36 AG11 Abstracts IP1 IP5 Algebraic Statistics for Social Network Models: Advances in Elliptic-Curve Cryptography Recent Results and Challenges The first part of this talk presents results on attacking In collaboration with Petrovic and Rinaldo, I have devel- elliptic-curve cryptography, in particular an ongoing effort oped algebraic statistical tools for the study of some dyadic to break the Certicom challenge ECC2K-130 and a detailed random graph models, including Markov bases, that have study on the correct use of the negation map in the Pollard- important implications for the existence of maximum like- rho method. The second part presents a signature scheme lihood estimation and other statistical problems. These which on a 390 USD mass-market quad-core 2.4GHz Intel tools do not extend in a simple fashion to more complex Westmere (Xeon E5620) CPU can create 108000 signatures models in the class of exponential random graph models. In per second and verify 71000 signatures per second on an this presentation, I explain why there are difficulties as we elliptic curve at a 2128 security level. Public keys are 32 move away from dyadic models and I describe some of the bytes, and signatures are 64 bytes. These performance fig- challenges for algebraic statistics in this area of research. ures include strong defenses against software side-channel attacks: there is no data flow from secret keys to array in- Stephen Fienberg dices, and there is no data flow from secret keys to branch Carnegie Mellon University conditions. fi[email protected] Tanja Lange Technische Universiteit Eindhoven IP2 Eindhoven Institute for the Protection of Systems and Complexity of the Separation of Variables and Re- Inform lated Observations [email protected] Abstract not available at time of publication. IP6 Leonid Gurvits B-splines: A Fundamental Tool for Analysis and Los Alamos National Laboratory Computation [email protected] Starting from their inception in approximation theory 60 years ago, the use of B-splines has blossomed in diverse IP3 areas in science, engineering and electronic arts. These in- Calculus and Constructible Sheaves clude animation, computational geometry, computer-aided design, computer-aided manufacturing, computer graphics, This talk describes an ingenious integral calculus based on control theory, geometric design, image analysis, medical Euler characteristic, stemming from work on constructible visualization, optimization, partial differential equations, sheaves due to MacPherson and Kashiwara in the 1970s, robotics, and statistics. More recently B-splines have been and connecting back further to classical integral geometry. assuming a fundamental role in the isogeometric analysis, The talk will emphasize (1) its novel utility in data man- an ambitious effort to unify shape representation and en- agement, particularly in aggregation of redundant data and gineering analysis. Honoring its interesting historical de- inverse problems over sensor networks; and (2) how issues velopment, this talk gives a mathematical introduction to of numerical computation, inspired by applications, leads splines and B-splines in one and several space dimensions to fascinating connections with Morse theory and compu- and also considers various generalizations. tational topology. Tom Lyche Robert W. Ghrist University of Oslo University of Pennsylvania Department of Informatics and CMA [email protected] tom@ifi.uio.no IP4 IP7 The Geometry of Cell Structures in Foams and Symbolic-numeric Algorithms for Computing the Metals Singular Solution of Polynomial Systems Networks of crystals in a metal and networks of bubbles In this talk we will describe some recent progress in in a foam are both examples cell complexes that arise in symbolic-numeric algorithms for computing the singular nature. Both systems evolve over time according to mathe- solution of polynomial systems. Using the local dual struc- matical equations. We believe that, for generic initial con- ture of the isolated singular solution with limited accuracy, ditions, they evolve towards a statistically universal state. we present modifications of Newton’s method to restore The topology and geometry of this universal state is great quadratic convergence. In particular, when the corank of interest in applications. This talk will present both math- the Jacobian matrix at the singular solution is one, based ematical results and computer investigations on these sys- on the regularized Newton iteration and the computation tems in two and three space dimensions. of partial differential conditions satisfied approximately at Robert MacPherson the singular solution, we develop a new approach to com- Institute for Advanced Study pute a proper direction and step size of the Newton itera- School of Mathematics tion to ensure the quadratic convergence. Finally, we will [email protected] show recent results on computing verified error bounds for singular solutions of polynomial systems based on Rump’s verification method. Lihong Zhi AG11 Abstracts 37 Academia Sinica CP1 [email protected] Extracting Topological Properties Using Manifold Learning Techniques IP8 High-dimensional, unordered data, is often difficult to an- Multiview Geometry alyze or visualize. In such cases, manifold learning tech- niques - like PCA, Isomap, Hessian LLE - are quite helpful The study of two-dimensional images of three-dimensional and efficient in extracting the low-dimensional representa- scenes is a foundational subject for computer vision, known tion of the data. When used individually, these methods as multiview geometry. We present recent work with Chris just give the low-dimensional representation of the data. Aholt and Rekha Thomas on the polynomials defining im- But owing to the complementary nature of the limitations ages taken by n cameras. Our varieties are threefolds that of these methods, performing a hierarchy of tests and com- vary in a family of dimension 11n-15 when the cameras are paring the outputs can reveal useful topological features moving. We use toric geometry and multigraded Hilbert like the presence of holes and (degree of) non-linearity of schemes to characterize degenerations of camera positions. the data. We develop and showcase such a hierarchical strategy on various material science data to provide insight into process-property-structure relationships. Bernd Sturmfels University of California , Berkeley Sai Kiranmayee Samudrala [email protected] Department of Mechanical Engineering, Iowa State University Ames, Iowa -50011 CP1 [email protected] Discrete Bounded-curvature Paths and Path Plan- ning Baskar Ganapathysubramanian In 1957 Dubins classified shortest bounded curvature paths Iowa State University in the plane into two classes. We generalize this result to [email protected] a new class of discrete polygonal paths. The properties and construction of discrete Dubins paths are discussed. CP1 Properties of discrete bounded-curvature motion are anal- ysed and applied to prove properties of smooth bounded- Lower Bounds on Stochastic Games curvature paths. In particular, the classification of discrete Shapley’s discounted stochastic games are classical models Dubins paths is used to obtain Dubins seminal result as a of game theory describing two-player zero-sum games of limiting case. Related to these problems the question of potentially infinite duration. We show, based on a gener- defining curvature for a polygonal path is addressed. Po- alization of Eisenstein criterion, that there exists a game tential research questions, such as generalizing Cauchy’s with N positions, m actions for each player in each posi- arm lemma, are discussed. tion, such that its value is an algebraic number of degree mN−1 Sylvester Eriksson-Bique . This strengthens a result of Etessami and Yan- nakakis who considered the case of m =2andproveda University of Helsinki Ω(N) New York University 2 lower bound. [email protected] Kristoffer A. Hansen Aarhus University, Denmark David Kirkpatrick [email protected] University of British Columbia, Canada [email protected] Michal Koucky Czech Academy of Sciences Valentin Polishchuck [email protected] University of Helsinki [email protected].fi Niels Lauritzen, Peter Bro Miltersen Aarhus University, Denmark CP1 [email protected], [email protected] On Middle Universal Weak and Cross Inverse Property Loops With Equal Length of Inverse Cy- Elias Tsigaridas cles University of Aarhus [email protected] This study presents an isotopism under which the weak inverse property(WIP) is isotopic invariant in loops. It is shown that under this isotopism, whenever n is a positive MS1 even integer, a finite WIPL has an inverse cycle of length Symmetric Determinantal Representations of Poly- n if and only if its isotope is a finite WIPL with an in- nomials verse cycle of length n. Explanations and procedures are givenonhowtheseresultscanbeusedtoapplyCIPLsto In a seminal paper (STOC 1979), Valiant expressed the cryptography. polynomial computed by an arithmetic formula as the de- terminant of a matrix whose entries are constants or vari- Temitope G. Jaiyeola ables. This result was then extended by Malod and Toda Obafemi Awolowo University to weakly-skew circuits. We are interested here in express- [email protected] ing formulas and weakly-skew circuits by determinants of symmetric matrices. In my talk, I will sketch some of these 38 AG11 Abstracts constructions for fields of characteristic different from 2. In tion which by definition accepts closed sets only. Also, we characteristic 2, the picture is quite