Georgia Institute of Technology College of Sciences

I will describe new techniques for computing the homology of braid groups with coefficients in certain exponential coefficient systems. An unexpected side of this story (at least to me) is a connection with the cohomology of certain braided Hopf algebras — quantum shuffle algebras and Nichols algebras — which are central to the classification of pointed Hopf algebras and quantum groups. We can apply these tools to get a bound on the growth of the cohomology of Hurwitz moduli spaces of branched covers of the plane in certain cases. This yields a weak form of Malle’s conjecture on the distribution of fields with prescribed Galois group in the function field setting. This is joint work with Jordan Ellenberg and TriThang Tran.

We will present the classical formulations (Gibbs, Maxwell, etc.) of the second law of thermodynamics and present the basics of the equilibrium statistical mechanics. The results are all classic and the presentation will be elementary, but we will try to point out some of the more subtle mathematical questions. The main goal of the lectures is to lay the groundwork to proceed to read "J. Dorfman: An introduction to chaos and non-equilibrium statistical mechanics". There will be cookies and some (sugar free) drinks.

We investigate Lipschitz maps, I, mapping $C^2(D) \to C(D)$, where $D$ is an appropriate domain. The global comparison principle (GCP) simply states that whenever two functions are ordered in D and touch at a point, i.e. $u(x)\leq v(x)$ for all $x$ and $u(z)=v(z)$ for some $z \in D$, then also the mapping I has the same order, i.e. $I(u,z)\leq I(v,z)$. It has been known since the 1960’s, by Courr\`{e}ge, that if I is a linear mapping with the GCP, then I must be represented as a linear drift-jump-diffusion operator that may have both local and integro-differential parts. It has also long been known and utilized that when I is both local and Lipschitz it will be a min-min over linear and local drift-diffusion operators, with zero nonlocal part. In this talk we discuss some recent work that bridges the gap between these situations to cover the nonlinear and nonlocal setting for the map, I. These results open up the possibility to study Dirichlet-to-Neumann mappings for fully nonlinear equations as integro-differential operators on the boundary. This is joint work with Nestor Guillen.

In this lecture series, held jointly (via video conference) with the University of Buffalo and the University of Arkansas, we aim to understand the lecture notes by Vincent Guirardel on geometric small cancellation. The lecture notes can be found here: https://perso.univ-rennes1.fr/vincent.guirardel/papiers/lecture_notes_pcmi.pdf This week we will compete the first of two steps in proving the small cancellation theorem (Lecture 3).

A knot is a smooth embedding of S^1 into S^3 or R^3. There is a natural way to "add" two knots, called the connected sum. Under this operation, the set of knots forms a monoid. We will quotient by an equivalence relation called concordance to obtain a group, and discuss what is known about the structure of this group.

In 1988, Penner conjectured that all pseudo-Anosov mapping classes arise up to finite power from a construction named after him. This conjecture was known to be true on some simple surfaces, including the torus, but has otherwise remained open. I will sketch the proof (joint work with Hyunshik Shin) that the conjecture is false for most surfaces.

Consider Hermitian matrices A, B, C on an n-dimensional Hilbert space such that C=A+B. Let a={a_1,a_2,...,a_n}, b={b_1, b_2,...,b_n}, and c={c_1, c_2,...,c_n} be sequences of eigenvalues of A, B, and C counting multiplicity, arranged in decreasing order. Such a triple of real numbers (a,b,c) that satisfies the so-called Horn inequalities, describes the eigenvalues of the sum of n by n Hermitian matrices. The Horn inequalities is a set of inequalities conjectured by A. Horn in 1960 and later proved by the work of Klyachko and Knutson-Tao. In these two talks, I will start by discussing some of the history of Horn's conjecture and then move on to its more recent developments. We will show that these inequalities are also valid for selfadjoint elements in a finite factor, for types of torsion modules over division rings, and for singular values for products of matrices, and how additional information can be obtained whenever a Horn inequality saturates. The major difficulty in our argument is the proof that certain generalized Schubert cells have nonempty intersection. In the finite dimensional case, it follows from the classical intersection theory. However, there is no readily available intersection theory for von Neumann algebras. Our argument requires a good understanding of the combinatorial structure of honeycombs, and produces an actual element in the intersection algorithmically, and it seems to be new even in finite dimensions. If time permits, we will also discuss some of the intricate combinatorics involved here. In addition, some recent work and open questions will also be presented.

Parallel algorithms study ways of speeding up sequential algorithms by splitting work onto multiple processors. Theoretical studies of parallel algorithms often focus on performing a small number of operations, but assume more generous models of communication. Recent progresses led to parallel algorithms for many graph optimization problems that have proven to be difficult to parallelize. In this talk I will survey routines at the core of these results: low diameter decompositions, random sampling, and iterative methods.

I will discuss a process called a cork twist for relating homeomorphic but not diffeomorphic smooth 4-manifolds. This involves finding a contractible submanifold of a given 4-manifold, removing it, and re-gluing by a diffeomorphism of the boundary. This is a surprisingly simple way of relating non-diffeomorphic manifold that was discovered in the 1990s but has recently been getting a lot of attention.