Georgia Institute of Technology College of Sciences

We will talk a little about realizing automorphisms of a free group as graph maps and how to use Stallings folds to study them.

Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra.  Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear.  In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions.  These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.  

Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher. 

Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra.  Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear.  In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions.  These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.  

Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher. 

We will introduce an analogue of big mapping class groups as defined by Algom-Kfir and Bestvina which hopes to answer the question: What is “Big Out(Fn)”? This group will consist of proper homotopy classes of proper homotopy equivalences of locally finite, infinite graphs. We will then discuss some classification theorems related to the coarse geometry of these groups. This is joint work with Hannah Hoganson and Sanghoon Kwak.

Gaussian processes (GPs) are widely employed as versatile modeling and predictive tools in spatial statistics, functional data analysis, computer modeling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on GPs over Riemannian manifolds in order to develop richer and more flexible inferential frameworks. While GPs have been extensively studied for asymptotic inference on Euclidean spaces using positive definite covariograms, such results are relatively sparse on Riemannian manifolds. We undertake analogous developments for GPs constructed over compact Riemannian manifolds. Building upon the recently introduced Matérn covariograms on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Matérn Gaussian random measures on compact manifolds to derive the microergodic parameters and formally establish the consistency of their maximum likelihood estimates as well as asymptotic optimality of the best linear unbiased predictor.

Each point x in Gr(r, n) corresponds to an r × n matrix Ax which gives rise to a matroid Mx on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|My = Mx} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals Ix of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of Ix geometrically when the combinatorics of the matroid is sufficiently rich.

A distinctive feature of nonlinear evolution equations is the possibility of breakdown of solutions in finite time. This phenomenon, which is also called singularity formation or blowup, has both physical and mathematical significance, and, as a consequence, predicting blowup and understanding its nature is a central problem of the modern analysis of nonlinear PDEs.

In this talk we concentrate on wave maps – a geometric nonlinear wave equation – and we discuss the existence and stability of self-similar solutions, as in all higher dimensions they appear to drive the generic blowup behavior. We outline a novel framework for studying global-in-space stability of such solutions; we then men-tion some long-awaited results that we thereby obtained, and, finally, we discuss the new mathematical challenges that our approach generates.

The chromatic index $\chi'(G)$ of a graph $G$ is the least number of colors assigned to the edges of $G$ such that no two adjacent edges receive the same color. The total chromatic number $\chi''(G)$ of a graph $G$ is the least number of colors assigned to the edges and vertices of $G$ such that no two adjacent edges receive the same color, no two adjacent vertices receive the same color and no edge has the same color as its two endpoints. The chromatic index and the total chromatic number are two of few fundamental graph parameters, and their correlation has always been a subject of intensive study in graph theory.

By definition, $\chi'(G) \le \chi''(G)$ for every graph $G$. In 1984,  Goldberg conjectured that for any multigraph $G$, if $\chi'(G) \ge \Delta(G) +3$ then $\chi''(G) = \chi'(G)$. We show that Goldberg's conjecture is asymptotically true. More specifically,  we prove that for a multigraph $G$ with maximum degree $\Delta$ sufficiently large, $\chi''(G) = \chi'(G)$ provided $\chi'(G) \ge \Delta + 10\Delta^{35/36}$.  When $\chi'(G) \ge \Delta(G) +2$, the chromatic index $\chi'(G)$ is completely determined by the fractional chromatic index. Consequently,   the total chromatic number $\chi''(G)$ can be computed in polynomial-time in this case.

We study reducible surgeries on knots in S^3, developing thickness bounds for L-space knots that admit reducible surgeries and lower bounds on the slice genus of general knots that admit reducible surgeries. The L-space knot thickness bounds allow us to finish off the verification of the Cabling Conjecture for thin knots. Our techniques involve the d-invariants and mapping cone formula from Heegaard Floer homology. This is joint work with Holt Bodish.

Let M be the 3-manifold obtained by r-surgery on the right handed trefoil knot. Classification of contact structures on such manifolds have been mostly understood for r \geq 1 and r=0. Etnyre-Min-Tosun has an upcoming work on the classification of the tight contact structures for all r. The fillability of contact structures on M is mostly understood if r is not between 0 and 1/2. In this talk, we will discuss the fillability of the contact structures M for 0

We will discuss two types of structured multi-objective optimization programs.  In the first, the goal is to minimize a sum of functions described by a graph: each function is associated with a vertex, and there is an edge between vertices if two functions share a subset of their variables.   Problems of this type arise in state estimation problems, including simultaneous localization and mapping (SLAM) in robotics, tracking, and streaming reconstruction problems in signal processing.  We will show that under mild smoothness conditions, these types of problems exhibit a type of locality: if a node is added to the graph (changing the optimization problem), the optimal solution changes only for variables that are ``close’’ to the added node, immediately giving us a quick way to update the solution as the graph grows.

In the second part of the talk, we will consider a multi-task learning problem where the solutions are expected to lie in a low-dimensional subspace.  This corresponds to a low-rank matrix recover problem where the columns of the matrix have been ``sketched’’ independently.  We show that a novel convex relaxation of this problem results in optimal sample complexity bounds.  These bounds demonstrate the statistical leverage we gain by solving the problem jointly over solving each individually.

Independent component analysis is a useful and general data analysis tool. It has found great successes in many applications. But in recent years, it has been observed that many popular approaches to ICA do not scale well with the number of components. This debacle has inspired a growing number of new proposals. But it remains unclear what the exact role of the number of components is on the information theoretical limits and computational complexity for ICA. Here I will describe our recent work to specifically address these questions and introduce a refined method of moments that is both computationally tractable and statistically optimal.

In this talk, using a technique introduced by P.~T.~Nam in 2012 and the Coulomb Uncertainty Principle, I will present the proof of new bounds on the excess charge for non relativistic  atomic systems, independent of the particle statistics. These new bounds are the best bounds to date for bosonic systems. This is joint work with Juan Manel González and Trinidad Tubino.

Join Zoom Meeting: https://gatech.zoom.us/j/94786316294

I will describe the arithmetic of polynomials over idylls and various division algorithms and rules. For instance, that arithmetic might capture a total order/sign or an absolute value. These division algorithms will relate, for instance, the number of positive roots of a polynomial to the signs of the coefficients (Descartes's Rule of Signs).

Morse theory and Morse homology together give a method for understanding how the topology of a smooth manifold changes with respect to a filtration of the manifold given by sub-level sets. The Morse homology of a smooth manifold can be expressed using an algebraic object called a persistence module. A persistence module is a module graded by real numbers, and in this setup the grading on the module corresponds to the aforementioned filtration on the smooth manifold.

This is the first of a series of talks that aims to explain the relationship between Morse homology and persistence modules. In the first talk, I will give a rapid review of Morse theory and a review of Morse homology. An understanding of singular homology will be assumed. 

With applications in the analysis of political districtings, Markov chains have become and essential tool for studying contiguous partitions of geometric regions. Nevertheless, there remains a dearth of rigorous results on the mixing times of the chains employed for this purpose. In this talk we'll discuss a sub-exponential bound on the mixing time of the Glauber dynamics chain for the case of bounded-size contiguous partition classes on certain grid-like classes of graphs.

https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09

The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. One of many unique features of the mammalian brain network is its spatial embedding and hierarchical organization. I will discuss effects of these structural characteristics on network dynamics as well as their computational implications with a focus on the flexibility between modular and global computations and predictive coding.