Georgia Institute of Technology College of Sciences

Many complex models from science and engineering can be studied in the framework of coupled systems of differential equations on networks. A network is given by a directed graph. A local system is defined on each vertex, and directed edges represent couplings among vertex systems. Questions such as stability in the large, synchronization, and complexity in terms of dynamic clusters are of interest. A more recent approach is to investigate the connections between network topology and dynamical behaviours. I will present some recent results on the construction of global Lyapunov functions for coupled systems on networks using a graph theoretic approach, and show how such a construction can help us to establish global behaviours of compelx models.

In this talk, I will present work on two very different problems, with the only common theme being a substantial departure from standard approaches. In the first part, I will discuss how the spread of many common contagions may be more accurately modeled with nonlocal approaches than with the current standard of local approaches, and I will provide a minimal mathematical foundation showing how this can be done. In the second part, I will present a new computational method for ranking items given only a set of pairwise preferences between them. (This is known as the minimum feedback arc set problem in computer science.) For a broad range of cases, this method appears to beat the current "world record" in both run time and quality of solution.

Theoretical aspects: If a smooth dynamical system on a compact invariant set is structurally stable, then it has the shadowing property, that is, any pseudo (or approximate) orbit has a true orbit nearby. In fact, the system has the Lipschitz shadowing property, that is, the distance between the pseudo and true orbit is at most a constant multiple of the local error in the pseudo orbit. S. Pilyugin and S. Tikhomirov showed the converse of this statement for discrete dynamical systems, that is, if a discrete dynamical system has the Lipschitz shadowing property, then it is structurally stable. In this talk this result will be reviewed and the analogous result for flows, obtained jointly with S. Pilyugin and S. Tikhomirov, will be described. Numerical aspects: This is joint work with Brian Coomes and Huseyin Kocak. A rigorous numerical method for establishing the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous system of ordinary differential equations is presented. Given a suitable approximate connecting orbit and assuming that a certain associated linear operator is invertible, the existence of a true connecting orbit near the approximate orbit and for a nearby parameter value is proved provided the approximate orbit is sufficiently ``good''. It turns out that inversion of the operator is equivalent to the solution of a boundary value problem for a nonautonomous inhomogeneous linear difference equation. A numerical procedure is given to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse (the latter determines how ``good'' the approximating orbit must be).

A notorious open problem in arithmetic geometry asks whether ranks ofelliptic curves are unbounded in families of quadratic twists. A proof ineither direction seems well beyond the reach of current techniques, butcomputation can provide evidence one way or the other. In this talk wedescribe two approaches for searching for high rank twists: the squarefreesieve, due to Gouvea and Mazur, and recursion on the prime factorization ofthe twist parameter, which uses 2-descents to trim the search tree. Recentadvances in techniques for Selmer group computations have enabled analysisof a much larger search region; a large computation combining these ideas,conducted by Mark Watkins, has uncovered many new rank 7 twists of$X_0(32): y^2 = x^3 - x$, but no rank 8 examples. We'll also describe aheuristic argument due to Andrew Granville that an elliptic curve hasfinitely many (and typically zero) quadratic twists of rank at least 8.

(Alon-Tarsi Conjecture): For even n, the number of even nxn Latin squares differs from the number of odd nxn Latin squares. (Stones-Wanless, Kotlar Conjecture): For all n, the number of even nxn Latin squares with the identity permutation as first row and first column differs from the number of odd nxn Latin squares of this type. (Aharoni-Berger Conjecture): Let M and N be two matroids on the same vertex set, and let A1,...,An be sets of size n + 1 belonging to both M and N. Then there exists a set belonging to both M and N and meeting all Ai. We prove equivalence of the first two conjectures and a special case of the third one and use these results to show that Alon-Tarsi Conjecture implies Rota's bases conjecture for odd n and any system of n non-singular real valued matrices where one of them is non-negative and the remaining have non-negative inverses.Joint work with Ron Aharoni.

A classical theorem of John Wermer asserts that the algebra of continuous functions on the circle with holomophic extensions to the disc is a maximal subalgebra of the algebra of all continuous functions on the circle. Wermer's theorem has been extended in numerous directions. These will be discussed with an emphasis on extensions to several complex variables.

An introduction for non-experts on real and finite Euler sums, also known as multiple zeta values.

Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this setup to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as "combinatorial shadows" of curves. This tropical relationship between graphs and algebraic curves has led to beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs.

The talk consists of two parts.The first part is devoted to results in Discrepancy Theory. We consider geometric discrepancy in higher dimensions (d > 2) and obtain estimates in Exponential Orlicz Spaces. We establish a series of dichotomy-type results for the discrepancy function which state that if the $L^1$ norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be very large in some other function space.The second part of the thesis is devoted to results in Additive Combinatorics. For a set with small doubling an order-preserving Freiman 2-isomorphism is constructed which maps the set to a dense subset of an interval. We also present several applications.

It is well known that many optimization problems, ranging from linear programming to hard combinatorial problems, as well as many engineering and economics problems, can be formulated as linear complementarity problems (LCP). One particular problem, finding a Nash equilibrium of a bimatrix game (2 NASH), which can be formulated as LCP, motivated the elegant Lemke algorithm to solve LCPs. While the algorithm always terminates, it can generates either a solution or a so-called ‘secondary ray’. We say that the algorithm resolves a given LCP if a secondary ray can be used to certify, in polynomial time, that no solution exists. It turned out that in general, Lemke-resolvable LCPs belong to the complexity class PPAD and that, quite surprisingly, 2 NASH is PPAD-complete. Thus, Lemke-resolvable LCPs can be formulated as 2 NASH. However, the known formulation (which is designed for any PPAD problem) is very complicated, difficult to implement, and not readily available for potential insights. In this talk, I’ll present and discuss a simple reduction of Lemke-resolvable LCPs to bimatrix games that is easy to implement and have the potential to gain additional insights to problems (including several models of market equilibrium) for which the reduction is applicable.

Applicability of Pearson's correlation as a measure of explained variance is by now well understood. One of its limitations is that it does not account for asymmetry in explained variance. Aiming to obtain broad applicable correlation measures, we use a pair of r-squares of generalized regression to deal with asymmetries in explained variances, and linear or nonlinear relations between random variables. We call the pair of r-squares of generalized regression generalized measures of correlation (GMC). We present examples under which the paired measures are identical, and they become a symmetric correlation measure which is the same as the squared Pearson's correlation coefficient. As a result, Pearson's correlation is a special case of GMC. Theoretical properties of GMC show that GMC can be applicable in numerous applications and can lead to more meaningful conclusions and decision making. In statistical inferences, the joint asymptotics of the kernel based estimators for GMC are derived and are used to test whether or not two random variables are symmetric in explaining variances. The testing results give important guidance in practical model selection problems. In real data analysis, this talk presents ideas of using GMCs as an indicator of suitability of asset pricing models, and hence new pricing models may be motivated from this indicator.

We discuss a technique, going back to work of Molchanov, for determining the small-time asymptotics of the heat kernel (equivalently, the large deviations of Brownian motion) at the cut locus of a (sub-) Riemannian manifold (valid away from any abnormal geodesics). We relate the leading term of the expansion to the structure of the cut locus, especially to conjugacy, and explain how this can be used to find general bounds as well as to compute specific examples. We also show how this approach leads to restrictions on the types of singularities of the exponential map that can occur along minimal geodesics. Further, time permitting, we extend this approach to determine the asymptotics for the gradient and Hessian of the logarithm of the heat kernel on a Riemannian manifold, giving a characterization of the cut locus in terms of the behavior of the log-Hessian, which can be interpreted in terms of large deviations of the Brownian bridge. Parts of this work are joint with Davide Barilari, Ugo Boscain, and Grégoire Charlot.

Ishwari will cover chapter 5 section 1 of Bounded Analytic Functions.

Since the 50s and Nash general proof of equilibrium existence in games it is well understood that even simple games may have many, even uncountably infinite, equilibria with different properties. In such cases a natural question arises, which equilibrium is the right one? In this work, we perform average case analysis of evolutionary dynamics in such cases of games. Intuitively, we assign to each equilibrium a probability mass that is proportional to the size of its region of attraction. We develop new techniques to compute these likelihoods for classic games such as the Stag Hunt game (and generalizations) as well as balls-bins games. Our proofs combine techniques from information theory (relative entropy), dynamical systems (center manifold theorem), and algorithmic game theory. Joint work with Georgios Piliouras

We present the proof of the Shannon-McMillan-Breiman Theorem. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.

The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: The University of Alabama at Birmingham, The Georgia Institute of Technology, Emory University, The University of Tennessee Knoxville. The following six speakers will give presentations on topics that include geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology: Robert Finn (Stanford University), Bo Guan (Ohio State University), John Harvey (University of Notre Dame), Fernando Schwartz (University of Tennessee), Henry Wente (Toledo, Ohio), Xiangwen Zhang (Columbia University) .