1. Course 1: Game Theory by Juan Manfredi 1.1

SITE REU PITTSBURGH IN PURE MATH PROJECT DESCRIPTION

MARTA LEWICKA AND JUAN MANFREDI

1. Course 1: Game Theory by Juan Manfredi 1.1. What is this course about. Mathematical Game Theory is concerned with formal- izing decision-making situations in a rigorous manner in order to find optimal choices; that is, ones that yield the highest expected payoff. Started by John Von Neumann and Oskar Morgenstern and popularized by John Nash, the field has recently focused the interest of not only mathematicians and economists but also biologists, social scientists, and engineers. We will begin by studying the game of Hex is a two-person, zero-sum game played on a symmetric board. We will follow the recent book by Yuval Peres Game Theory, Alive for the discussion of some of the basic theorems traditionally associated with Game Theory such as various fixed point theorems, Nash and correlated equilibria, and Von Neumann’s Minimax Theorem. The necessary background in Probability will be provided. 1.2. Undergraduate research vision. Probability is key to understand many of the modern developments in Mathematics. Recently Peres-Schramm-Sheffield-Wilson (2008, JAMS) have discovered an explicit connection between martingales and non linear pdes. This can explained in the frame of mean value properties that hold only locally. GameTheory play the role of Nonlinear Probability. This theory has just begun its development. Undergrad- uate students can understand the basic notions in a discrete setting, where heavy measure theory is not needed, and simulations are relatively easy to set up. 1.3. Applications and broader impact. Game Theory is often taught in Economics departments given its immediate application to economic behavior. Increasingly the techniques of Game Theory have been used in Engineering, especially in Signal Processing and in Electrical Networks. Students who get a good understanding of Probability and Game Theory, will be very prepared to attend graduate school in a variety of subjects ranging from Economics to Biostatistics. 1.4. Suggested projects. (i) Analysis of the Factory Game: Suppose three factories are located by a lake. Each year, they have a choice of either polluting it or purifying it. If they choose to pollute the lake, there is no cost to the factories. However, if two or more factories pollute the lake, the water becomes unusable, and each factory has to pay b units to obtain water from another source. To purify the lake, each factory must pay a units. If two or more factories choose to purify the lake, then the water is still usable, and there is no additional cost to the factories (you may assume b¿a¿0). Consider extensions to n factories. (ii) Tug-of-War on graphs, discrete ∞-Laplace operator. 1 2 MARTA LEWICKA AND JUAN MANFREDI

(iii) Escaping to the boundary. First start with biased random walks and then consider biased random walks plus tug-of-war. This project will ﬁrst run simulations to get ideas on we can attempt to prove rigorously.

2. Course 2: PDES on graphs by Juan Manfredi 2.1. What is this course about. In this course we will ﬁrst present Calculus on graphs in detail. Derivative, Integrals, and the Fundamental Theorem of Calculus in graphs. Of course these notions are much easier than their counterparts in the continuous setting. Our ﬁrst PDE to be studied in detail will be the Laplacian, followed by the ∞-Laplacian and the p-Laplacian. We will use a probability approach to prove existence and uniqueness of solutions both in the linear case (classical) and in the non-linear case (recent developments).

2.2. Undergraduate research vision. Probability is key to understand many of the modern developments in Mathematics. Recently Peres-Schramm-Sheﬃeld-Wilson (2008, JAMS) have discovered an explicit connection between martingales and non linear pdes. This can explained in the frame of mean value properties that hold only locally in the continuous case, but are much easier to understand in the discrete case. GameTheory play the role of Nonlinear Probability. This theory has just begun its development. Undergraduate students can understand the basic notions in a discrete setting, where heavy measure theory is not needed, and simulations are relatively easy to set up.

2.3. Applications and broader impact. PDEs lie at the foundation of scientiﬁc computing. The discrete case can be understood without heavy use of Measure Theory and Functional Analysis. Students will be able to follow the details from day one. We will consider applications to Electrical Networks and to Signal Processing, considering in both case non-linear analogues of well-established classical theories.

2.4. Suggested projects. (i) Non-linear Kirchoﬀ laws. The classical Kirchoﬀ’s circuit law give rise to the Laplace equation. We will study examples of non-linear circuits. (ii) The discrete p-Laplace operator. (iii) Numerical approaches to solve non-linear PDEs based on mean value properties.

2.5. The tools of the trade for Courses 1 and 2. (i) Internet tools Use Google and Wikipedia, but don’t believe anything you get from the web without double checking and crossreferencing. (ii) LATEX LATEXis the de-facto standard for mathematical typesetting. It is available free for all major computer platforms. MikTex is the most popular implementation for Windows: http://miktex.org/ MacTex is the most popular implementation for Mac OS: http://www.tug.org/mactex/2009/ For Ubuntu and other Linux systems: https://help.ubuntu.com/community/LaTeX 3

(iii) Mathematica One of several software programs to do mathematics by computer. It is an expensive professional program which happens to be available to University of Pittsburgh undergraduates for $5. See http://technology.pitt.edu/software/browse/mathematica.html (iv) SAGE This is the future of doing mathematics by computer. It is a free open source alternative to the commercial programs Mathematica, Maple, and Matlab. http://www.sagemath.org/ 2.6. References. [L] Lawler, Gregory F., Topics in Probability and Analysis, Lectures from the Univer- sity of Chicago REU Program, 2009, http://www.math.uchicago.edu/ lawler/reu.pdf [MOS] J.J. Manfredi, A. Oberman, and A. Sviridov, Nonlinear Elliptic Partial Differential Equations and p-harmonious Functions on Graphs. Preprint 2012. [MS2] A. P. Maitra, and W. D. Sudderth, Discrete gambling and stochastic games. Ap- plications of Mathematics 32, Springer, New York, 1996. [MPR] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value char- acterization of p-harmonic functions. Proc. Amer. Math. Soc., 138:881–889, 2010. [MPR2] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions. Preprint. [MPR3] J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise. Preprint [O] A. M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions. Math. Comp. 74(251):1217– 1230, 2005. [PSSW] Y. Peres, O. Schramm, S. Sheffield and D. Wilson; Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc., 22:167–210, 2009. [PS] Y. Peres, S. Sheffield; Tug-of-war with noise: a game theoretic view of the p- Laplacian. Duke Math. J., 145(1):91–120, 2008.

3. Course 3: Geometric rigidity by Reza Pakzad 3.1. What is this course about. In this course the rigidity of geometric objects will be studied from essentially two points of view. After presenting the purely geometric idea in the most general framework of metric spaces, we will first discuss the rigidity of structures made of linkages and hinges, or equivalently the question of isometric embeddings of graphs in the plane and in space. Several variations of the concept: isometric rigidity, bending/flexing rigidity and infinitesimal rigidity will be explained and discussed. Among the topics discussed in this part of the program are Cauchy’s theorem and some of its variants such as Dehn’s rigidity theorem and Alexandrov’s rigidity theorem, Connelly and Bricard’s example of flexible polydehra and Bellows conjecture. We will move forward to the contin- uum case of smooth surfaces and will discuss several classical theorems, including Hilbert’s rigidity theorem for spheres (and closed convex surfaces in general), developability of flat surfaces and finally Nash and Kuiper results on non-rigidity of C1 isometric immersions. A major open problem in this field is whether a closed surface in space in unbendable. The 4 MARTA LEWICKA AND JUAN MANFREDI notions of infinitesimal isometries and Killing vector-fields will be explored. Finally, some of the applications of these results in the theory of elasticity will be discussed. 3.2. Undergraduate research vision. The main idea is to introduce the students to the geometric complexity of the problem and the various tools which have been used to tackle this problem. Even though some of the proofs of the presented theorems might the too technical for an undergraduate level student, the concept of rigidity is easy to grasp and intuitional and many basic examples such as Cauchy’s theorem or developabilty theorems for flat surfaces could be easily worked out in detail with an elementary background in mathematics. The first part of the course has a combinatorial flavor. One of the objectives of this program is to introduce the students to some problems regarding rigidity in the theory of elasticity. All the discussed topics are lower dimensional and could even be illustrated by simple physical demonstrations. Prerequisites: Calculus 1-3, Elementary linear algebra, elementary ODEs and PDEs, elementary differential Geometry. 3.3. Applications and broader impact. The question of rigidity of elastic structures is a longstanding problem in engineering, solid mechanics and mathematical theory of elasticity. Recent developments in the theory of elasticity has lead to new nonlinear theories of rods, plates and shells involving isometries and infinitesimal isometries and a natural question in this context is whether a given surface is geometrically rigid under a given set of isometric transformations, i.e. whether there are such transformations which are not rigid motions. Structural rigidity of graphs and lattices has recently found applications in [computer] network stability. Finally, a longstanding and challenging problem about rigidity of closed surfaces is a source of new explorations in the theory of Riemannian manifolds. 3.4. Suggested Projects. (i) Infinitesimal rigidity of closed surfaces: It is classically shown that any smooth closed convex surface will only admit trivial first order infinitesimal isometries. Cohn-Vossen showed the existence of a closed rotationally symmetric surface which admits non-trivial C2 displacements in this class. The object of the research will be look for a closed surface in this particular class for which the C∞ infinitesimal isometries are all trivial. This would imply that smoother infinitesimal isometries are not dense in the class of less regular ones. We will learn in the course of project that a curve in R2 is not geometrically rigid, but a full 2 dimensional domain is. The following 3 projects considers variations of this problem. A geodesic triangulation of surface is a triangulation whose edges are (global) geodesic paths on the surface. (ii) Rigidity of geodesic triangulations on flat torus. A definition of the flat torus and a classification of its isometric automorphisms will be given. We assume two geodesic triangulations on the 2d flat Torus are given such that they are isometrical as metric spaces. The questions to be treated in the paper are the following: Are they necessarily identical modulo an isometric automorphism of the torus? If not, is there any condition on the triangulation which implies its geometric rigidity. (iii) Rigidity of geodesic triangulations on the 2d sphere. We assume two geodesic triangulations on a sphere are given such that they are isometrical as metric spaces. 5

Are they necessarily identical modulo an isometric automorphism of the torus? If not, we seek condition or conditions on the triangulation implying its geometric rigidity in the above sense. (iv) Fractal rigidity. Fractal sets in the plane can admit non-integer Haussdorf dimension. The purpose of this project is to find example of rigidity or non-rigidity of Hausdorff dimension 1 < d < 2. The main question is whether they behave like 1 dimensional or like two dimensional subsets of the plane. (v) Rigidity of convex closed surfaces. This will be a review paper of the notion of convexity of closed surfaces. The main theorem, which uses Herglotz formula, states that any isometric immersion of a closed convex surface in space is a rigid motion. A main problem in the literature about the bending-rigidity of closed surfaces in the space is then discussed. A simpler question will then be evaluated: Whether a closed surface which admits a Killing vector field is necessarily rotationnally symmetric. 3.5. References. (i) Geometry and the Imagination, David Hilbert. (ii) A Comprehensive Introduction to Differential Geometry, Vol 5, Michael Spivak. (iii) R. Connelly, ”Rigidity”, in Handbook of Convex Geometry, vol. A, 223271, North- Holland, Amsterdam, 1993.

4. Course 4: Calculus of Variations by Reza Pakzad 4.1. What is this course about. This course will be an introduction to elementary concepts and examples of calculus of variations, with a view of introducing the students to the theory of lower dimensional elastica. The fundamental concepts and examples to be covered include the idea of functionals, the local and global extrema, the Euler-Lagrange equations, the direct method, geodesics, Fermat’t principle, the principle of least action, Laplace equation, the Dirichlet principle in two dimensions and the Plateau problem, Eigen- value problems and Sturm-Liouville’s theorem. Some more advanced topics to be discussed are constrained problems and Lagrange multipliers, isoperimetric problem, Second variations and Legendre condition, Invariance of functionals and Noether’s theorem. At the final stages of the course, some problems in elasticity such as linear membranes and plates, variational problem of buckling, Euler’s elastica and Kirchhoff model for plates will be discussed. 4.2. Undergraduate research vision. Calculus of Variations is a fundamental branch of mathematical analysis which is absent from almost all the undergraduate mathematical curricula in North America. This is surprising since many of the fundamental notions and basic examples could be coherently presented and explained to undergraduate students with even only a background in one variable Calculus. The students will be engaged with many concrete examples of functionals from Mechanics, Physics and Geometry where they can combine an intuitive grasp of the matter with rigorous calculations based on their elementary knowledge in one or several variable calculus. It is hoped that this could prepare them for reflecting on more involved but still lower-dimensional problems which will be presented to them as the course projects. 6 MARTA LEWICKA AND JUAN MANFREDI

Pre-requisites: Calculus 1-3, Elementary linear algebra, elementary ODEs and PDEs, some familiarity with Elementary differential Geometry might help. 4.3. Applications and broader impact. Calculus of Variations is one of the most lively branches of mathematical analysis today. It interacts with various other topics in mathematics, physics, mechanics and economy. Non-covexity or nonlinear constraints have made many of a problem challenging and rich in perspectives and sometimes has lead to surprising new discoveries. Those students who follow this course will potentially be able to follow their interests in any of these various modern topics. Examples include Optimal Transporta- tion, Harmonic Mappings, Curvature functionals and Willmore conjecture, Phase diffusion, pattern formation and microstructures. Among others, a rigorous variational approach to nonlinear elasticity has recently lead to the discovery of new potential nonlinear theories for rods, plates and shells. This new approach to elasticity can be very helpful in understanding various phenomena, ranging from DNA folding and elasticity of nanotubes and graphene sheets to mechanic behavior of growing tissues and pre-enginneered non-euclidean gels. Many questions in these domain remain unanswered. The students with a background in differential geometry (see Topic 1) will be oriented to tackle some of the more accessible problems in this field. 4.4. Suggested projects. (i) Euler’s Elastica: Euler’s celebrated paper on nonlinear elastic rods, is a rich source of classical results and observations. The task will consist in reviewing the paper, now appeared in english, and to re-phrase its contents using a modern and accessible language. (ii) Energy scaling for the confinement of Euler’s elastica in a small 2d ball and the minimal configuration: An elastic bar of length 1, modeled by the curvature L2- norm functional, is confined within a ball of radius r > 0. Under various boundary conditions, we seek the minimizing configuration and the scaling of the minimal energy with respect to r > 0. (iii) Energy scaling for the confinement of non-self-intersecting Euler’s elastica in a small 3d ball and the minimal configuration: A nonlinear elastic rod of length 1 is confined within a ball of radius r > 0. Under various boundary conditions, and under a no-self-intersecting constraint, we seek the minimizing configuration and the scaling of the minimal energy with respect to r > 0. (iv) Configuration of a pinched cylinder: A contraction applied to one side of the boundary of a long paper cylinder propagates to the other end, making a 90 degrees ro- tation. The goal is to justify this phenomenon using the Kirchhoff theory of shells. Since a cylinder is a developable surface, the problem is essentially one-dimensional and accessible to undergraduate students. (v) Euler-Plateau problem: Euler-Plateau problem consists in minimizing the surface area of the surface alongside the elastic energy of its boundary. A linearized model will first be studied for existence and regularity of solutions. The next step is to consider these questions for the full non linear problem. 4.5. References. (i) Geometry and the Imagination David Hilbert 7

(ii) Methods of Mathematical Physics, Vol. 1 , Courant and Hilbert (iii) Leonhard Euler’s Elastic Curves W. A. Oldfather, C. A. Ellis and Donald M. Brown Isis Vol. 20, No. 1 (Nov., 1933), pp. 72-160 Published by: The University of Chicago Press. (iv) Minimal surfaces bounded by elastic lines L. Giomi, L. Mahadevan School of En- gineering and Applied Sciences, Department of Physics, Harvard University.

5. Course 5: Foundations of Differential Geometry by Reza Pakzad 5.1. What is this course about. This course will focus on the geometry of curves in planes and space and of 2d surfaces from from a qualitative and quantitative point of view. Topics include curvature of plane curves, curvature and torsion of space curves, the Frenet frames and formulas, curvature integrals on closed curves, the Gauss map, the Gauss curvature, the Weingarten map, the fundamental forms, isometric embeddings, Theorema Egregium, the integral of the curvature over a geodesic triangle, geodesics on a surface, isometric immersions and bendings, infinitesimal isometries. Some applications in Elastic theories of rods, plates and surfaces will be briefly reviewed. The purpose is to familiarize the students first with these basic concepts of differential geometry from an intuitive point of view and instructing them in applying their basic mathematical knowledge to engage rigorously with these topics and obtain proficiency in basic differential geometric calculations.

5.2. Undergraduate research vision. The basic concepts of differential geometry are easily accessible to undergraduate students. An intuitive approach will first be adapted to engage the students in imaginative and creative mathematical thinking before proceed- ing to translating the concepts into more rigorous calculations. An undergraduate student familiar with the representation of curves and surfaces in plane and space as taught in a typical Calculus III course and an elementary background in linear algebra and differential equations will then be able to follow the course up to its more detailed and rigorous presentations. There are naturally a multitude of accessible examples available in lower dimensions which will demonstrate clearly the notions discussed in class. Pre-requisites: Calculus 1-3, Elementary linear algebra and differential equations

5.3. Applications and broader impact. From geometric analysis and geometric function theory to the study of metric structures on Riemannian manifolds, from nonlinear elasticity to relativity theory, Differential Geometry is omni-present in contemporary mathematical research. It is hoped that a basic grasp of fundamental notions of this topic will motivate the students to engage in studying this topic in deep in order to get involved in these advanced subjects in analysis and geometry in future. Recently, rigorous study of elastic rods, plates and shells in the framework of the nonlinear elasticity theory has lead to new questions regarding the geometric properties of curves and surfaces in plane and in space and isometric immersions of 2d Riemannian manifolds in the 3-space. This new approach to elasticity of lower dimensional objects can be very helpful in understanding various phenomena, ranging from DNA folding and elasticity of nanotubes and graphene sheets to mechanic behavior of growing tissues and pre-enginneered non-euclidean gels. This course on differential geometry will be conceived in view of familiarizing the students 8 MARTA LEWICKA AND JUAN MANFREDI with these applications. In particular, this course will set up the background for the two other courses to be offered in the same program: Calculus of Variations with an emphasis on the theories of rods and plates, and geometric rigidity.

5.4. Suggested projects. (i) Theorema Egregium: Gauss’s Theorema Egregium will be described, proved and examples will be provided. The other focus of the paper will be to give a proof of the inverse theorem in the local case: If two surfaces admit the same Gaussian curvature, then they are locally isometric. (ii) Bertrand-Diquet-Puiseuc Theorem: This theorem expresses the Gaussian curvature of a surface in terms of the area of a geodesic disc or its perimeter. A proof will be presented and some applications and generalization to higher dimensions will be explored. (iii) Gauss-Bonnet Theorem: The Gauss Bonnet theorem, which connects the geometry and topology of a surface will be presented and proved. The special case of a geodesic triangle on a given surface is of interest. Analytically, the vortices can be interpreted as jumps in the anti-derivative of the geodesic curvature (see also project on Discrete Differential Geometry). Through a triangulation of the surface, we deduce the global Gauss-Bonnet theorem for a closed surface from this special case. (iv) Ricci and Riemann curvature in higher dimensions: The notions of Riemann, Ricci and scalar curvature on a Riemannian manifold will be defined and explored. In each case, a suitable intuitive interpretation will be provided and justified. (v) Gauss-Codazzi equations: A long-standing problem in differential geometry is the question of existence of isometric embeddings of a given 2d Riemannian manifold into the 3 dimensional space. The solvability of Gauss-Codazzi equations for a given Riemannian metric on the unit disk is equivalent to the existence of an isometric imbedding of the disk equipped with that metric into R3. This theorem will be proved in the paper and some simple applications will be explored. (vi) Discrete differential geometry: The notion of curvature measures for discrete curves and surfaces will be defined and it will be established that the discrete versions of the integral curvature formulas such as Gauss-Bonnet formula (for surfaces) will remain valid.

5.5. References: (i) A Comprehensive Introduction to Diﬀerential Geometry, Vols 2 and 5, Michael Spivak. (ii) Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics), Singer and Thorpe

6. Course 6: Sobolev inequalities by Piotr Hajlasz (4 weeks 18 hours of lectures and 6 hours of seminars related to the research topics) 9

6.1. What is this course about. At the beginning of XXth century, with a classical notion of differentiable functions, the theory of partial differential equations and calculus of variations came to a dead end. The further development was only possible with a suitable generalization of notion of the derivative. This led to the discovery of the Sobolev spaces. This theory is a single most important tool in studying nonlinear partial differential equations, both in its theoretical aspects and numerical implementation. The course will be an introduction to the vast area of integral and geometric inequalities: Sobolev inequalities, Hardy inequalities, Poincaréinequalities, isoperimetric inequalities. Part of the course will be devoted to the theory of Sobolev spaces. Notes for the course will be posted online. 6.2. Undergraduate research vision. The theory of Sobolev spaces and geometric inequalities is a tool of fundamental importance that anyone interested in analysis should know. Many problems leading to the original research can be formulated in a way accessible for undergraduate students. Individual investigation of new problems is the best way to learn the subject. Students will write reports and prepare presentations based on their research. Prerequisites. Students need to know multivarible calculus with rigorous proofs. It would be helpful if they know basic measure theory and functional analysis. 6.3. Applications and broader impact. The theory of Sobolev spaces is an important tool in the theory of nonlinear partial differential equations and numerical analysis. The students will broader their knowledge in the subject and will learn how to work on research projects. 6.4. Suggested Projects. (i) Many results for functions defined in Rn can be reformulated for functions on graphs. The aim of this project is to discover and prove suitable generalizations of versions versions of Sobolev inequalities for functions defined on graphs. This project leads to interesting connections to potential theory and discrete partial differential equations. (ii) Generalization of the classical trace theorem for Sobolev spaces to the case of Sobolev functions defined on domains in discrete spaces. (iii) It was proved in 2008 that the boundary of a domain for which the Sobolev embedding theorem is valid has Lebesgue measure zero. The aim of this project is to show that the result is optimal by providing a construction of an s-Johan domain with boundary of positive measure. (iv) It was proved in 2007 that the Hardy-Littlewood maximal operator is continuous in the Sobolev space. The aim of this project is to prove that a similar result is true for the spherical maximal function of Stein.

7. Course 7: Smooth functions by Piotr Hajlasz 7.1. What is this course about. Multivariable calculus is a subject that all students interested in exact sciences and engineering have to learn. In the course we will go far beyond the standard multivariable calculus and we will present such important results as the Sard theorem, the Whitney extension theorem, the Hestens-Seeley extension theorem, 10 MARTA LEWICKA AND JUAN MANFREDI the Borel theorem and the Peetre theorem just to name a few. Notes for the course will be posted online. Prerequisites. Students need to know multivarible calculus with rigorous proofs.

7.2. Undergraduate research vision. There will be a number of problems accessible to undergraduate students related to the results presented in the course. The students will work on individual projects, they will write reports and prepare presentations based on their research.

7.3. Applications and broader impact. The multivariable calculus is an important tool in the applications to engineering and physics. The students will broader their knowledge in the subject and will learn how to work on research projects.

7.4. Suggested projects. (i) A celebrated theorem of Marcinkiewicz and Zygmund is known as the converse theorem to the Taylor formula. The aim of this project is to find a new proof based on the Calderónpointwise inequalities. (ii) Peetre characterized differential operators with smooth coefficients in purely algebraic terms. The aim of the project is to look for various generalizations of this theorem. (iii) Smooth surfaces can be defined implicitly through the equation F (x, y, z) = 0. The aim of the project is to use this approach to construct interesting surfaces. For example the union of a torus and a sphere can be described by a polynomial equation of degree 6. Can it be described by a polynomial equation of a smaller degree? This research will require familiarity with programs like Mathematica.

8. Course 8: An introduction to topological degree by Marta Lewicka 8.1. What is this course about. This course aims to provide a self-contained introduction to the theory of topological degree in Euclidean spaces of finite dimension (so-called Brouwer degree) and infinite dimensions (the Leray-Schauder degree). The final form of the course will be adjusted according to the students’ preparation and motivation. The general outline is as follows: (i) Motivation and examples. Review of differentiation in Rn and the concept of Lebesgue measure. Solutions to equations and Sard’s lemma. (ii) The Brouwer degree. Motivation and properties: normalization, additivity, homo- topy invariance. Construction of the degree. Direct consequences of the fundamental properties. (iii) First topological applications. Degree of a complex polynomial, fundamental theorem of algebra. The Brouwer fixed point theorem, Kakutani’s example. (iv) Applications to nonlinear systems of equations. Existence of solutions, number of solutions, bifurcation - examples from [BFPS]. 11

8.2. Undergraduate research vision. The notion of degree is classical and fundamental in nonlinear analysis. It shows beautiful and deep connections between analysis and topology and it can be easily visualised. The proofs are intiuitive and geometric. Because of its elementary nature, the subject of the course does not require much background and is accessible to a wide range of audience. The course is intended for students mostly interested in analysis. During the lectures many exercises and problems will be assigned and a close collaboration with the instructor running the related problem sessions is expected. The ﬁnal form of the course will be adjusted according to the students’ preparation and motivation.

8.3. Applications and broader impact. The topological degree serves, primarily, to es- timate the number of solutions to an equation. When one solution is easily found, degree theory can often be used to prove existence of a second, nontrivial solution. Hence the applications in practically any ﬁeld of mathematics or when rigorous arguments in engineering models are needed, are ubiquitous.

8.4. Suggested projects. (i) Research the concept of a retract and prove the Borsuk theorem (the sphere is not a retract of the unit ball), and Bohl’s theorem (any continuous function f defined on a unit ball and taking values in the sphere must have a fixed point, i.e. f(x) = x for some x). (ii) The odd mapping theorem states that any continuous map f on a unit ball which does not collapse antipodal points on the boundary (i.e. f(x) 6= f(−x) for any x on the sphere) must have a zero: f(x) = 0 for some x. This result has many famous and useful applications. Perhaps the most classical one is the ’ham-sandwich theorem’ stating that given n bounded, measurable subsets of Rn, there exists a hyperplane which divides all of them into halfs. Another one is the antipodal theorem: every continuous function on a sphere maps some pair of antipodal points to the same point. The project is to research these results, find (more than one) independent proofs, check how far can one relax the assumptions, and decribe some amusing applications with a geometrical flavor. (iii) Applications to finding periodic solutions to nonlinear systems of ODEs. Poincare’ maps. Extensions to cases without uniqueness of solutions, which calls for extending the notion of topological degree to multivalued operators.

8.5. References. [L] Lloyd N.G., Degree theory, Cambridge Tracts in Math. 73, Cambridge University Press, 1978. [BFPS] Benevieri P., Furi M., Pera M.P., Spadini M., An introduction to topological degree in Euclidean spaces, chapter 1 of the book. [DG] Dugundji J., Granas A., Fixed point theory, Warszawa, 1982. [A] Amann H., Lectures on some ﬁxed point theorems. 12 MARTA LEWICKA AND JUAN MANFREDI

9. Course 9: Optimal control and viscosity solutions to Hamilton-Jacobi equations by Marta Lewicka 9.1. What is this course about. The objective of this course is to give a compact introduction to optimal control theory and Hamilton-Jacobi equations. These are important and lively fields of mathematical analysis: There is a close relationship between these fields, as well as applications to differential games theory, calculus of variation and systems of con- servation laws. The classical methods of optimal control are largely due to the work of L. Pontryagin and his collaborators in the Soviet Union and R. Bellman in the United States. In a simplest example, consider a car traveling on a straight line through a hilly road. The question is, how should the driver press the accelerator pedal in order to minimize the total traveling time? The term ’control law’ refers specifically to the way in which the driver presses the accelerator and shifts the gears. The ’system’ consists of both the car and the road, and the optimality criterion is the minimization of the total traveling time. Another optimal control problem is to find the way to drive the car so as to minimize its fuel consumption, given that it must complete a given course in a time not exceeding some amount. Yet another control problem is to minimize the total monetary cost of completing the trip, given prices for time and fuel. During the course, many examples and exercises will be provided. The final form of the course will be adjusted according to the students’ preparation and motivation. The general outline is as follows: (i) Basics on Optimal Control: the Pontryagin Maximum Principle (the derivation and geometric explanation), the Mayer problem with terminal constraints. (ii) Basics on Calculus of Variations: motivation and examples, derivation of the Euler- Lagrange equations. (iii) Basics on the viscosity solutions to the Hamilton-Jacobi equations.

9.2. Undergraduate research vision. No prerequisites beyond some standard analysis in Rn and basic ODEs are necessary. The course should thus be accessible to everybody. During the lectures many exercises and problems will be assigned and a close collaboration with the instructor running the related problem sessions is expected. The ﬁnal form of the course will be adjusted according to the students’ preparation and motivation.

9.3. Applications and broader impact. Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. In mathematics but also in computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller in complexity, yet the ﬁnal gain is much larger.

9.4. Suggested projects. (i) Dynamic programming. Reserach the dynamic programming principle of Bellman and ﬁnd the intricate connection between all three subjects covered in the course: The value function of the optimal control problem is the unique viscosity solution 13

of the Hamilton-Jacobi-Bellman equation. On the other hand, trajectories satis- fying the Pontryagin Maximum Principle provide the characteristic curves for the Hamilton-Jacobi equations of dynamic programming. (ii) Applications of dynamic programming: Dijkstra’s algorithm for the shortest path problem, Fibonacci sequence, tower of Hanoi puzzle. (iii) Differential games. In an optimal control problem there is single control and a single criterion to be optimized; while the differential game theory generalizes this to two controls and two criteria, one for each player. Each player attempts to control the state of the system so as to achieve his goal and the system responds to the inputs of both players. Examples of differential games in modern economy. Nash equilibrium. Pareto optimality as the minimal notion of efficiency. 9.5. References. [B] Bressan: Viscosity solutions of Hamilton-Jacobi equations and optimal control problems. [BC] Bardi and Capuzzo-Dolcetta: Optimal control and viscosity solutions of Hamilton- Jacobi-Bellman equations. [F] Friedman: Differential games.

10. Course 10: Convex polytopes and systems of algebraic equations by Kiumars Kaveh 10.1. What is this course about. This course has two goals: (1) to cover basic material in convex geometry (from the point of view of Brunn-Minkowski), and (2) show the beautiful connection between the geometry/combinatorics of convex polytopes and algebraic geometry. The main motivating problem is counting the number of solutions of a system of polynomial equations. One of the central objects of study in the course is a semigroup of integral points. A main theorem to be discussed asserts that one can describe the asymptotic behavior of general (even non-finitely generated) semigroups of integral points. More precisely, given n an (additive) semigroup S ⊂ N×Z , the function HS(k) which counts the number of points in a level {k}×Zn (as k goes to infinity) has polynomial growth. Moreover one can naturally associate a convex body ∆ to S such that the degree and coefficient of growth of HS is given by the dimension and volume of ∆ respectively. We will see how this combinatorial theorem gives a beautiful proof of the Bernstein-Khovanskii-Kushnirenko theorem on the number of (complex) solutions of a system of polynomial equations in terms of volumes of their Newton polytopes. This is the beginning of the theory of Newton polytopes and toric varieties, which is an important part of combinatorial algebraic geometry and commutative algebra. Towards the end we will briefly touch the newly emerged theory of Newton- Okounkov bodies which attempts to far generalize the notion of Newton polytope of a polynomial. 10.2. Undergraduate resarch vision. Because of its elementary nature, the subject of the course does not require much background and is accessible to a wide range of audience. Firstly, the course familiarizes the students with fundamental notions from convex geometry which naturally appear in many areas of mathematics. Secondly, it serves as a beautiful 14 MARTA LEWICKA AND JUAN MANFREDI invitation to classical as well as modern algebraic geometry, and touches the cutting edge of research in this field. The approach is geometric and intuitive and gives students an excellent research experience in algebraic geometry which (unfortunately) sometimes is believed to be very demanding, inaccessible and abstract. 10.3. Applications and broader impact. Convexity plays a fundamental role in mathematics, and its ubiquity in optimization makes it of crucial importance in many domains of application such as linear and semi-definite programming and combinatorial optimization with numerous applications in engineering. On the other hand, toric algebraic geometry and in particular the Bernstein-Kushnirenko-Khovanskii theorem plays an important role in computational algebra, specially in developing efficient algorithms to find solutions of systems of polynomial equations with wide range of applications such as in chemistry (molecular modeling), robotics and computer graphics. 10.4. Topics covered. (i) Convex polytopes: Basic definitions (Minkowski sum of convex bodies), Vector space of virtual convex bodies and volume polynomial, Mixed volume, Geometric inequalities (Brunn-Minkowski and Alexandrov-Fenchel inequalities), Support function, Number of faces of a convex polytope (f-vector). (ii) Semigroups of integral points and systems of algebraic equations: Semigroups of integral points and their asymptotic behavior, Erhart function of a polytope, Hilbert function of a graded algebra and Hilbert’s theorem, Newton polytope of a polynomial, Projective toric varieties, Systems of algebraic equations and the Bernstein- Khovanskii-Kushnirenko theorem, Newton-Okounkov bodies. 10.5. Suggested projects. (i) During the course we will prove a beautiful theorem about the asymptotic behavior of semigroups of integral points. This project attempts to extend this to non- commutative semigroups. We introduce the notion of a ”twisted” semigroup of integral points which is a semigroup of the semi-direct product Zn o N where N acts on Zn by a permutation of coordinates on Zn. We expect that, as in the commutative case, we can define a convex body whose dimension and volume describe the asymptotic behavior. Understanding the asymptotic behavior of such semigroups right away gives important information about the Hilbert functions of so-called ”twisted homogeneous coordinate rings” which are important examples of graded algebras appearing in non-commutative geometry. To our knowledge, not much is known about the Hilbert functions of twisted homogeneous coordinate rings. (ii) Let ∆ be a polytope in Rn with integral vertices. The number of integral points in k∆ (regarded as a function of k) is the so-called Erhart function of ∆. It is a polynomial for large values of k. Similarly, for a finite set A of points in Zn, the number of points in the k-fold sum A + ··· + A is also a polynomial for large k, and moreover this polynomial has the same asymptotic as the Erhart polynomial of the convex hull of A. Both these functions/polynomials are important and have been subject of study in algebraic combinatorics and combinatorial commutative algebra for the past few decades. They are related to Hilbert polynomials of projective 15

toric varieties. In this project we attempt to relate these two polynomials. More precisely, we try to find a formula for the number of points in A + ··· + A in terms of the the Erhart polynomial of convex hull of A and its faces. We expect that a complete answer to this problem in general is rather difficult. The project will address the two dimensional case of this problem (i.e. for polygons). (iii) The celebrated ”localization formula” in equivariant cohomology, applied to smooth projective toric varieties, implies an interesting combinatorial formula for the volume of a (simplicial) polytope in terms of edge vectors of its vertices. More precisely it expresses the volume as a sum over vertices of a certain expression in terms of edge vectors. In this project we attempt to give a combinatorial proof of this, and also investigate efficiency of algorithms for computing volume (of convex hull of a finite set of points) based on this formula.

11. Course 11: Enumerative geometry and Schubert calculus by Kiumars Kaveh 11.1. What is this course about. How many circles are tangent to 3 given circles? How many lines in space intersect four given lines? Problems of this kind are subject of enumerative geometry which has roots in antiquity. In ground breaking work [S] Schubert showed that the answer to many questions in enumerative geometry lies in understanding the geometry and topology of ”Grassmanian varieties” and ”flag varieties”. Generalizing the notion of projective space, every point in a Grassmanian Gr(n, k) is represented by a k- dimensional vector subspace in the complex vector space Cn. The points in the flag variety are flags of vector subspaces in Cn. Schubert’s work motivated a huge amount of research and inspired modern branches of mathematics such as geometric representation theory. One of the famous Hilbert problems asked to give a precise foundation for enumerative geometry and Schubert calculus. Attempting to answer the Hilbert problem enumerative geometry has seen immense development in the last century. This course will cover basic material about the geometry of the Grassmanians and flag varieties. Investigating geometry of the flag variety involves facts from linear algebra and matrices such as Bruhat decomposition. We will discuss Schubert varieties inside Grassma- nian and flag variety and will see examples of how understudying the geometry/topology of the flag variety and its Schubert varieties answers questions from enumerative geometry. It will naturally be related with the combinatorics of the group of permutations, simple transpositions, its Bruhat graph and root system. At the end we will talk about the Gelfand-Cetlin polytopes which are polytopes associated to the flag varieties and encode interesting information about their geometry/topology [KST]. 11.2. Undergraduate research vision. The course will expose the students to concepts from several central and intertwining branches of mathematics. While the motivating problems have roots in classical geometry, the topic serves as a very nice introduction to modern areas. The main objects of study are flag varieties and Grassmanians whose study involves interplay between linear algebra, geometry, combinatorics, and some convex geometry. Be- cause of the combinatorial nature of the methods used there are many interesting problems of research importance which are in the scope of a good undergraduate student in mathematics. 16 MARTA LEWICKA AND JUAN MANFREDI

11.3. Applications and broader impact. The course is an invitation to (geometric) representation theory, Lie groups and algebraic geometry. Representation theory and the combinatorics involved has a wide range of applications in physics and quantum chemistry (where it is used to describe the group of symmetries of a molecule). On the other hand, notions from geometric representation theory and enumerative geometry such as the Gromov-Witten invariants play an important role in string theory, a promising candidate for the ”theory of everything” in modern physics. The concepts and ideas in geometric representation theory (which have deep connections with symplectic geometry and classical mechanics) are crucial in extending the classical theories of physics to quantum versions.

11.4. Topics covered. Flag variety and Grassmanian, Pl¨ucker embedding in projective space, Bruhat decomposition and Schubert varieties, Some Schubert calculus, Weyl group and Bruhat graph, A touch of representation theory of GL(n, C), Gelfand-Cetlin polytopes.

11.5. Suggested projects. (i) A famous problem of Schubert asserts that: the number of space quadrics tangent to 9 quadrics in general position is 666, 841, 088. Compute this number by computing mixed volumes of Gelfand-Cetlin polytopes. This gives a new and remarkably nice solution to a classical problem. (ii) Compute number of vertices and edges of the Gelfand-Cetlin polytopes. This project is a ﬁrst step towards understanding the combinatorics of Gelfand-Cetlin polytopes which are expected to play an important role in geometric representation theory, enumerative geometry and several related areas. (iii) Compute topological invariants (e.g. Euler characteristic) of hypersurfaces in the ﬂag variety using the combinatorics of Bruhat graph and Gelfand-Cetlin polytopes.

11.6. References. [KST] Kiritchenko, V.; Smirnov, E.; Timorin, V. Schubert calculus and Gelfand-Zetlin polytopes. arXiv:1104.1089 [S] Schubert, H. Lösungdes Charakteristiken-Problems fürlineare Räumebeliebiger Dimension Mitt. Math. Gesellschaft Hamburg , 1 (1889), 134–155

12. Course 12: Nilpotence (2 weeks course 12hrs) by Bogdan Ion 12.1. What is this course about. The lectures will be addressing the classiﬁcation and properties of conjugacy classes of nilpotent matrices (or, more generally, of nilpotent orbits in semisimple Lie algebras) and rudiments of representation theory of nilpotent Lie groups focusing on the Heisenberg group. This type of structure emerges from, plays a fundamental role in, and provides connections between several mathematical ﬁelds (Lie theory, geometry, topology, harmonic analysis). Highlights of the course will include: parametrization, geometry and topology of nilpotent orbits (i.e. closures, symplectic structure, fundamental group), the Jacobson-Morozov theorem, the Dynkin-Kostant theorem, (weighted) Dynkin diagrams, Coxeter groups, the Heisenberg group, the Stone-von Neumann theorem, the metaplectic group. 17

12.2. Undergraduate research vision. The lectures require only minimal prerequisites in linear algebra (nilpotent matrices, the Jordan canonical form) and the results to be discussed are amenable to elementary techniques. The more advanced concepts will be introduced mainly through examples, which are abundant and in most cases exhaustive. Early on, combinatorial parametrizations will be set in place and used heavily to encode the ensuing structure. Many connections with other subjects will be brieﬂy addressed. One of the goals of the course is to emphasize the unity and richness of mathematics. 12.3. Applications and broader impact. The subject matter serve as a launching pad for some of the most signiﬁcant developments of the past few decades: the Springer correspondence, the orbit method, McKay correspondence, the Lie theoretic construction of simple singularities, primitive ideals in universal enveloping algebras, associated varieties, q-weight multiplicities, the spherical unitary dual, etc. It will also lead to improved undergraduate STEM education and, consequently, to the development of a globally competitive STEM workforce. 12.4. Suggested projects. (i) Kleinian sigularities [The subregular nilpotent orbit, the transversal slice, Brieskorn’s theorem, explicit illustration in type A] (ii) Springer correspondence [Statement of the type A correspondence, illustrating com- putations, comparison with other constructions of irreducible representations for the symmetric group] (iii) The orbit method [Kirilov’s construction, details for the Heisenberg group, other examples] (iv) The Weil/Segal-Shale/oscillator representation [Construction, properties of the Fourier transform derived from the heat kernel for the harmonic oscillator] (v) Theta functions [Basic theory, interpretation in terms of the Weil representation, Weil’s proof of quadratic reciprocity]

13. Course 13: Introduction to Toric Varieties and Discrete Convex Geometry by Alexander Borisov 13.1. What this course is about. The basic objects of Discrete Convex Geometry are integer solutions of systems of linear inequalities. These questions are almost as old as mathematics itself, and, perhaps more importantly, they naturally appear in a diverse array of mathematics-related disciplines, most notably in Operations Research and Economics. Toric varieties are some particular examples of algebraic varieties that can be constructed from Discrete Convex Geometry objects. By virtue of their deﬁnition, they relate Discrete Convex Geometry and Algebraic Geometry, but they also have deep connections to other branches of Mathematics (Number Theory, Geometry of Numbers, Combinatorics, to name a few) as well as Mathematical Physics, Operations Research and Computer Science ([Bor4], [LZ], [E], [G], [KS], [S]). Since Algebraic Geometry requires a background in Commutative Algebra, which most undergraduates lack, our approach to the subject will be elementary. Rather than proudly developing the general theory, we will be focusing on the series of examples of increasing level of diﬃculty. Perhaps the main goal of this course is to foster the development of the 18 MARTA LEWICKA AND JUAN MANFREDI students’ geometrical intuition. We will also outline some of the many connections of this subject to other areas of human knowledge.

13.2. Undergraduate Research Vision. In my opinion, an ideal area for undergraduate research in pure mathematics, must have the following general features: 1) It must contain many problems that can be easily explained to an undergraduate; 2) These problems should be solvable without (or with limited use of) complicated mathematical machinery; 3) These problems must be unsolved, or not solved in a satisfactory manner; 4) The area must be connected to many active areas of mathematics and related disciplines; 5) The insight developed by working in this area can help the students in many other research endeavors. Toric varieties satisfy all of the above. In fact, I can personally attest to this, as my ﬁrst research, as an undergraduate, was on toric Fano varieties ([BB]). I have since worked in many areas of pure mathematics, but toric varieties still hold a special place in my heart, and the geometric intuition that they inspired has served me well. I hope to share this passion with the new generation of researchers.

13.3. Applications and broader impact. As mentioned above, Discrete Convex Geom- etry is the mathematical basis for a number of real-life problems in Operations Research, Economics, and other areas. However, no standard courses help the students to develop the geometric intuition for these problems. The geometry is, in general, a weak link for many students, since it is usually not adequately studied in high school and is almost never oﬀered at the college level beyond the Linear Algebra. The hands-on experience with the Discrete Convex Geometry objects will boost the students’ intuition and conﬁdence, that will serve them well in graduate studies in several mathematics-related disciplines.

13.4. Suggested projects. (i) Classification of 4-dimensional non-cyclic canonical toric singularities. This is unfinished business from my recent paper [BBBK]. No deep background is required, the result would be publishable. (ii) Toric surface singularities of minimal log discrepancy at least 1/2 (more advanced: at least 1/3). These are of interest for birational geometry ([Bor2], [R]). (iii) Nash resolution conjecture for Toric Varieties. A very intriguing relatively recent conjecture. The research will involve numerical experiments using computers ([ALPPT]). (iv) Accumulation conjecture for minimal log discrepancies of toric singularities. This is unfinished business from my 1997 paper [Bor1], that has received renewed attention recently. (v) Effective theorem of Lawrence. This most challenging project is about an important and beautiful paper of Jim Lawrence [L], which is, unfortunately, not effective. The goal is to use the ideas of Lagarias and Ziegler [LZ] to rewrite the proof, making explicit estimates. This project requires some background and interest in Number Theory. 19

13.5. Prerequisites. For most projects the prerequisites include Linear Algebra, Introduc- tion to Analysis, and Introduction to Algebra (at the level of Pitt courses Math 1180, Math 0413 and Math 0430). Depending on the course load in mathematics, these prerequisites are normally satisﬁed by mathematics majors after second or third year in college. Some experience with computer software (Maple or similar software, or a general programming language) is desired. The last project does not require any computer skills, but it requires some knowledge of Number Theory.

14. Course 14: Birational Geometry of Rational Surfaces by Alexander Borisov 14.1. What is this course about. This course is primarily about rational surfaces. These surfaces are relatively easy to understand, yet they provide a great introduction into birational geometry, which has been the fastest growing area of algebraic geometry in the last several decades. They also hold the key to the Jacobian conjecture, which is one of the longest-standing conjectures in algebraic geometry. The usual way to introduce students to Algebraic Geometry is through Commutative Algebra. However for the rational surfaces one can bypass a lot of the general theory by using explicit equations of the varieties involved. The course is designed to provide the students with motivation for the more advanced study of Algebraic Geometry as well as to foster their intuition. We will also explore other related areas of mathematics, especially the combinatorics of weighted trees and their morphisms.

14.2. Undergraduate research vision. In my opinion, an ideal area for undergraduate research in pure mathematics, must have the following general features: 1) It must contain many problems that can be easily explained to an undergraduate; 2) These problems should be solvable without (or with limited use of) complicated mathematical machinery; 3) These problems must be unsolved, or not solved in a satisfactory manner; 4) The area must be connected to many active areas of mathematics and related disciplines; 5) The insight developed by working in this area can help the students in many other research endeavors. A number of open problems on rational algebraic surfaces have a very combinatorial nature, related to certain weighted trees. They can be easily described to undergraduate students with no background in Algebraic Geometry, and they can be approached, using a computer, without formally invoking any really complicated theorems. At the same time, these problems provide a good starting point for deeper learning. As a bonus, some of the research may help resolve one of the longest standing conjecture in Algebraic Geometry: the 70-some year old Jacobian Conjecture of Keller.

14.3. Applications and broader impact. The combinatorics of weighted trees is very rich. These objects appear in Mathematical Physics, Probability, Computer Science, even Genetics. The students will get to really understand some of the related objects, like the matrix of the tree, and the discrete Laplacian. They will also get a ﬁrst-hand experience 20 MARTA LEWICKA AND JUAN MANFREDI

in using computers to approach complicated problems, which is a growing trend in pure mathematics and a well-established approach in many other sciences.

14.4. Suggested projects. (i) Graphs of exceptional curves on rational surfaces containing an affine plane. Goal: create a database. (ii) Morphisms of weighted graphs. Goal: use computer to find interesting examples of maps of weighted graphs that appear in the Jacobian Conjecture and its variations ([Or]). (iii) Invariants of divisorial valuations at infinity. Goal: understand possible values of recently introduced invariants ([Bor3]). (iv) Minimal log discrepancies and virtual minimal log discrepancies of quotient singularities. Minimal log-discrepancies are very important invariants of non-smooth (singular) varieties. The virtual minimal log discrepancies is a new invariant, that can be effectively calculated in the surface case. Goal: find the values of these invariants for all quotient singularities of surfaces ([K]).

14.5. Prerequisites. For most projects the prerequisites include Linear Algebra, Intro- duction to Analysis, and Introduction to Algebra (at the level of Pitt courses Math 1180, Math 0413 and Math 0430). Depending on the course load in mathematics, these prerequisites are normally satisﬁed by mathematics majors after second or third year in college. Some experience with computer software (Maple or similar software, or a general programming language) is desired. The ﬁrst two projects require some knowledge of C++ or other general programming languages.

14.6. References. [ALPPT] A. Atanasov, C. Lopez, A. Perry, N. Proudfoot, M. Thaddeus. Resolving toric varieties with Nash blow-ups. Exp. Math. 20 (2011), no. 3, 288-303. [BBBK] M. Barile, D. Bernardi, A. Borisov, J.-M. Kantor. On empty lattice simplices in dimension 4. Proc. Amer. Math. Soc. 139 (2011), no. 12, 42474253. [B] V. V. Batyrev. Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), no. 3, 493–535. [Bor1] A. Borisov. Minimal discrepancies of toric singularities. Manuscripta Math. 92 (1997), no. 1, 33–45. [Bor2] A. Borisov. On classification of toric singularities, Algebraic geometry, 9. J. Math. Sci. (New York) 94 (1999), no. 1, 1111–1113. [Bor3] A. Borisov. On two invariants of divisorial valuations at infinity, preprint (2012). [Bor4] A. Borisov. Quotient singularities, integer ratios of factorials, and the Riemann Hypothesis. Int. Math. Res. Not. IMRN 2008, no. 15, Art. ID rnn052. [BB] A. Borisov, L. Borisov. Singular toric Fano varieties, Acad. Sci. USSR Sb. Math. 75 (1993), no. 1, 277–283. [E] Günter Ewald. Combinatorial convexity and algebraic geometry. Graduate texts in mathematics, 168 Springer-Verlag, New-York. 1996. [F] William Fulton, Introduction to toric varieties. Annals of Mathematics Studies, 131. Princeton University Press, Princeton, NJ, 1993. 21

[G] Peter Gritzmann, JörgWills, Lattice points. Handbook of convex geometry, Vol B, 765–797, North Holland, Amsterdam, 1993. [K] JánosKollár,et al. Flips and abundance for algebraic threefolds. Astérisque 211 (1992). [KS] M. Kreuzer, H. Skarke. Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys. 4 (2000), no. 6, 1209-1230 [L] Jim Lawrence. Finite unions of closed subgroups of the n−dimensional torus. Applied geometry and discrete mathematics, 433–441, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 4, Amer. Math. Soc., Providence, RI, 1991. [LZ] Jeffrey C. Lagarias, Günter M Ziegler. Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Canad. J. Math 43 (1991), no. 5, 1022–1035. [O] Tadao Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Translated from Japanese. Springer-Verlag, Berlin-New York, 1988. [Or] S. Yu. Orevkov, Counter-examples to the ”Jacobian Conjecture at Infinity”, Trudy Mat. Inst. im. V.A.Steklova, 235 (2001), 181–210. Translated in Proc. Steklov Math. Inst., 235 (2001), 173–201. [R] Miles Reid. Decomposition of toric morphisms. Arithmetic and geometry. Pap. dedic. I. R. Shafarevich, Vol II, Progr. Math., 36 (1983), 395–418. [S] Herbert E. Scarf. Integral polyhedra in three space. Mathematics of Operations Research 10 (1985), no. 3, 403–438.

15. Course 15: Delaunay tesselations by Jason DeBlois 15.1. What this course is about. Given a collection S of points in space, the Delau- nay tesselation canonically solves the meshing problem, of producing a collection of non- overlapping polyhedra (the mesh) with vertex set S. It has a beautifully simple charac- terization via the “empty circumcircles” condition, can be eﬃciently implemented on a computer, and has a number of optimality properties. For these and other reasons the Delaunay tesselation has proven useful across a broad spectrum of mathematics both pure and applied. The course and projects will focus on understanding and applying the Delaunay tesselation in the context of geometry. We intend to introduce it in the simplest possible context, that of two-dimensional Euclidean geometry, and move from there to research-level problems in spherical and hyperbolic geometry, as well as 3-dimensional Euclidean space. Syllabus includes: Introduction to the constant-curvature geometries: Euclidean, spherical, and hyperbolic. The Delaunay tesselation, the Voronoi tesselation, and duality. Com- puter implementation using Python.

15.2. Sugested projects. (i) Sliver elimination in R3. A “sliver” is a tetrahedron that is contained in a small neighborhood of a planar rectangle. Slivers damage the Delaunay tesselation’s usefulness in dimension three, but their occurrence cannot be easily predicted by 22 MARTA LEWICKA AND JUAN MANFREDI

obvious features of the original collection S. We will explore strategies for elimi- nating slivers by changing the meshing strategy or augmenting S with additional points. Computer data collection will be involved. (ii) Regular tetrahedra sharing a vertex. A simple question, related to a famously incorrect assertion of Aristotle, is this: how many regular tetrahedra in R3 can share a vertex? The answer is either 20 (a known configuration), 21, or 22. This is really a question about spherical geometry: a unit-radius sphere centered at the shared vertex intersects each tetrahedron in an equilateral triangle with sides of length of π/3, and the upper bound of 22 on the number of such triangles follows from the spherical law of cosines and area considerations. This project will work toward determining the optimal configuration of equilateral triangles with side length π/3 on the sphere. The Delaunay tesselation is a fundamental tool for attacking such packing problems, and computer case analysis is often used as well. Some of my own work may also apply. (iii) Moduli spaces of hyperbolic surfaces. A surface of genus g ≥ 2 admits a non- compact, (6g − 6)-dimensional moduli space of hyperbolic structures up to isome- try. This project will address the question, ”How well can one know a hyperbolic surface using only a well-chosen, finite set of points?” In particular, we will identify constants g,k, for k ∈ N, such that the set Cg,k, of genus-g hyperbolic surfaces that possess a collection of k points with injectivity radius greater than g,k, is compact. We will investigate the behavior of g,k for fixed g as k → ∞, and vice versa, and determine whether the Cg,k give an exhaustion of the moduli space by compact sets. We will use my own work on the Delaunay tesselation.

16. Course 16: Character varieties and geometric structures by Jason DeBlois and Kiumars Kaveh 16.1. What this course is about. A geometric structure on a manifold X often deter- mines a representation of π1X into a Lie group. For instance, a Riemann surface X with genus at least two admits a large family of metrics locally modeled on that of the hyperbolic plane, each determining a “holonomy” representation into PSL(2, R). The same holds true for, say, complex projective structures or ﬂat connections on such X, with PSL(2, R) respectively replaced by PSL(2, C) or SU (n). For a manifold X and Lie group G like the above, the collection V of all representations of π1X into G has the structure of an aﬃne algebraic set. This is because G is the solution set to a system of equations on a space of matrices — for instance, PSL(2, R) = a b 4 {( c d ) ∈ R | ad − bc = 1} — and a presentation for π1X translates into a further system of equations cutting out V . There is a rich tradition of relating topological and geometric properties of X to algebro- geometric properties of its representation variety V , or to properties of associated spaces such as the character variety or moduli spaces. The course below aims to introduce this tradition in the context of 2- and 3-manifolds, and the projects explore it in more depth.

16.2. Course Outline. Two to four weeks. In two parts, one taught by DeBlois and one by Kaveh. 23

(i) Geometric structures: Introduction to manifolds, some 2- and 3-dimensional examples. Hyperbolic geometry and hyperbolic structures. Riemannian geometry and connections. Projective structures. (ii) Algebraic Geometry” Aﬃne algebraic sets and projective completions. Smooth models. Representation and character varieties. Moduli spaces. 16.3. Suggested projects.

(i) Identifying character varieties. For a given 3-manifold X, a presentation for π1X leads to a set of polynomials defining the PSL(2, C) -character variety V of X as an affine algebraic set. However it may not be trivial to use this data to identify the projective completion of V as an algebraic-geometric object. This is of cur- rent interest in the study of 3-manifolds; for example, the character variety of the Whitehead link complement was recently identified as a rational surface isomorphic to P2 blown up at 10 points. Subtle issues here include smoothness and the difference between birational equivalence and isomorphism. This project will attempt to resolve these subtleties for knots in the tables. (ii) Degenerations of character varieties. This project explores embeddings of these moduli spaces into projective spaces and their degenerations into so-called ”toric varieties”. Roughly speaking, if we have an embedding Φ of a variety V into a projective space, where components of Φ are given by polynomials, a ”toric degeneration” of (V, Φ) is a (well-behaved) algebraic deformation of (components of) Φ into a map whose components are monomials, and hence much simpler to study. The focus of the project will be on Riemann surfaces of genus 1, i.e. elliptic curves. Even in genus 1 case and when the group is SU(2) or PSL(2, C), the question of toric degeneration of the moduli space is subtle and important. The character variety of a Riemann surface (modulo conjugation) appears in many contexts such as mathematical physics (Yang-Mills theory), gauge theory, algebraic geometry and topology. They can alternatively be regarded as moduli space of flat connections on the Riemann surface. Understanding degenerations of these moduli spaces is expected to shed much light on several fundamental problems. Their toric degenerations are suggested by the construction of Jeffrey-Weitzman completely integrable system on the moduli space of flat connections. (iii) The action of Mod(S) on the PSL(2, R)-character variety. For a surface S and a Lie group G, the outer automorphism group Mod(S) of π1S acts naturally on its G-character variety. When G = PSL(2, K) for K = R or C, this action is “nice” — properly discontinuous — on the subset (S) consisting of characters of geometrically-determined representations. However the action is still not well- understood on the complement of A. In particular, the following question is open: does Mod(S) act ergodically on the complement of A? We will investigate.

References