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Series: Geometry Topology Seminar

Mapping classes are the natural topological symmetries of surfaces. Their study is often restricted to the orientation-preserving ones, which form a normal subgroup of index two in the group of all mapping classes. In this talk, we discuss orientation-reversing mapping classes. In particular, we show that Lehmer's question from 1933 on Mahler measures of integer polynomials can be reformulated purely in terms of a comparison between orientation-preserving and orientation-reversing mapping classes.

Friday, November 16, 2018 - 15:05 ,
Location: Skiles 156 ,
Sergio Mayorga ,
Georgia Tech ,
Organizer: Jiaqi Yang

In this talk I will begin by discussing the main ideas of mean-field games and then I will introduce one specific model, driven by a smooth hamiltonian with a regularizing potential and no stochastic noise. I will explain what type of solutions can be obtained, and the connection with a notion of Nash equilibrium for a game played by a continuum of players.

Series: Combinatorics Seminar

Let P be a system of unique shortest paths through a graph with real edge weights (i.e. a finite metric). An obvious fact is that P is "consistent," meaning that no two of these paths can intersect each other, split apart, and then intersect again later. But is that all? Can any consistent path system be realized as unique shortest paths in some graph? Or are there more forbidden combinatorial intersection patterns out there to be found?
In this talk, we will characterize exactly which path systems can or can't be realized as unique shortest paths in some graph by giving a complete list of new forbidden intersection patterns along these lines. Our characterization theorem is based on a new connection between graph metrics and certain boundary operators used in some recent graph homology theories. This connection also leads to a principled topological understanding of some of the popular algebraic tricks currently used in the literature on shortest paths. We will also discuss some applications in theoretical computer science.

Series: Geometry Topology Seminar

I will explain some connections between the counting of incompressible surfaces in hyperbolic 3-manifolds with boundary and the 3Dindex of Dimofte-Gaiotto-Gukov. Joint work with N. Dunfield, C. Hodgson and H. Rubinstein, and, as usual, with lots of examples and patterns.

Series: Algebra Seminar

Let K be a non-trivially valued non-Archimedean field, R its valuation subring. A formal Gubler model is a formal R-scheme that comes from a polyhedral decomposition of a tropical variety. In this talk, I will present joint work with Sam Payne in which we show that any formal model of any compact analytic domain V inside a (not necessarily projective) K-variety X can be dominated by a formal Gubler model that extends to a model of X. This result plays a central role in our work on "structure sheaves" on tropicalizations and our work on adic tropicalization. If time permits I will explain some of this work.

Series: ACO Student Seminar

Consider a linear combination of independent identically distributed random variables $X_1, . . . , X_n$ with fixed weights $a_1, . . . a_n$. If the random variablesare continuous, the sum is almost surely non-zero. However, for discrete random variables an exact cancelation may occur with a positive probability. Thisprobability depends on the arithmetic nature of the sequence $a_1, . . . a_n$. We will discuss how to measure the relevant arithmetic properties and how to evaluate the probability of the exact and approximate cancelation.

Series: School of Mathematics Colloquium

please note special time!

Random matrices arise naturally in various contexts ranging from theoretical physics to computer science. In a large part of these problems, it is important to know the behavior of the spectral characteristics of a random matrix of a large but fixed size. We will discuss a recent progress in this area illustrating it by problems coming from combinatorics and computer science:
Condition number of “full” and sparse random matrices. Consider a system of linear equations Ax = b where the right hand side is known only approximately. In the process of solving this system, the error in vector b gets magnified by the condition number of the matrix A. A conjecture of von Neumann that with high probability, the condition number of an n × n random matrix with independent entries is O(n) has been proven several years ago. We will discuss this result as well as the possibility of its extension to sparse matrices.
Random matrices in combinatorics. A perfect matching in a graph with an even number of vertices is a pairing of vertices connected by edges of the graph. Evaluating or even estimating the number of perfect matchings in a given graph deterministically may be computationally expensive. We will discuss an application of the random matrix theory to estimating the number of perfect matchings in a de- terministic graph.
Random matrices and traffic jams. Adding another highway to an existing highway system may lead to worse traffic jams. This phenomenon known as Braess’ paradox is still lacking a rigorous mathematical explanation. It was recently explained for a toy model, and the explanation is based on the properties of the eigenvectors of random matrices.

Series: Stochastics Seminar

(Based on joint work with Cécile Mailler)Consider a stochastic process that behaves as a d-dimensional simple and symmetric random walk, except that, with a certain fixed probability, at each step, it chooses instead to jump to a given site with probability proportional to the time it has already spent there. This process has been analyzed in the physics literature under the name "random walk with preferential relocations", where it is argued that the position of the walker after n steps, scaled by log(n), converges to a Gaussian random variable; because of the log spatial scaling, the process is said to undergo a "slow diffusion". We generalize this model by allowing the underlying random walk to be any Markov process and the random run-lengths (time between two relocations) to be i.i.d.-distributed. We also allow the memory of the walker to fade with time, meaning that when a relocations occurs, the walker is more likely to go back to a place it has visited more recently. We prove rigorously the central limit theorem described above by associating to the process a growing family of vertex-weighted random recursive trees and a Markov chain indexed by this tree. The spatial scaling of our relocated random walk is related to the height of a typical vertex in the random tree. This typical height can range from doubly-logarithmic to logarithmic or even a power of the number of nodes of the tree, depending on the form of the memory.

Thursday, November 15, 2018 - 13:30 ,
Location: Skiles 006 ,
Marcel Celaya ,
Georgia Tech ,
Organizer: Trevor Gunn