Georgia Institute of Technology College of Sciences

The extremal function of a class of matroids maps each positive integer n to the maximum number of elements of a simple matroid in the class with rank at most n. We will present a result concerning the role of finite groups in minor-closed classes of matroids, and then use it to determine the extremal function for several natural classes of representable matroids. We will assume no knowledge of matroid theory. This is joint work with Jim Geelen and Peter Nelson.

In the Instanton and Heegaard Floer theories, a nearly fibered knot is one for which the top grading has rank 2. Sivek-Baldwin and Li-Ye showed that the guts (ie. the reduced sutured manifold complement) of a minimal genus Seifert surface of a nearly fibered knot has of one of three simple types.We show that nearly fibered knots with guts of two of these types have handle number 2 while those with guts of the third type have handle number 4.  Furthermore, we show that nearly fibered knots have unique incompressible Seifert surfaces rather than just unique minimal genus Siefert surfaces. This is joint work in progress with Fabiola Manjarrez-Gutierrez.

We will describe an elegant construction of potential counterexamples to the Smooth 4-Dimensional Poincaré Conjecture whose input is a fibered, homotopy-ribbon knot in the 3-sphere. The construction also produces links that are potential counterexamples to the Generalized Property R Conjecture, as well as balanced presentations of the trivial group that are potential counterexamples to the Andrews-Curtis Conjecture. We will then turn our attention to generalized square knots (connected sums of torus knots with their mirrors), which provide a setting where the potential counterexamples mentioned above can be explicitly understood. Here, we show that the constructed 4-manifolds are diffeomorphic to the 4-sphere; but the potential counterexamples to the other conjectures persist. In particular, we present a new, large family of geometrically motivated balanced presentations of the trivial group. Along the way, we give a classification of fibered, homotopy-ribbon disks bounded by generalized square knots up to isotopy and isotopy rel-boundary. This talk is based on joint work with Alex Zupan.

When can we find perfect matchings in hypergraphs whose vertices represent group elements and whose edges represent solutions to systems of linear equations? A prototypical problem of this type is the Hall-Paige conjecture, which asks for a characterisation of the groups whose multiplication table (viewed as a Latin square) contains a transversal. Other problems expressible in this language include the toroidal n-queens problem, Graham-Sloane harmonious tree-labelling conjecture, Ringel's sequenceability conjecture, Snevily's subsquare conjecture, Tannenbaum's zero-sum conjecture, and many others. All of these problems have a similar flavour, yet until recently they have been approached in completely different ways, using algebraic tools ranging from the combinatorial Nullstellensatz to Fourier analysis. In this talk we discuss a unified approach to attack these problems, using tools from probabilistic combinatorics. In particular, we will see that a suitably randomised version of the Hall-Paige conjecture can be used as a black-box to settle many old problems in the area for sufficiently large groups.  Joint work with Alexey Pokrosvkiy

The classical Hele-Shaw flow describes the motion of incompressible viscous fluid, which occupies part of the space between two parallel, nearby plates. With source and drift, the equation is used in models of tumor growth where cells evolve with contact inhibition, and congested population dynamics. We consider the flow with Hölder continuous source and Lipschitz continuous drift. We show that if the free boundary of the solution is locally close to a Lipschitz graph, then it is indeed Lipschitz, given that the Lipschitz constant is small. This is joint work with Inwon Kim.

Have you ever wanted to marry topology, hyperbolic geometry, and geometric group theory, all at once?* Bowden-Hensel-Webb do this and more when they embark on their study of Diff0(S). In this talk, we will discuss the main theorems of Bowden-Hensel-Webb's paper, the most notable of which is (arguably) the lack of uniform perfection of Diff0(S). We will then summarize the main tools they use to prove these results. (Note: the perspectives on Diff0(S) in this talk will DIFFer greatly from those used in the diffeomorphism groups class.) 

*If you answered "yes" for your personal life as opposed to your academic life: that's ok, I won't judge (if you don't tell me).

I discuss a sharp quantitative stability result for the Sobolev inequality with explicit constants. Moreover, the constants have the optimal behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant.

The problem of estimation of smooth functionals of unknown parameters of statistical models will be discussed in the cases of high-dimensional log-concave location models (joint work with Martin Wahl) and infinite dimensional Gaussian models with unknown covariance operator. In both cases, the minimax optimal error rates have been obtained in the classes of H\”older smooth functionals with precise dependence on the sample size, the complexity of the parameter (its dimension in the case of log-concave location models or the effective rank of the covariance in the case of Gaussian models)  and on the degree of smoothness of the functionals. These rates are attained for different types of estimators based on two different methods of bias reduction in functional estimation.

We will discuss work of Michael Aizenman, Francois Germinet, Abel Klein, and Simone Warzel from 2007 on optimal Bernoulli decompositions of random variables and applications thereof. We will briefly discuss the basic properties of such decompositions, and demonstrate the existence of decompositions for which the contribution of the Bernoulli disorder is optimized in various ways.

We will then go through a proof of almost sure spectral localization (at the bottom of the spectrum) for continuous random Schroedinger operators with arbitrary bounded disorder. This proof relies on a Bernoulli decomposition of the disorder combined with a slightly stronger variant of the 2005 result from Jean Bourgain and Carlos Kenig showing such localization when the disorder is Bernoulli.

In the minimum eigenvalue problem we are given a collection of rank-1 symmetric matrices, and the goal is to find a subset whose sum has large minimum eigenvalue, subject to some combinatorial constraints. The constraints on which subsets we can select, could be cardinality, partition, or more general matroid base constraints. Using pipage rounding and a matrix concentration inequality, we will show a randomised algorithm which achieves a (1- epsilon) approximation for the minimum eigenvalue problem when the matrices have constant size, subject to any matroid constraint.

The bulk of the talk will be background on “pipage rounding, pessimistic estimators and matrix concentration” adapted from the paper with that title by Nicholas J. A. Harvey and Neil Olver. The application to the minimum eigenvalue problem is joint work with Aditi Laddha and Mohit Singh.

In 1955, Mark Kac introduced a simple model to study the evolution of a gas of particles undergoing pairwise collisions. Although extremely simplified in such a way to be rigorously treatable, the model maintains interesting aspects of gas dynamics. In recent years, we worked with M. Loss and others to extend the analysis to more "realistic" versions of the original model.

I will introduce the Kac model and present some standard and more recent results. These results refer to a system with a fixed number of particles and at fixed kinetic energy (micro canonical ensemble) or temperature (canonical ensemble). I will introduce a "Grand Canonical" version of the Kac system and discuss new results on it.