Georgia Institute of Technology College of Sciences

Given an action by a loop space on a structured ring spectrum we describe how to produce its associated quotient ring spectrum. We then describe how this structure may be leveraged to produce intermediate Hopf-Galois extensions of ring spectra, analogous to the way one produces intermediate Galois extensions from normal subgroups of a Galois group. We will give many examples of this structure in classical cobordism spectra and in particular describe an entirely new construction of the complex

In Watchareepan Atiponrat's thesis the properties of decomposable exact Lagrangian codordisms betweenLegendrian links in R^3 with the standard contact structure were studied. In particular, for any decomposableexact Lagrangian filling L of a Legendrian link K, one may obtain a normal ruling of K associated with L.Atiponrat's main result is that the associated normal rulings must have an even number of clasps.

In a 1958 paper, Milnor observed that then new work by Bott allowed him to show that the n sphere admits a vector bundle with non-trivial top Stiefel-Whitney class precisely when n=1,2,4, 8. This can be interpreted as a calculation of the mod 2 Hurewicz map for the classifying space BO, which has the structure of an infinite loopspace.

The commutator length of an element g in the commutator subgroup [G,G] of agroup G is the smallest k such that g is the product of k commutators. WhenG is the fundamental group of a topological space, then the commutatorlength of g is the smallest genus of a surface bounding a homologicallytrivial loop that represents g. Commutator lengths are notoriouslydifficult to compute in practice. Therefore, one can ask for asymptotics.This leads to the notion of stable commutator length (scl) which is thespeed of growth of the commutator length of powers of g.

This thesis studies the effect of transverse surgery on open books, the Heegaard Floer contact invariant, and tightness. We show that surgery on the connected binding of a genus g open book that supports a tight contact structure preserves tightness if the surgery coefficient is greater than 2g-1. We also give criteria for when positive contact surgery on Legendrian knots will result in an overtwisted manifold.

A trisection of a smooth, oriented, compact 4-manifold X is a decomposition into three diffeomorphic 4-dimensional 1-handlebodies with certain nice intersections properties. This is a very natural 4-dimensional analog of Heegaard splittings of 3-manifolds. In this talk I will define trisections of closed 4-manifolds, but will quickly move to the case of 4-manifolds with connected boundary. I will discuss how these "relative trisections" interact with open book decompositions on the bounding 3-manifold.

In this talk, we will discuss: (1) How geometry plays a role in machine learning/data science? (2) What it's like being a mathematician at a software company.

Topological full groups are naturally associated with semigroups of local homeomorphisms: iterations of a single homeomorphism, holonomy groupoids of laminations, groupoids of local isomorphisms of quasiperiodic sets (for example Penrose tilings), etc. Some of these groups have interesting properties from the point of view of group theory. For instance, they provide first examples of amenable infinite simple finitely generated groups (by a result of K. Juschenko and N. Monod) and first examples of simple amenable groups of Burnside type.

A trisection is a decomposition of a four-manifold into three trivial pieces and serves as a four-dimensional analogue to a Heegaard decomposition of a three-manifold. In this talk, I will discuss an adaptation of the theory of trisections to the relative setting of knotted surfaces in the four-sphere that serves as a four-dimensional analogue to bridge splittings of classical knots and links. I'll show that every such surface admits a decomposition into three standard pieces called a bridge trisection.

I will describe the results of a joint project with Mike Mandell on the algebraic K-theory of the sphere spectrum, focusing on recent work that describes the fiber of the cyclotomic trace using a spectral lift of Tate-Poitou duality.