the implicit equation of the surface as a factor, mu-bases [email protected] for rational surfaces are difficult to compute. Moreover, mu-bases for a rational surface often have high degrees, Jan Draisma so these resultants generally contain many extraneous fac- Eindhoven University of Technology, The Netherlands tors. In this talk we develop fast algorithms to implicitize Department of Mathematics and Computer Science rational tensor product surfaces by computing the resul- [email protected] tant of three moving planes corresponding to three syzy- gies with low degrees. These syzygies are easy to compute, and the resultants of the corresponding moving planes gen- MS93 erally contain fewer extraneous factors than the resultants Data with Infinitely Many Critical Points of the moving planes corresponding to mu-bases. We show how to predict and compute all the possible extraneous The generic (finite) number of critical points of the Eu- factors that may appear in these resultants. Examples will clidean distance function from a data point to a variety is be provided to clarify and illuminate the theory. called the Euclidean distance degree. In this talk we inves- tigate the special locus of data points where the number of Ron Goldman critical points is not finite. This is the non-finiteness set Rice University of the projection function from the ED correspondence to Computer Science the data space. [email protected] Emil Horobet Department of Mathematics and Computer Science MS92 Sapientia Hungarian University of Transilvania Implicitizing Rational Tensor Product Surfaces us- [email protected] ing the Resultant of Three Moving Planes

Implicitizing rational surfaces is a fundamental computa- MS93 tional task in Computer Graphics and Computer Aided Critical Point Computations on Smooth Varieties: 136 AG17 Abstracts

Degree and Complexity Bounds [email protected], [email protected]

Let V ⊂ Cn be an equidimensional algebraic set and g be an n-variate polynomial with rational coefficients. Com- MS94 puting the critical points of the map that evaluates g at the Simplicial Manifolds with Small Valence points of V is a cornerstone of several algorithms in real algebraic geometry and optimization. Under the assump- A foam, regarded as a cell complex, is dual to a trian- tion that the critical locus is finite and that the projective gulation of space. Of particular interest in applications closure of V is smooth, we provide sharp upper bounds on are those foams that are tetrahedrally close packed (TCP), the degree of the critical locus which depend only on deg(g) that is, the dual is a tiling with tetrahedra that are close and the degrees of the generic polar varieties associated to to regular. A first step in understanding these structures V . Hence, in some special cases where the degrees of the is the study of manifold triangulations that are positively generic polar varieties do not reach the worst-case bounds, curved in a certain discrete sense. We entirely – combi- this implies that the number of critical points of the eval- natorially as well as geometrically – classify these trian- uation map of g is less than the currently known degree gulations, and show how metric geometry can be used to bounds. We show that, given a lifting fiber of V , a slight understand their properties. variant of an algorithm due to Bank, Giusti, Heintz, Lecerf, Matera and Solern´o computes these critical points in time which is quadratic in this bound up to logarithmic factors, Florian Frick linear in the complexity of evaluating the input system and Cornell University polynomial in the number of variables and the maximum ff[email protected] degree of the input polynomials. Frank H. Lutz Mohab Safey El Din Technische Universit¨at Berlin University Pierre and Marie Curie [email protected] [email protected] John Sullivan Department of Mathematics, TU Berlin, Germany MS93 [email protected] Distance Degree in Semialgebraic Geometry MS94 In a normed space (X, ν), a best approximation of x ∈ X Voronoi Geometry and Applications to Structure in a closed set C ⊂ X is an y ∈ C such that ν(x − y)= Classification dist(x, C). In this talk we consider problems related to best approximation in the case when C is an algebraic set in the n Many physical systems can be modeled as large sets of space R endowed with a strictly convex semi-algebraic 3 norm. We define the critical point correspondence and atom-like particles in R . Understanding how points can be prove that it is a semialgebraic set. Then we introduce arranged in space is thus a very natural problem, though ef- the distance degree as the (suitably defined) degree of this fective and tractable descriptions of these arrangements are semialgebraic set. We also sketch a possible approach to often difficult to come by. We suggest an unconventional thecasewhenC is semialgebraic. approach, based on Voronoi tessellations, for describing the arrangements of points in space, and will aim to highlight why this approach can avoid theoretical limitations of con- Margaret Stawiska-Friedland ventional methods. We also consider several short applica- Mathematical Reviews tions of this work in studying high temperature materials [email protected] and mechanisms.

Shmuel Friedland Emanuel A. Lazar Department of Mathematics ,University of Illinois, University of Pennsylvania Chicago Department of Materials Science and Engineering [email protected] [email protected]

MS94 MS94 Geometry of Flat Origami Triangulations On the Geometry of Steel

Rigid origami structures consist of a network of folds and Polycrystalline materials, such as metals, are composed of vertices joining unbendable plates. While the generic con- crystal grains of varying size and shape. Typically, the figuration space of such a structure can be arbitrarily com- occurring grain cells have the combinatorial types of 3- plicated, one can often gain insight into the allowed de- dimensional simple polytopes, and together they tile 3- formations by linearizing the geometric constraints. The dimensional space. We will see that some of the occurring singularity at the flat state makes this impossible. We grain types are substantially more frequent than others - study the infinitesimal second-order rigidity of such flat where the frequent types turn out to be ”combinatorially origami structures. We also present numerical statistical round”. Here, the topological classification of grain types results when their fold patterns are derived from Poisson- gives us a new starting point for the geometric microstruc- Delaunay triangulations. ture analysis of steel.

Bryan G. Chen, Christian Santangelo Frank H. Lutz University of Massachussetts Technische Universit¨at Berlin AG17 Abstracts 137

[email protected] [email protected]

MS95 MS95 Shapes of the Simplest Minimal Free Resolutions On the Resolution of Fan Algebras in P 1 × P 1 We construct explicitly a resolution of a fan algebra of prin- The study of graduate syzygies for 3 homogeneous poly- cipal ideals over a Noetherian ring for the case when the nomials is very well known, but the general (multihomo- fan is a proper rational cone in the plane. Under some mild geneous) context is of great interest but involves further conditions on the initial data, we show that this resolution difficulties and is very much unknown. In (2) the au- is minimal. thors have analysed, from a geometric and an algebraic perspective, the minimal free resolution of an ideal given Teresa Cortadellas by 3 bihomogeneous polynomials of bidegree d =(2, 1) Universitat de Barcelona, Facultat d’Educaci´o in R = k[X1,X2][X3,X4]. In particular, an accurate de- [email protected] scription of the non-Koszul syzygies is obtained from an 1 application of K¨unneth formula and Serre duality for P Carlos D’Andrea (3). In this talk, our goal is to give a detailed descrip- Universitat de Barcelona tion of the (multi)graded minimal free resolution of an [email protected] ideal I of R, generated 3 bihomogeneous polynomials de- fined by f =(f1,f2,f3) with bidegree (d1,d2), di > 0 Florian Enescu P1 × P1 andsuchthatV (I)isemptyin . We will pre- Georgia State University cise the shape of the resolution in degree d =(1,n), and [email protected] explain how non-genericity (factorization) of the fi’s de- termine the resolution. (1) Botbol, N. and Chardin, M. (2016). Castelnuovo-Mumford regularity with respect to MS95 multigraded ideals. arXiv:1107.2494, to appear in J. Al- Syzygies of Tensor Product Surfaces with Base- gebra. (2) Cox, D. A., Dickenstein, A., and Schenck, H. points (2007). A case study in bigraded commutative algebra, Lect. Notes Pure Appl. Math., 254:67–111. (3) Weyman, A tensor product surface is the image of a map λ : P1 × J. and Zelevinsky, A. Multigraded formulae for multigraded P1 → P3, such surfaces arise in geometric modeling and in resultants. J. Algebr. Geom., 3(4):569–597, 1994. this context it is important to know their implicit equation. Currently, syzygies are one of the main tools to find im- Nicolas Botbol, Alicia Dickenstein plicit equations of parameterized curves and surfaces. In Universidad de Buenos Aires this talk I will overview the main techniques to find im- [email protected], [email protected] plicit equations of tensor product surfaces using syzygies. Additionally, I will present recent results on the structure Hal Schenck of the syzygies that determine the implicit equation for Mathematics Department tensor product surfaces. It turns out that for tensor prod- University of Illinois uct surfaces with basepoints the degree of the syzygies that [email protected] determine the implicit equations is directly related to the geometry of the base locus of λ.

MS95 Eliana M. Duarte Bounding the Degrees of a Minimal μ-Basis for a University of Illinois at Urbana-Champaign Rational Surface Parametrization [email protected]

For an arbitrary rational surface parametrization MS96 ∈ F 4 P (s, t)=(a1(s, t),a2(s, t),a3(s, t),a4(s, t)) [s, t] On the Computation of Sparse Resultants over an infinite field F, we show the existence of a μ-basis 33 Recently, the notion of sparse resultant associated to n +1 with polynomials bounded in degree by O(d ), where Laurent polynomials with supports in finite subsets of Zn has been redefined and studied using multiprojective elimi- d =max(deg(a1), deg(a2), deg(a3), deg(a4)). nation theory. This approach produces more uniform state- ments and allows to extend known results to arbitrary col- Our proof depends on obtaining a homogeneous ideal in lections of supports. In this talk, we will present a new F[s, t, u] by homogenizing a1,a2,a3,a4. Making additional recursive construction to produce Macaulay style formulas assumptions on this homogeneous ideal we can obtain lower to compute the sparse resultant as a quotient of the de- bounds: terminant of a Sylvester type matrix by one of its minors • If the projective dimension of this homogeneous ideal without imposing any conditions on the support sets, sim- is one, then the bound for the μ-basis is d. plifying and generalizing the formulas in previous work by • If this homogeneous ideal has height three, then the D’Andrea (2002). bound for the μ-basis is in the order of O(d22). Carlos D’Andrea • When this homogeneous ideal is a general Artinian Universitat de Barcelona almost complete intersection, then the bound for the [email protected] μ-basis is in the order of O(d12). Gabriela Jeronimo Yairon Cid Ruiz Universidad de Buenos Aires and Conicet, Argentina University of Barcelona [email protected] 138 AG17 Abstracts

Martin Sombra modules Universitat de Barcelona [email protected] We investigate the MacWilliams extension theorem for codes over alphabets that have the structure of a finite Frobenius bimodule over a finite ring. More precisely, MS96 for various weight functions we study whether weight- Hilbert’s Nullstellensatz and Combinatorial Opti- preserving linear maps between such codes extend to mization Problems weight-preserving automorphisms on the entire ambient space. In order to do so we need to consider partitions Systems of polynomial equations have a surprisingly pro- induced by the given weight function and introduce their ductive historical relationship with combinatorial opti- character-theoretic dual. While the former is a partition of mization problems. In this talk, we survey models of com- M n, the ambient space of the code, the latter is a partition binatorial problems as systems of polynomial equations, in the character-module of Rn,whereR is the underlying and the associated theoretical and practical results of com- finite ring. The resulting duality theory of these partitions bining these models with Hilbert’s Nullstellensatz and lin- allows us to answer the extension theorem in the affirma- ear algebra. tive for various weight functions.

Susan Margulies Heide Gluesing-Luerssen Department of Mathematics University of Kentucky United States Naval Academy [email protected] [email protected] Tefjol Pllaha University of Kentucky MS96 Department of Mathematics A Local Verification Method for the Solutions of a [email protected] Zero-dimensional Polynomial Systems

In this talk, I will propose a symbolic-numeric algorithm MS97 to verify the existence of solutions of a polynomial system within a local region. That is, given a zero-dimensional Codes for Distributed Storage from 3-regular n Graphs polynomial system f1 = ···= fn =0inC and a polydisk ∈ Cn Δofradiusr centered at a point m , we propose an al- In this talk we consider distributed storage systems (DSSs) gorithm that aims to count the number k of solutions (with from a graph theoretic perspective. A DSS is constructed multiplicity) of the given system within Δ. The method by means of the path decomposition of a 3-regular graph might fail, however, in case of success, it yields the correct into P4 paths. The paths represent the disks of the DSS result under guarantee. Our algorithm always succeeds if r and the edges of the graph act as the blocks of storage. We is sufficiently small and Δ is well-isolating for a solution of deduce the properties of the DSS from a related graph and the system. A key ingredient of our approach is the compu- show their optimality. tation of a suitable lower bound of a polynomial mapping on the boundary of a polydisk in n-dimensional space. For Shuhong Gao, Fiona Knoll, Felice Manganiello,Gretchen this, we combine resultant computation and recent results L. Matthews on the arithmetic Nullstellensatz. Our bit complexity anal- Clemson University ysis indicates that the complexity of our approach mainly [email protected], [email protected], man- depends on local parameters such as the size of m and r,or [email protected], [email protected] the number k of the solutions in Δ. We are thus convinced that our method considerably improves upon known exact methods in cases where k is small compared to the degrees MS98 of the input polynomials. We implemented our algorithm Test Sets for Nonnegativity of Reflection-invariant for the special case of a bivariate system, and our experi- Polynomials ments seem to support the above claim. In this talk we will show that, among real polynomials Michael Sagraloff which are invariant under the action of a finite reflection MPII, Germany group, a large family of them, either of low degree or sparse, [email protected] are globally nonnegative provided they are nonnegative on the hyperplane arrangement (our test set) associated to the MS97 reflection group. We explore stronger conditions on the de- gree and the sparsity that reduce the size of the test set, On Semigroup Ideals and Generalized Hamming as it happens for the particular case of symmetric polyno- Weights of AG Codes mials. Some results related to ideals of numerical semigroups will Jose Acevedo be discussed and then their implications to the so-called Georgia Tech Feng-Rao numbers and the Hamming weights of AG-codes [email protected] will be explained. Maria Bras-Amor´os, Kwankyu Lee Mauricio Velasco Universitat Rovira i Virgili Universidad de los Andes [email protected], [email protected] [email protected]

MS97 MS98 Extension Theorems for Codes over Frobenius Bi- Determinantal Representations of Hyperbolic AG17 Abstracts 139

Plane Curves with Cyclic Invariance [email protected], [email protected]

Given a determinantal representation by means of a cyclic weighted shift matrix with complex entries, one can show MS99 the resulting polynomial is hyperbolic and invariant under Certification of Approximate Roots of Polynomial the action of the cyclic group. By properly modifying a Systems determinantal representation construction of Dixon (1902), we show for every hyperbolic polynomial with cyclic invari- The first part of this work is concerned with certifying that ance there exists a determinantal representation admitted a given point is near an exact root of an overdetermined via some cyclic weighted shift matrix with complex entries. polynomial system with rational coefficients. Our certifica- tion is based on hybrid symbolic-numeric methods to com- pute an exact rational univariate representation (RUR) of a component of the input system from approximate roots. Lillian F. Pasley The accuracy of the RUR is increased via Newton itera- North Carolina State University tions until the exact RUR is found, which we certify using [email protected] exact arithmetic. Since the RUR is well-constrained, we can use that to certify the given approximate roots using Konstantinos Lentzos α-theory. The second part focuses on certifying isolated Technical University of Dortmund singular roots of well-constrained polynomial systems with [email protected] rational coefficients. We use a determinantal form of the isosingular deflation. The resulting polynomial system is overdetermined, but the roots are now simple, thereby re- MS98 ducing the problem to the overdetermined case.

Symbolic Computation for Hyperbolic Polynomials Tulay Ayyildiz Akoglu, Agnes Szanto North Carolina State University Hyperbolic polynomials generalize univariate polynomials [email protected], [email protected] with only real roots to the multivariate case. Hyperbolic programming (HP) is the problem of computing the infi- Jon Hauenstein mum of a linear function when restricted to the hyperbol- University of Notre Dame icity cone of a hyperbolic polynomial, a generalization of [email protected] semidefinite programming. The talk will discuss an ap- proach based on symbolic computation for solving general instances of HP, relying on the multiplicity structure of the MS99 algebraic boundary of the cone, without the assumption Probabilistic Enumerative Geometry of determinantal representability of the hyperbolic poly- nomial. This allows us to design exact algorithms able In this talk I will discuss a novel approach to the enumer- to certify the multiplicity of the solution and the optimal ative geometry of flats (e.g. Schubert problems), adopting value of the linear function. a probabilistic approach. The motivation for a random study comes from Real Algebraic Geometry, where there is Naldi Simone,DanielPlaumann no “generic” situation and where the outcome of an enu- TU Dortmund merative problem strongly depends on the configuration [email protected], of the constraints (e.g. on a real cubic surface there are [email protected] 27 complex lines, but how many of these lines are real?). When working over the real numbers, the technique I will introduce will compute the√ average answer to the enumer- − MS98 ativeproblem(thereare6 2 3 real lines on a random cubic surface!) Over the complex numbers, where “ran- Slack Ideals of Polytopes dom” means “generic”, this technique offers a new way for computing in the cohomology ring of the Grassmannians In this talk we discuss a new tool for studying the real- (intersection numbers of Schubert cycles are expressed in ization spaces of polytopes, namely the slack ideal associ- terms of expected determinants of certain gaussian matri- ated to the polytope. These ideals were first introduced ces). to study PSD rank of polytopes, and their structure also (This is based on joint works with S. Basu, P. B¨urgisser, encodes other important polytopal properties, gives us a K. Kozhasov, E. Lundberg and C. Peterson) new way to understand important concepts such as projec- tive uniqueness, and suggests connections with the study Antonio Lerario of other algebraic and combinatorial objects (toric ideals SISSA and graphs, for example). [email protected]

Joao Gouveia MS99 Universidade de Coimbra Numerical Irreducible Decomposition for Multi- [email protected] projective Varieties

Antonio Macchia Subvarieties of a product of projective spaces (multipro- Universit`a degli Studi di Bari jective varieties) have natural representations in numerical [email protected] algebraic geometry through multiprojective witness sets, which have significantly lower complexity than a classi- Rekha Thomas, Amy Wiebe cal witness set for these varieties. Algorithms based on University of Washington multiprojective witness sets take advantage of the inherent 140 AG17 Abstracts

structure of multiprojective varieties. This talk will de- thresholds for events involving monomial ideals with given scribe numerical irreducible decomposition in this setting, Hilbert function, Krull dimension, first graded Betti num- paying attention to how it exploits the structure of multi- bers, and present several experimentally-backed conjec- projective varieties. It represents joint work with Leykin tures about regularity, projective dimension, strong gener- and Rodriguez. icity, and Cohen-Macaulayness of random monomial ideals.

Frank Sottile Texas A&M University Lily Silverstein [email protected] University of California, Davis [email protected]

MS99 A Numerical Method for Solving Real Bi- MS100 parametric Systems Randomization in Computational and Commuta- tive Algebra We consider all the real solutions of a bi-parametric poly- nomial system with parameter values restricted to a finite Randomization has been used in many fields to generate rectangular region, under certain assumptions. A curve typical objects and study their properties. It has also been called “the border curve” divides the region into connected harnessed to design algorithms that can solve problems subsets such that in each subset the real zero set of the more efficiently and more simply than known determin- polynomial system defines finitely many smooth functions, istic algorithms. We kick off this minisymposium by sur- whose graphs are disjoint. We propose a numerical method veying some results utilizing randomness in computational to compute real witness points on the border curve, which and commutative algebra. can generate the whole curve by path tracking. A set of multivariate polynomials together with its Jacobian matrix Despina Stasi multiplied by a vector of auxiliary variables often appears Illinois Institute of Technology in many applications. Such structure is the so-called Ja- [email protected] cobian structure. Interestingly, a recursive Jacobian struc- ture appears in our formulation to obtain the real witness points. An estimation of root counting is presented in this MS100 paper which leads to a homotopy continuation method. Random Numerical Semigroups The experimental results of a preliminary C++ implemen- tation of the proposed method are also given to illustrate A numerical semigroup is a subset of the natural numbers the effectiveness. Meanwhile, a numerical error estimation that contains zero and is closed under addition. Numerical of the computed border curve is also provided. semigroups have been studied for over 150 years and have connections to number theory, combinatorics, commutative Wenyuan Wu algebra, and algebraic geometry. In this brief talk, I will Chongqing Institute of Green and Intelligent Technology introduce a model for randomly generating numerical semi- Chinese Academy of Sciences groups, present some results that describe the structure of [email protected] numerical semigroups arising from the model, and provide some motivation for studying numerical semigroups from a Changbo Chen, Yong Feng probabilistic perspective. This is joint work with Jesus De CIGIT,CAS Loera and Chris O’Neill. [email protected], [email protected] Dane Wilburne Illinois Institute of Technology MS100 [email protected] Syzygies of Random Stanley-Reisner Ideals Jesus De Loera We study the syzygies of Stanley-Reisner ideals associated University of California, Davis to random flag complexes. We use these to provide fur- [email protected] ther examples of asymptotic nonvanishing of syzygies, in the mannar of Ein and Lazarsfeld. In addition, we use Chris O’Neill this construction to produce the first higher dimensional UC Davis family of examples where the syzygies become ”normally [email protected] distributed”, in the sense conjectured by Ein, Erman, and Lazarsfeld. MS101 Daniel M. Erman Truncated Moment Problems for Unital Commu- University of Michigan tative Real Algebras [email protected] In this talk I introduce some infinite dimensional version of the classical truncated moment problem which naturally MS100 arise in several applied fields. The general theoretical ques- Random Monomial Ideals tion addressed is whether a linear functional L defined on a linear subspace B of a unital commutative real algebra Inspired by the study of random graphs and simplicial com- A admits an integral representation with respect to a non- plexes, and motivated by the need to understand aver- negative Radon measure supported on a fixed closed sub- age behavior of ideals, we propose and study probabilis- set K of the character space of A. I present a recent joint tic models of random monomial ideals. We prove theo- work with M. Ghasemi, S. Kuhlmann and M. Marshall rems about the probability distributions, expectations and where we determine a class of linear functionals which ad- AG17 Abstracts 141

mit such integral representations, using solutions to contin- tinian Gorenstein algebras and Hankel operators of finite uous full moment problems on seminormed algebras. I will rank, a generalization of Kronecker theorem, decompo- also sketch some applications of this general result to the sition algorithms to recover the frequencies and weights classical finite dimensional case and to problems occurring of a polynomial-exponential functions from its first mo- e.g. in mathematical physics and in optimization. ments and some applications to different problems such as the decomposition of measures in weighted sums of Dirac Maria Infusino measures, sparse interpolation, fast decoding of algebraic University of Konstanz codes, and tensor decomposition. The relaxation of this [email protected] type of problems into convex optimization problems in super-resolution approaches will be discussed.

MS101 Bernard Mourrain Moment Problems for Symmetric Algebras of Lo- INRIA Sophia Antipolis cally Convex Spaces [email protected]

In this talk I will explain how a locally convex (lc) topology τ on a real vector space V extends to a locally multiplica- MS102 tively convex (lmc) topology τ on the symmetric algebra Persistence of Convex Codes, Inferring the Rank S(V ). This allows an application of the results on lmc of Non-linear Matrix Factorizations and the Space topological algebras obtained by Ghasemi, Marshall and of Smells myself to get representations of τ-continuous linear func- tionals L : S(V ) → R satisfying L( S(V )2d) ⊆ [0, ∞) Many neural systems in the brain generate neural activity (more generally, L(M) ⊆ [0, ∞)forsome2d−power mod- that can be characterized mathematically as convex codes. ule M of S(V )) as integrals with respect to uniquely de- [n] termined Radon measures supported by special sorts of A convex code is a subset of the power set 2 that arises closed balls in the dual space of V. This result is simulta- from intersection patterns of convex sets in a Euclidean neously more general and less general than the correspond- space. These codes retain topological features of the under- ing result of Berezansky, Kondratiev and Sifrin. It is more lying space of stimuli. What combinatorial properties of a general because V can be any lc topological space (not code determine its convexity and its embedding dimension just a separable nuclear space), the result holds for arbi- is only partially understood. In this talk I will explain trary 2d−powers (not just squares), and no assumptions of how hyperplane codes (a special class of convex codes) are quasi-analyticity are required. It is less general because it related to the problem of detecting a hidden matrix fac- is necessary to assume that L : S(V ) → R is τ−continuous torization, hidden by an unknown monotone non-linearity. (not just continuous on each homogeneous part of S(V )). This relationship is accomplished via a zigzag of neural codes, that is similar to the Dowker complex filtration. I Salma Kuhlmann will then present an algebraic approach and computational University of Konstanz , Germany results for estimating the dimension of the space of smells, [email protected] as perceived by an animal, from experimental data. Vladimir Itskov MS101 The Pennsylvania State University Structured Matrices Arising in Multivariate Poly- [email protected] nomial Problems

In this talk we will consider some linear algebra problems MS102 having a structure derived from problems in multivariate Grobner Bases of Neural Ideals polynomial systems. Examples include rank and nullify computations along with fast algorithms. The brain processes information about the environment via neural codes. These codes often come from the case in George Labahn which each neuron has a region of stimulus space, called University of Waterloo its receptive field, in which it fires at a high rate. Much [email protected] research has focused on understanding what features of re- ceptive fields can be extracted directly from a neural code. Bernhard Beckermann In particular, Curto, Itskov, Veliz-Cuba, and Youngs re- Universit´e de Lille cently introduced the concept of neural ideal, which is an Math´ematiques algebraic object that encodes the full combinatorial data of [email protected] a neural code. Every neural ideal has a particular generat- ing set, called the canonical form, that directly encodes a Evelyne Hubert minimal description of the receptive field structure intrinsic INRIA M´editerran´ee to the neural code. On the other hand, for a given mono- [email protected] mial order, any polynomial ideal is also generated by its unique (reduced) Grbner basis with respect to that mono- mial order. This talk pertains to the question of how these MS101 two types of generating sets canonical forms and Grb- Decomposition of Polynomial-exponential Series ner bases are related. Our main result states that if the from Moments canonical form of a neural ideal is a Grbner basis, then it is the universal Grbner basis (that is, the union of all The decomposition of multivariate polynomial-exponential reduced Grbner bases). Furthermore, we prove that this functions from truncated series of moments is a prob- situation occurs precisely when the universal Grbner basis lem which appears in many contexts. We will present contains only pseudo-monomials (certain generalizations of an algebraic setting, giving a correspondence between Ar- monomials). Along the way, we develop a representation 142 AG17 Abstracts

of pseudo-monomials as hypercubes in a Boolean lattice. ting, the stimulus space can be visualized as subset of R2 covered by a collection U of convex sets such that the ar- Rebecca Garcia rangement U forms an Euler diagram for C.Thereare Departme of Mathematics and Statistics some methods to determine whether such a convex real- Sam Houston State University ization U exists; however, these methods do not describe [email protected] how to draw a realization. In this work, we look at the problem of algorithmically drawing Euler diagrams for neu- Luis David Garcia Puente ral codes using two polynomial ideals: the neural ideal, a Sam Houston State University pseudo-monomial ideal; and the neural toric ideal, a bi- [email protected] nomial ideal. In particular, we study how these objects are related to the theory of piercings in information visu- Ryan Kruse alization, and we show how minimal generating sets of the Central College ideals reveal whether or not a code is 0, 1, or 2-inductively [email protected] pierced.

Jessica Liu Nora Youngs Bard College Colby College [email protected] Department of Mathematics and Statistics [email protected] Dane Miyata Willamette University Elizabeth Gross [email protected] San Jose State University [email protected] Ethan Petersen Rose-Hulman Institute of Technology MS103 [email protected] Algorithm for Computing μ-Bases of Univariate Kaitlyn Phillipson Polynomials St. Edwards University [email protected] We present a new algorithm for computing a μ-basis of the syzygy module of n polynomials in one variable over K Anne Shiu an arbitrary field . The algorithm is conceptually dif- Texas A&M University ferent from the previously-developed algorithms by Cox, [email protected] Sederberg, Chen, Zheng, and Wang for n =3,andby Song and Goldman for an arbitrary n. The algorithm in- volves computing a “partial’ reduced row-echelon form of a MS102 (2d +1)× n(d +1)matrixover K,whered is the maximum Real Multigraded Modules as Data Structures for degree of the input polynomials. The proof of the algo- Fly Wings rithm is based on standard linear algebra and is completely self-contained. Comparison with the Song-Goldman algo- We use two-parameter persistent homology to produce rithm will be given. data usable for statistical analysis from pictures of fly wings. The goal of the analysis is to give an example of Zachary Hough, Hoon Hong, Irina Kogan continuously evolving traits crossing a threshold to cause North Carolina State University a large or discrete morphological jump. We hypothesize [email protected], [email protected], iako- that topological novelty in wing vein structure arises when [email protected] directional selection pushes continuous variation in a de- velopmental program beyond a certain threshold. The fly wing persistence modules are not like those MS103 from one-parameter persistence: they are infinitely Computing Singularities of Rational Curves and (co)generated and are modules over a non-Noetherian ring. Surfaces using their Mu-bases To deal with such difficulties, we discuss a replacement for the Noetherian chain condition – finite encoding – and pro- pose a two-parameter version of the barcode – QR code – The technique of mu-bases is originated from the theory of to summarize the fly wings for statistical analysis. moving curves and moving surfaces, which provides a con- nection between the parametric forms and implicit forms Justin Curry, Ezra Miller, Ashleigh Thomas of curves and surfaces due to their special algebraic and Duke University geometric properties. Mu-bases are new applications of Department of Mathematics the theory of Syzygy module in Computer Aided Geomet- [email protected], [email protected], ath- ric Design and Geometric Modeling, which facilitate many [email protected] problems such as implicitization, inversion formula, as well as singularity computation of rational curves and ratio- nal surfaces. We shall investigate the intrinsic connection MS102 between mu-bases and singularities for rational curves and Neural Ideals and Stimulus Space Visualization surfaces and develop symbolic algorithms of computing sin- gularities and their orders of rational curves and surfaces. AneuralcodeC is a collection of binary vectors of a given length n that record the co-firing patterns of a set of neu- rons. Our focus is on neural codes arising from place cells, Xiaohong Jia neurons that respond to geographic stimulus. In this set- Chinese Academy of Science AG17 Abstracts 143

[email protected] University of California, Berkeley [email protected]

MS103 Jose I. Rodriguez Degree-optimal Moving Frame for Rational Curves Department of Statistics University of Chicago We present an algorithm that, for a given vector a of n rel- [email protected] atively prime polynomials in one variable over an arbitrary × field, outputs an n n invertible matrix P with polynomial Serkan Hosten entries, such that it forms a degree-optimal moving frame Department of Mathematics for the rational curve defined by a. The first column of San Francisco State University the matrix P consists of a minimal-degree B´ezout vector (a [email protected] minimal-degree solution to the univariate effective Nullstel- lensatz problem) of a,andthelastn − 1 columns comprise an optimal-degree basis, called a μ-basis, of the syzygy MS104 module of a. To develop the algorithm, we prove several Maximum Likelihood Estimation for Cubic and new theoretical results on the relationship between opti- Quartic Canonical Toric Del Pezzo Surfaces mal moving frames, minimal-degree B´ezout vectors, and μ-bases. In particular, we show how the degrees bounds This article is concerned with the problem of Maximum of these objects are related. Comparison with other algo- Likelihood Estimation for algebraic statistical models with rithms for computing moving frames and B´ezout vectors singularities, in particular those which correspond to toric will be given, however, we are currently not aware of an- del Pezzo surfaces with Du Val singular points. The im- other algorithm that produces an optimal degree moving portance of algebraic statistical models corresponding to frame or a B´ezout vector of minimal degree. toric varieties is due to their relation to log-linear statisti- cal models which are widely used in Statistics and areas like Irina Kogan, Hoon Hong, Zachary Hough natural language processing when analysing cross-classified North Carolina State University data in multidimensional contingency tables. Another rea- [email protected], [email protected], [email protected] son for studying the MLE for such algebraic statistical models, is that they correspond to singular varieties. Sin- Zijia Li gularities play an important role in statistical inference as Johannes Kepler University the commonly assumed smoothness of algebraic statisti- [email protected] cal models is very restrictive and is almost never satisfied for models of statistical relevance. For instance, algebraic statistical models of binary symmetric phylogenetic three- MS104 valent trees are proven to be Fano varieties with Gorenstein Maximum Likelihood Degrees for Discrete Random singularities, which is another motivation for studying del Models Pezzo surfaces with Du Val singularities. In this article a closed-form for the Maximum Likelihood Estimate of al- The goal of this talk is to highlight a theorem proved during gebraic statistical models which correspond to cubic and the Mathematical Research Community: Algebraic Statis- quartic toric del Pezzo surfaces with Du Val singular points tics summer program by the Likelihood Geometry group. is given. This group was led by Serkan Hosten and Jose Israel Ro- driguez. In the talk, I will focus on the application of the Dimitra Kosta theorem to graphical models. The University of Edinburgh [email protected] Courtney Gibbons Hamilton College [email protected] MS104 Introduction to Maximum Likelihood Degrees

MS104 Maximum likelihood estimation is a fundamental problem Topological Invariants and the Maximum Likeli- in statistics. By studying the geometry of this problem hood Degree one is able to gain new insights into the problem. In this introductory talk, the maximum likelihood degree will be Let XA be the projective toric variety defined by an integer defined. With a running example, we will show how to d×n matrix A of rank d and let c be a general element of the determine the ML degree by solving a system of equations associated dimension n complex algebraic torus. We show and by computing an Euler characteristic. that the maximum likelihood (ML) degree of the variety c · XA obtained by the torus action of c on XA can be seen Jose Rodriguez as a coefficient of a particular component of the Chern- University of Chicago Mather class of XA. This realization allows us to determine [email protected] a, so called, ML discriminant for c · XA.Inparticularwe show that if the principal A-determinant, EA,ofXA does not vanish at c we have that the ML degree of c · XA is MS105 given by the normalized volume of the polytope obtained Periodic Auxetics: Elliptic Curves, Convexity Con- by taking the convex hull of the matrix A.Usingthiswe straints and Framework Generation also confirm a known relation between the ML degree of c · XA and the Euler characteristic of an associated very Crystalline framework materials and synthetic structures affine variety. exhibit auxetic behavior when stretching in some direction entails lateral widening. We show that all periodic bar-and- Martin Helmer joint frameworks with auxetic deformations are subject to 144 AG17 Abstracts

rigorous convexity constraints and then use these neces- [email protected] sary characteristics to generate an infinite virtual catalog of auxetic designs. For three degrees of freedom, we discuss a recognition algorithm based on elliptic curves. MS105 Auxetic and Expansive: Beyond Two Dimensions Ciprian S. Borcea Department of Mathematics Periodic frameworks with auxetic behavior possess the un- Rider University usual property of becoming wider, rather than thinner, [email protected] whenstretchedinsomedirection.Inanexpansiveperi- odic framework, all pairwise distances between joints in- Ileana Streinu crease simultaneously, and hence expansive implies aux- Department of Computer Science etic. A complete (combinatorial and geometric) character- Smith College ization of expansive frameworks in dimension two as peri- [email protected] odic pseudo-triangulations has led to a simple method for generating such structures. Yet in higher dimensions the characterization of expansive frameworks remains elusive MS105 and the design problem (both for auxetic and expansive Defects in Three Dimensional Smectics and their structures) appears to be much harder: only a handful of Combination Rules such examples can be found in the materials science litera- ture. We present recent advances which lead, in particular, As materials with broken translational symmetry, defects to infinitely large families of such designs in arbitrary di- in smectic liquid crystals do not follow the traditional ho- mensions. motopy theoretic classification scheme, and a more geo- metrical approach is required. Using methods from sin- Ciprian S. Borcea gularity theory we study the topological classification and Department of Mathematics combination rules for point and line defects in (two and) Rider University three dimensional smectics. We give a local classification [email protected] of defect structures for both point and line defects using a graph theoretical representation, and construct the path- Ileana Streinu dependent algebra of defect combinations in terms of cer- Department of Computer Science tain binary operations on the graphs. In particular, our Smith College work shows that defect morphology is surprisingly rich in [email protected] three dimensional smectics, with traditional invariants fail- ing to distinguish many topologically distinct defects. MS106 Thomas Machon Castelnuovo-Mumford Regularity vs. Virtual Co- Physics&Astronomy homological Dimension University of Pennsylvania [email protected] This talk is based on a surprising connection between a question in commutative algebra regarding Castelnuovo- Mumford regularity, and a question of Gromov about hy- MS105 perbolic Coxeter groups. We show that the virtual co- Twisted Topological Tangles or: The Knot Theory homological dimension of a Coxeter group is essentially of Knitting the same as the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of its nerve. Using this, we modify At first glance, knitted fabrics have a the symmetry of a a construction of Osajda in group theory to find for every rectangular lattice, yet upon closer inspection, the filamen- positive integer r a monomial ideal generated in degree two, tous character of the yarn imbues the fabric with a much with linear syzygies, and regularity of the quotient equal to richer geometric and topological structure. The topologi- r. Previously known examples had regularity less than 5. cal nature of knitted stitches derives its richness from knot For Gorenstein ideals we prove that the regularity of their theory, with the added complication that the structure is quotients can not exceed four, thus showing that for d>4 periodic. As with traditional molecular crystals, topolog- every triangulation of a d-manifold has a hollow square ical defects in the knitted lattice act as point sources of or simplex. We also show that for most monomial ideals surface curvature. However, the multiplicity of defects ex- generated in degree two and with linear syzygies the regu- tends to both achiral and chiral knots. These additional larity is O(log(log(n)), where n is the number of variables, contributions couple with the in-plane elasticity intrinsic improving in this case a bound found by Dao, Huneke and to the fabric to create more complex curvatures. We aim Schweig. All results are in collaboration with with Thomas to use knot theory to classify all possible topological de- Kahle and Matteo Varbaro. fects in knitted fabrics. Using elasticity theory, the ideal configuration of a knit stitch gives insight into the local ef- Alexandru Constantinescu fective curvature. In addition to the theoretical endeavour, University of Genova we use principles from chiral liquid crystalline defects to [email protected] understand the directed self-assembly of interwoven struc- tures. Our ultimate goal is to solve the inverse problem by creating an algorithm to stitch defects together to gener- MS106 ate a pattern for a fabric that will clothe an object of any Resolutions Associated to a Matrix of Linear Forms geometry. which is Annihilated by a Vector of Indeterminates

Elisabetta A. Matsumoto Let R = k[T1,...,Tf ] be a standard graded polynomial Applied Mathematics ring over the field k and Ψ be an f × g matrix of linear Harvard University forms from R,where1≤ g

and that gradeIg(Ψ) is exactly one short of the maximum Anna Lorenzini possible grade. We resolve R = R/Ig(Ψ), prove that R University of Perugia has a g-linear resolution, record explicit formulas for the h- [email protected] vector and multiplicity of R, and prove that if f −g is even, then the ideal Ig(Ψ) is unmixed. Furthermore, if f − g is Juan Migliore odd, then we identify an explicit generating set for the un- Notre Dame University unm unm mixed part, Ig(Ψ) ,ofIg(Ψ), resolve R/Ig(Ψ) ,and [email protected] unm record explicit formulas for the h-vector of R/Ig(Ψ) . unm (The rings R/Ig(Ψ) and R/Ig(Ψ) automatically have Uwe Nagel the same multiplicity.) These results have applications to University of Kentucky the study of the blow-up algebras associated to linearly [email protected] presented grade three Gorenstein ideals. Justina Szpond Andrew Kustin Pedagogical University of Cracow University of South Carolina [email protected] [email protected] Adam Van Tuyl MS106 McMaster University [email protected] Properties of Symmetric Ideals

Ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group MS107 arise in various contexts, including algebraic statistics and Chebyshev Multivariate Sparse Interpolation representation theory. Often it is advantageous to consider the larger class of ideals that are invariant under the action Given a polynomial that can be evaluated at chosen points, of the monoid of strictly increasing functions. We discuss sparse interpolation offers to first recover the support of recent results describing properties of such ideals. They this polynomial, the cardinal of which is known, or at least maybeviewedaspartofanefforttodevelopcommutative bounded. We examine the situation where this support algebra with an eye towards symmetry. consists of generalized multivariate Chebyshev polynomi- als. Uwe Nagel University of Kentucky Evelyne Hubert [email protected] INRIA M´editerran´ee [email protected] MS106 Michael Singer When are the Symbolic Powers of an Ideal Equal North Carolina State University to its Ordinary Powers? [email protected] The symbolic powers of a radical ideal are the sets of poly- nomial functions vanishing to a prescribed order on the MS107 variety defined by the ideal. It is generally difficult to de- scribe generators for the symbolic powers of an ideal and it Irredundant Decomposition of Radical Ideals via is interesting to describe the precise relationship between Triangular Sets the symbolic powers and the ordinary powers of an ideal. In this talk, we are concerned with the simplest possible This is joint work with Agnes Szanto. Representing the relationship, namely when the symbolic powers of an ideal radical of a given polynomial ideal or, equivalently, the coincide with its ordinary powers. We classify the codimen- corresponding affine variety, is one of fundamental prob- sion two locally complete intersection ideals that satisfy lems in a computational algebraic geometry. One of stan- this strong property. The key tool to prove this classifi- dard approaches to this problem is via triangular sets. For cation is the ability to construct graded minimal free res- recent applications in theory of algebraic differential equa- olutions for the ordinary powers of the ideal under these tions, see [Ovchinnikov, Pogudin, Vo, ”Effective differential hypotheses. elimination and Nullstellensatz”, arXiv:1610.04022]. In her Ph.D. thesis, Agnes Szanto proposed an algorithm for com- Alexandra Seceleanu puting such representation, which extensively used multi- University of Nebraska - Lincoln variate projective resultants and univariate subresultants. [email protected] For an ideal generated by polynomials in n variables of total degree at most d with prime components of codimen- Susan Cooper sion at most m, it computes a representation such that North Dakota State University the degrees of elements of the triangular sets does not ex- ( 3) [email protected] ceed ndO m and the number of the triangular sets does 2 not exceed nO(m)dO(m ). We improve this algorithm. The Giuliana Fattabi main features of the improved algorithm are 1. degrees of 2 University of Perugia the polynomials in the output are bounded by ndO(m ); [email protected] 2. the resulting representation is irredundant in the sense that every component of every triangular set in the output Elena Guardo does not lie in any other triangular set; 3. the previous University of Catania property implies that the number of triangular sets in the [email protected] output does not exceed the number of irreducible compo- 146 AG17 Abstracts

nents of the initial variety, so is bounded by dm. [email protected]

Gleb Pogudin Johannes Kepler University, Linz PP1 [email protected] Metric Reconstruction Via Optimal Transport

Given a sample of points X in a metric space M and a MS107 scale r>0, the Vietoris–Rips simplicial complex VR(X; r) Sylvester Double Sums and Subresultants in the is a standard construction to attempt to recover M from X General Case up to homotopy type. A deficiency of this approach is that VR(X; r) is not metrizable if it is not locally finite, and Sylvester doubles sums, introduced first by Sylvester, are thus does not recover metric information about M.We symmetric expressions of the roots of two polynomials. attempt to remedy this shortcoming by defining a met- This classical definition of double sums makes no sense in ric space thickening of X,whichwecalltheVietoris–Rips the presence of multiple roots, since the definition is in- thickening VRm(X; r), via the theory of optimal trans- volving denominators that vanish when there are multiple port. When M is a complete Riemannian manifold, we roots. The aim of this paper is to give a general definition show the the Vietoris–Rips thickening satisfies Hausmann’s of Sylvester double sums, which coincide (up to a sign) with theorem (VRm(X; r)  M for r sufficiently small) with the classical definition, as well as a general proof of the re- a simpler proof: homotopy equivalence VRm(X; r) → M lationship between double sums and subresultants: they is canonically defined as a center of mass map, and its are equal up to a constant. In the simple case root case, homotopy inverse is the (now continuous) inclusion map proofs of this property were known (see [Lascoux, Pragacz, M → VRm(X; r). Furthermore, we describe the homo- Double Sylvester sums for subresultants and multi-Schur topy type of the Vietoris–Rips thickening of the n-sphere fonctions, JSC 2003], [d’Andrea, Hong, Krick, Szanto, An at the first positive scale parameter r where the homotopy Elementary Proof of Sylvester’s Double Sums for Subre- type changes. sultants, JSC 2007], [Roy, Szpirglas, Sylvester double sums and subresultants, JSC 2011] ). The general proof given Henry Adams here is by induction on the length of the remained sequence Stanford University of P and Q and is based on the link between double sums [email protected] of two P and Q and double sums of Q and R,whereR is opposite of the remainder of P and Q, this link being Florian Frick similar to the link between subresultants of P and Q and Cornell University subresultants of Q and R. A generalization of Hermite In- ff[email protected] terpolation is introduced for the proof of this link. Joint work with M.-F. Roy (Universit´edeRennes) Michal Adamaszek Aviva Szpirglas University of Copenhagen Universit´edePoitier [email protected] [email protected] PP1 MS107 Identifying Species Hybridization Network Fea- Sylvester Sums in Multiple Roots and Subresul- tures from Gene Tree Quartets tants A phylogenetic tree is not always enough to describe the Given two univariate polynomials with single roots, relation between species, in particular, when hybridization, the well-known relationships between subresultants and horizontal gene transfer or gene flow occurs. Phylogenetic Sylvester sums give formulas for the subresultant in terms networks are the objects used to represent the relation be- of the sets of the roots of the polynomials. Here we gener- tweenspeciesthatadmitsuchevents.Wefocusonphylo- alize Sylvester single sums to multisets (sets with repeated genetic networks whose cycles do not share edges, known elements) and show that these sums compute subresultants as level-1 networks. Under the coalescent model on a level- of the polynomials as a function of their roots indepen- 1 network, the probabilities of gene tree quartets can be dently of their multiplicity structure. This is a closed for- computed in terms of the probabilities arising on simpli- mula for subresultants in terms of roots that works for ar- fied networks. Using only the natural descending order in bitrary polynomials. Our extension involves in some cases the real line of such probabilities for each 4-taxon set, we confluent Schur polynomials. can generically identify all cycles of size greater than 3, and hybrid nodes in the cycles of size greater than 4, in Marcelo A. Valdettaro the unrooted species network. We also show that we can- Universidad de Buenos Aires, Argentina not identify the hybrid node of a 4-cycle by this approach. [email protected] Hector Banos Carlos D’Andrea University of Alaska, Fairbanks Universitat de Barcelona [email protected] [email protected]

Teresa Krick PP1 Universidad de Buenos Aires and IMAS, CONICET Matroidal Root Structure of Skew Polynomials [email protected] over Finite Fields

Agnes Szanto A skew polynomial ring R = K[x; σ, δ] is a ring of polyno- North Carolina State University mials with non-commutative multiplication. This creates a AG17 Abstracts 147

difference between left and right divisibility, and thus a con- is referred to as the steady-state variety in algebraic sys- cept of left and right evaluations and roots. A polynomial tems biology. This allows us to study certain properties of in such a ring may have more roots than its degree, which the steady-state variety, such as its degree. Furthermore, leads to the concepts of closures and independent sets of ReactionNetworks provides a bank of common motifs and roots. There is also a structure of conjugacy classes on the methods for determining specific properties and computing roots. In R = Fqm [x, σ], this leads to matroids of right in- various invariants of a given reaction network. dependent and left independent sets. These matroids are isomorphic via the extension of the map φ :[1]→ [1] de- qi−1−1 Cvetelina Hill fined by φ(a)=a q−1 . Additionally, extending the field Georgia Institute of Technology of coefficients of R results in a new skew polynomial ring [email protected] S of which R is a subring, and if the extension is taken to include roots of an evaluation polynomial of f(x), then all roots of f(x)inS are in the same conjugacy class. Elizabeth Gross San Jose State University Travis A. Baumbaugh, Felice Manganiello [email protected] Clemson University tbaumba@clemson,edu, [email protected] Timothy Duff Georgia Institute of Technology tduff[email protected] PP1 The Positive Bergman Complex of An Oriented MatroidwithValuation PP1 We study a generalization of the positive Bergman com- plex of Ardila, Klivans, and Williams, and the tight span of Dress or tropical linear space of Speyer, for oriented ma- Sixty-four Curves of Degree Six troids with valuation. We characterize those valuations of an oriented matroid M which yield a regular subdivision of the matroid polytope of M whose faces correspond to In this paper, we have done a computational study of oriented matroids which inherit the orientation of M. smooth curves of degree six in the real projective plane. There are 56 known topological types of sextic curves. Marcel Celaya, Josephine Yu These 56 types are refined into 64 rigid isotopy classes. School of Mathematics For each of these 64 classes we have constructed a repre- Georgia Institute of Technology sentative polynomial. These polynomials are verified using [email protected], [email protected] a fast CAD sextic classifier coded in Mathematica. We also looked at the empirical probability distributions on the var- ious types. Degree six planar curves have 324 bitangents. PP1 Of these 324 bitangents we studied the real bitangents and Sampling and Persistence Diagrams the possible numbers of those bitangents for each type. In the projective plane, lines that do not intersect a given When presented with a point cloud sampled noisily from sextic form the avoidance locus of the said curve. This is a a smooth manifold, a fundamental objective is to learn as union of upto 46 convex regions which is bounded by the much about the morphology of the manifold as possible. dual curve of degree 30 of the given sextic. Inferring the structure of such a surface, given only a raw On a large sample for each of the types we computed the point cloud, is essential to overcoming common machine real inflection points of a sextic which can atmost be 24. learning challenges such as classification and prediction. We also computed real tensor eigenvectors and what are Persistence diagrams have emerged as valuable tools in this the possible numbers of tensor eigenvectors each type can pursuit. The persistence diagram is a multi-set that serves attain. We studied the tentative real tensor rank for each as a coarse descriptor of topological features, in essence ho- of the representatives and also looked at the construction mology groups, of the underlying surface. The distribution of quartic surfaces. according to which the observed points are sampled factors into the fidelity of the reconstruction of the manifold. We discuss existing results regarding the question: does the Nidhi Kaihnsa, Mario Kummer probability measure on a manifold induce an interesting Max Planck Institute, Mathematics in the Sciences, probability measure on the space of persistence diagrams? Leipzig [email protected], [email protected] Kathryn H. Heal Harvard University SEAS Daniel Plaumann [email protected] TU Dortmund [email protected] PP1 ReactionNetworks.m2 - A Package for Algebraic Mahsa Sayyary Systems Biology Max Planck Institute, Mathematics in the Sciences, Leipzig The Macaulay2 package ReactionNetworks provides the [email protected] necessary tools to study chemical reaction networsk, mod- elsusedinsystemsbiology,fromanalgebraicpointof Bernd Sturmfels view. In particular, our package provides a quick and easy Max Planck Institute, Mathematics in the Sciences, way to construct the steady-state and conservation equa- Leipzig tions for a reaction network; such equations define what UC, Berkeley 148 AG17 Abstracts

[email protected] ing for Toric Models

The study of maximum likelihood estimates (MLEs) for PP1 toric models has been one of the successes of Algebraic Tensor Diagrams and Chebyshev Polynomials Statistics. In the setting of a “twisted” toric model, which corresponds to a scaling of the original parameterization Given a complex vector space V , consider the ring Ra,b(V ) map, we prove that it is possible to track a homotopy path of polynomial functions on the space of configurations of whose endpoints correspond to the different MLEs of two a vectors and b covectors which are invariant under the twisted models that may have distinct ML degrees. We natural action of SL(V ). Rings of this type play a central illustrate with examples drawn from loglinear models, and role in representation theory, and their study dates back present how this idea could be applied to help compute to Hilbert. Over the last three decades, different bases MLEs for toric models efficiently. of these spaces with remarkable properties were found. To explicitly construct, as well as to compare, some of Evan Nash these bases remains a challenging problem, already open The Ohio State University when V is 3-dimensional. Recent developments in the 3- [email protected] dimensional setting of this theory will be presented. Lisa Lamberti Carlos Amendola ETHZ TU Berlin [email protected] [email protected]

Nathan Bliss PP1 University of Illinois at Chicago Solving Polynomial System Using Package Chicago, Illinois USA MonodromySolver [email protected]

We present the Macaulay2 package MonodromySolver Serkan Hosten which finds the solution set of a generic polynomial sys- Department of Mathematics tem with parametric coefficients using monodromy action San Francisco State University in the parameter space. Given a generic polynomial sys- [email protected] tem, a finite undirected graph is created to generate the ho- motopy paths implicitly along the edges of the graph. The Jose I. Rodriguez implementation is competitive with existing solvers imple- Department of Statistics mented in other software. The total number of homotopy University of Chicago paths that the package’s algorithms track is roughly linear [email protected] in the number of solutions. This makes the monodromy technique particularly well-suited to problems, for which known bounds on the number of solutions are much larger PP1 than the actual number. Computations over Local Rings in Macaulay2 Kisun Lee Georgia Institute of Technology Local rings are ubiquitous in algebraic geometry. Not only [email protected] are they naturally meaningful in a geometric sense, but also they are extremely useful as many problems can be at- tacked by first reducing to the local case. Here we present PP1 a software package for performing computations over lo- The Covering Problem via Convex Algebraic Ge- calizations of polynomial rings with respect to prime ide- ometry als. The main tools and procedures here involve homologi- cal properties, such as the flatness property of localization This study is focused on the so-called ”sphere covering and the existence of minimal free resolutions for finitely problem”, which consists in finding the optimal covering generated modules over local rings, which follows from of a given object using spheres. The classical approach Nakayama’s lemma. The procedures presented here are is based on reducing it to a point covering problem us- described as pseudocodes and implemented in Macaulay2, ing discretizations, which can incur in high computational a computer algebra software specializing in algebraic ge- cost if a high quality solution is needed. We expose a ometry and commutative algebra. The main motivation discretization-free theoretical approach to this problem, ex- for this work is enabling mathematicians to computation- ploiting its geometric structure and using Real Algebraic ally study the local properties of algebraic varieties near Geometry tools, such as the Positivstellensatz and its vari- irreducible components of higher dimension, such as the ations, to express the covering problem solution in terms intersection multiplicity of higher dimensional varieties. of a sequence of ”representation in sums of squares” deci- sion problems, which can be computed with Semidefinite Mahrud Sayrafi Programming. UC Berkeley Leonardo M. Mito, Gabriel Haeser [email protected] University of S˜ao Paulo FAPESP Michael Stillman [email protected], [email protected] Cornell University [email protected]

PP1 David Eisenbud Maximum Likelihood Estimate Homotopy Track- MSRI AG17 Abstracts 149

[email protected] line. In this research, we propose an approach based on numerical algebraic geometry and nonparametric regres- sion for identifying clusters of in-control and out-of-control PP1 parts in manufacturing processes. The proposed approach S-Arithmetic Groups in Quantum Computing approximates the process data with smoothing splines, and solves the resulted systems using numerical algebraic ge- The tasks of compiling an algorithm for a quantum com- ometry. The identified local maxima of the resulted poly- puter (exact synthesis) and setting up the quantum com- nomial systems correspond to the in-control and out-of- puter for a computation (state preparation) are two major control clusters of manufacturing parts. We validate the problems we face in handling such devices. These tasks proposed approach based on a real dataset from a major obviously highly depend on the architecture of the given automotive manufacturing company assembly line. quantum device. However, for most of the commonly sug- gested architectures they surprisingly turn out to be num- Sara Shirinkam, Adel Alaeddini ber theoretic in nature. In our work we primarily concern University of Texas, San Antonio ourselves with the currently most popular quantum gate set [email protected], [email protected] known as Clifford+T . The set of exactly executable algo- rithms on a device equipped with this gate set forms an S- Elizabeth Gross arithmetic subgroup of the unitary group and we can thus San Jose State University reformulate the tasks of exact synthesis and state prepa- [email protected] ration as constructive membership and constructive orbit membership with respect to this group. We suggest algo- rithms dealing with both exact synthesis and state prepa- PP1 ration based on the action of the S-arithmetic group on the Mixing Behavior of a Family of Random Walks affine Bruhat-Tits building. For a low number of qubits we Corresponding to an Integer Sequence can show that we obtain solutions this way whose quality is a significant improvement over known methods, sometimes We will discuss certain random walks on a finite abelian even obtaining provably optimal solutions. Moreover, the group that arise from integer sequences satisfying modest employed methods generalize well to other important gate conditions. We will give a formula to compute the eigenval- sets (e.g. the qutrit-Clifford+R gate set). ues of the transition matrix. We will also discuss attempts to bound the mixing time of these random walks, explore Sebastian Schoennenbeck some concrete examples, and compare the mixing behavior RWTH Aachen University to that of other well-known random walks on groups and [email protected] graphs.

Vadym Kliuchnikov Caprice Stanley Microsoft Research Redmond North Carolina State University [email protected] [email protected]

PP1 Gram Determinants of Real Binary Tensors

A binary tensor consists of 2n entries arranged into hyper- cube format 2 × 2 ×···×2. There are n ways to flatten suchatensorintoamatrixofsize2× 2n−1.Foreach flattening, M, we take the determinant of its Gram ma- trix, det(MMT ). We consider the map that sends a tensor to its n-tuple of Gram determinants. We propose a semi- algebraic characterization of the image of this map. This offers an answer to a question raised by Hackbusch and Uschmajew concerning the higher-order singular values of tensors.

Anna Seigal UC Berkeley [email protected]

PP1 Identifying Clusters of in-Control and Out-of- Control Parts in Manufacturing Processes Using Numerical Algebraic Geometry and Nonparamet- ric Regression

Model based clustering is one of multivariate data analysis methods dealing with grouping objects based on their sim- ilarities. It has many real-world applications ranging from economy to healthcare to manufacturing. Most manufac- turing systems require effective approaches for clustering large scale and multi-dimensional data that are automat- ically generated from hundreds of units in the production