A central theme in 4-dimensional topology is the search for exotic 4-manifolds, i.e. families of smooth manifolds that are homeomorphic not diffeomorphic. We will survey some basic results in this area.
There is no pre-seminar this time.
The classical theorems of Perron and Frobenius, which explore spectral properties of nonnegative matrices, have been extensively examined and generalized from various perspectives, including a cone-theoretic (geometric) viewpoint. Concurrently, in the past decade, there has been a notable effort to fuse the techniques of algebraic geometry and combinatorics in an exploration of Lorentzian polynomials by Brändén and Huh, also known as completely log-concave polynomials (CLC) by Anari et.al. or strongly log-concave polynomials by Gurvits.
In this talk, I will discuss my ongoing joint work with Greg Blekherman regarding the class of polynomials with Lorentzian signature (PLS) defined over closed convex cones. This class encompasses various special polynomials, including Lorentzian polynomials over the nonnegative orthant and hyperbolic polynomials over hyperbolicity cones. We establish a compelling connection between PLS over a self-dual cone K and the generalized Perron Frobenius theorem over K. This connection enables us to provide an alternative necessary and sufficient condition to characterize the Lorentzian polynomials.
We discuss new methods for using the Heegaard Floer homology of hypersurfaces to distinguish between smooth closed 4-manifolds that are homeomorphic but non-diffeomorphic. Specifically, for a 4-manifold X with b_1(X)=1, the minimum rank of the reduced Heegaard Floer homology of any embedded 3-manifold X representing a generator of H_1(X) gives a diffeomorphism invariant of X. We use this invariant to distinguish certain infinite families of exotic 4-manifolds that cannot be distinguished by previously known techniques. Using related ideas, we also provide the first known examples of (non-simply-connected) exotic 4-manifolds with negative definite intersection form. This is joint work with Tye Lidman and Lisa Piccirillo.
In recent years, algebraization theorems arising from model theory, in particular o-minimality, have been a crucial ingredient in several breakthroughs in arithmetic geometry and Hodge theory. In this talk, I'll present some of my recent work on p-adic versions of these model theoretic algebraization criteria, with a focus on two different applications of this circle of ideas. The first being an algebraization theorem in the context of Shimura varieties, which are vaguely speaking moduli spaces of Hodge structures. The second being in the context of non-abelian Hodge theory, in the setting of moduli spaces of flat connections and local systems.
Zoom: https://gatech.zoom.us/j/95425627723
The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface. A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3. We will explain work of Ershov-He and Church-Ershov-Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1. Time permitting, we will also discuss some extensions of these ideas. In particular, we will explain how to show that the terms of the Johnson filtration are finitely presented assuming the Torelli group is finitely presented.
This will be an expository talk which aims to introduce some problems in harmonic analysis and geometric measure theory concerning the geometry of a measure for which an associated integral operator is well behaved. As an example, we shall prove a result of Mattila and Preiss concerning the relationship between the rectifiability of a measure and the existence of the Riesz transform in the sense of principle value.
