Georgia Institute of Technology College of Sciences

While the existence and properties of the SRB measure for the billiard map associated with a periodic Lorentz gas are well understood, there are few results regarding other types of measures for dispersing billiards. We begin by proposing a naive definition of topological entropy for the billiard map, and show that it is equivalent to several classical definitions. We then prove a variational principle for the topological entropy and proceed to construct a unique probability measure which achieves the maximum. This measure is Bernoulli and positive on open sets. An essential ingredient is a proof of the absolute continuity of the unstable foliation with respect to the measure of maximal entropy. This is joint work with Viviane Baladi.
 

A discussion of Khovanov-Lee homology, how to extract some invariants of braid closures from the homology theory, and motivation for studying both the homology theory and the invariants.

Consider a stochastic process (such as a stochastic differential equation) arising from applications. In practice, we are interested in many things like the invariant probability measure, the sensitivity of the invariant probability measure, and the speed of convergence to the invariant probability measure. Existing rigorous estimates of these problems usually cannot provide enough details. In this talk I will introduce a few data-driven computational methods that solve these problems for a class of stochastic dynamical systems, including but not limited to stochastic differential equations. All these methods are driven by the simulation data, and are less affected by the curse-of-dimensionality than traditional grid-based methods. I will demonstrate a few high (up to 100) dimensional examples in my talk.

Annular Rasmussen invariants are invariants of braid closures which generalize the Rasmussen s invariant and come from an integer bifiltration on Khovanov-Lee homology. In this talk we will explain some connections between the annular Rasmussen invariants and other topological information. Additionally we will state theorems about restrictions on the possible values of annular Rasmussen invariants and a computation of the invariants for all 3-braid closures, or conjugacy classes of 3-braids. Time permitting, we will sketch some proofs.

Originally introduced independently by Hassler Whitney and Takeo Nakasawa, matroids are a combinatorial way of axiomatizing the notion of linear independence in vector spaces. If $K$ is a field and $n$ is a positive integer, any linear subspace of $K^n$ gives rise to a matroid; such matroid are called representable over $K$. Given a matroid $M$, one can ask over which fields $M$ is representable. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the universal partial field of a matroid $M$, which governs the representations of $M$ over all partial fields. Unfortunately, most matroids (asymptotically 100%, in fact) are not representable over any partial field, and in this case, the universal partial field gives no information.

Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the foundation of a matroid. The foundation of $M$ is a type of algebraic object which we call a pasture; pastures include both hyperfields and partial fields. Pastures form a natural class of field-like objects within Lorscheid's theory of ordered blueprints, and they have desirable categorical properties (e.g., existence of products and coproducts) that make them a natural context in which to study algebraic invariants of matroids. The foundation of a matroid $M$ represents the functor taking a pasture $F$ to the set of rescaling equivalence classes of $F$-representations of $M$; in particular, $M$ is representable over a pasture $F$ if and only if there is a homomorphism from the foundation of $M$ to $F$. (In layman's terms, what we're trying to do is recast as much as possible of the theory of matroids and their representations in functorial ``Grothendieck-style'' algebraic geometry, with the goal of gaining new conceptual insights into various phenomena which were previously understood only through lengthy case-by-case analyses and ad hoc computations.)

As a particular application of this point of view, I will explain the classification which Lorscheid and I have recently obtained of all possible foundations for ternary matroids (matroids representable over the field of three elements). The proof of this classification theorem relies crucially on Tutte's celebrated Homotopy Theorem. 

Lattice polytopes play an important role in combinatorics due to their intricate geometric structure as well as their enumerative properties. In this talk, we will discuss several instances in which lattice point enumeration of lattice polytopes relates to problems in algebraic combinatorics, particularly the representation theory of GL(n) and related groups. We will also see how certain types of algebraic constructions have polytopal counterparts. This talk is based on joint work with Karola Mészáros and Avery St. Dizier.

The relationship between biodiversity and ecological stability has long interested ecologists. The ongoing biodiversity loss has led to the increasing concern that it may impact ecosystem functioning, including ecosystem stability. Both early conceptual ideas and recent theory suggest a positive relationship between biodiversity and ecosystem stability. While quite a number of empirical studies, particularly experiments that directly manipulated species diversity, support this hypothesis, exceptions are not uncommon. This raises the question of whether there is a general positive diversity-stability relationship.

Literature survey shows that species diversity may not necessarily be an important determinant of ecosystem stability in natural communities. While experiments controlling for other environmental variables often report that ecosystem stability increases with species diversity, these other environmental variables are often more important than species diversity in influencing ecosystem stability. Studies that account for these environmental covariates tend to find a lack of relationship between species diversity and ecosystem stability. An important goal of future studies is to elucidate mechanisms driving the variation in the importance of species diversity in regulating ecosystem stability.

A useful way of studying contact 3 manifolds is by looking at their open book decompositions. A result of Akbulut-Ozbagci, Ghiggini, and Loi-Piergallini showed that the manifold is filled by a Stein manifold if and only if the monodromy of an open book can be factorised as the product of positive Dehn twists. Then, the problem of classifying minimal fillings of contact 3 manifolds, or answering questions about which manifolds can be realised by Legendrian surgery, becomes questions about finding factorisations for a given mapping class. This talk will be expository and expand upon how these mapping classes come up, and also discuss known results, techniques, and future directions for research.

Decoupling is a Fourier analytic tool that  has repeatedly proved its extraordinary potential for a broad range of applications to number theory (counting solutions to Diophantine systems, estimates for the growth of the Riemann zeta), PDEs (Strichartz estimates, local smoothing for the wave equation, convergence of solutions to the initial data), geometric measure theory (the Falconer distance conjecture)  and harmonic analysis (the Restriction Conjecture). The abstract theorems are formulated and proved in a continuous framework, for arbitrary functions with spectrum supported near curved manifolds. At this level of generality, the proofs involve no number theory, but rely instead on  wave packet analysis and incidence geometry related to the Kakeya phenomenon.   The special case when the spectrum is localized near lattice points leads to unexpected  solutions of conjectures once thought to pertain to the realm of number theory. 

In recent years political parties have more and more expertly 
crafted political districtings to favor one side or another, while at 
the same time, entirely new techniques to detect and measure these 
efforts are being developed.

I will discuss a rigorous method which uses Markov chains---random 
walks---to statistically assess gerrymandering of political districts 
without requiring heuristic validation of the structures of the Markov 
chains which arise in the redistricting context.  In particular, we will 
see two examples where this methodology was applied in successful 
lawsuits which overturned district maps in Pennsylvania and North Carolina.

Many functions of interest are in a high-dimensional space but exhibit low-dimensional structures. This work studies regression of a $s$-Hölder function $f$ in $\mathbb{R}^D$ which varies along an active subspace of dimension $d$ while $d\ll D$. A direct approximation of $f$ in $\mathbb{R}^D$ with an $\varepsilon$ accuracy requires the number of samples $n$ in the order of $\varepsilon^{-(2s+D)/s}$. In this work, we modify the Generalized Contour Regression (GCR) algorithm to estimate the active subspace and use piecewise polynomials for function approximation. GCR is among the best estimators for the active subspace, but its sample complexity is an open question. Our modified GCR improves the efficiency over the original GCR and leads to a mean squared estimation error of $O(n^{-1})$ for the active subspace, when $n$ is sufficiently large. The mean squared regression error of $f$ is proved to be in the order of $\left(n/\log n\right)^{-\frac{2s}{2s+d}}$, where the exponent depends on the dimension of the active subspace $d$ instead of the ambient space $D$. This result demonstrates that GCR is effective in learning low-dimensional active subspaces. The convergence rate is validated through several numerical experiments.

This is a joint work with Wenjing Liao.

We  show  that  if  $P\neq NP$,  then  a  wide  class  of  TSP heuristics fail to approximate the length of the TSP to asymptotic 
optimality, even for random Euclidean instances.  Previously, this result was not even known for any heuristics (greedy, etc) used in practice.  As an application, we show that when  using  a  heuristic from  this  class,  a  natural  class  of  branch-and-bound algorithms takes exponential time to find an optimal tour (again, even on a random point-set),  regardless  of  the  particular  branching  strategy  or lower-bound algorithm used.