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 mod- ern 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 tech- niques 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 1-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 1-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 con- tinuous 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 com- puting. 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: con- struction 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 struc- tures 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, bend- ing/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.