Georgia Institute of Technology College of Sciences

Introduced by Hendricks and Manolescu in 2015, Involutive Heegaard Floer homology is a variation of the 3-manifold invariant Heegaard Floer homology which makes use of the conjugation symmetry of the Heegaard Floer complexes. This theory can be used to obtain two new invariants of homology cobordism. This talk will involve a brief overview of general Heegaard Floer homology, followed by a discussion of the involutive theory and some computations of the homology cobordism invariants. 

A group is said to be torsion-free if it has no elements of finite order.  An example is the group, under composition, of self-homeomorphisms (continuous maps with continuous inverses) of the interval I = [0, 1] fixed on the boundary {0, 1}.  In fact this group has the stronger property of being left-orderable, meaning that the elements of the group can be ordered in a way that is nvariant under left-multiplication.  If one restricts to piecewise-linear (PL) homeomorphisms, there exists a two-sided (bi-)ordering, an even stronger property of groups.

I will discuss joint work with Danny Calegari concerning groups of homeomorphisms of the cube [0, 1]^n fixed on the boundary.  In the PL category, this group is left-orderable, but not bi-orderable, for all n>1.  Also I will report on recent work of James Hyde showing that left-orderability fails for n>1 in the topological category.  

(This is a joint event of the Combinatorics Seminar Series and the ACO Student Seminar.)

In this talk we will prove a conjecture of Talagrand, which is a fractional version of the “expectation-threshold” conjecture of Kalai and Kahn. This easily implies various difficult results in probabilistic combinatorics, e.g. thresholds for perfect hypergraph matchings (Johansson-Kahn-Vu) and bounded-degree spanning trees (Montgomery). Our approach builds on recent breakthrough work of Alweiss, Lovett, Wu, and Zhang on the Erdos-Rado “Sunflower Conjecture.” 

This is joint work with Keith Frankston, Jeff Kahn, and Bhargav Narayanan.

This talk concerns a naturally occurring family of Calabi-Yau manifolds that degenerates in the sense of metric geometry, algebraic geometry and nonlinear PDE. A primary tool in analyzing their behavior is the recently developed regularity theory. We will give a precise description of arising singularities and explain possible generalizations.