Georgia Institute of Technology College of Sciences

I will discuss knot concordances in 3-manifolds. In particular I will talk about knot concordances of knots in the free homotopy class of S^1 x {pt} in S^1 x S^2. It turns out, we can use some of these concordances to construct Akbulut-Ruberman type exotic 4-manifolds. As a consequence, at the end of the talk we will see absolutely exotic Stein pair of 4-manifolds. This is joint work with Selman Akbulut.

The importance of analyzing big data and in particular very large networks has shown that the traditional notion of a fast algorithm, one that runs in polynomial time, is often insufficient. This is where property testing comes in, whose goal is to very quickly distinguish between objects that satisfy a certain property from those that are ε-far from satisfying that property. It turns out to be closely related to major developments in combinatorics, number theory, discrete geometry, and theoretical computer science. Some of the most general results in this area give "constant query complexity" algorithms, which means the amount of information it looks at is independent of the input size. These results are proved using regularity lemmas or graph limits. Unfortunately, typically the proofs come with no explicit bound for the query complexity, or enormous bounds, of tower-type or worse, as a function of 1/ε, making them impractical. We show by entirely new methods that for permutations, such general results still hold with query complexity only polynomial in 1/ε. We also prove stronger results for graphs through the study of new metrics. These are joint works with Jacob Fox.

The classical degree-genus formula computes the genus of a nonsingular algebraic curve in the complex projective plane. The well-known Thom conjecture posits that this is a lower bound on the genus of smoothly embedded, oriented and connected surface in CP^2. The conjecture was first proved twenty-five years ago by Kronheimer and Mrowka, using Seiberg-Witten invariants. In this talk, we will describe a new proof of the conjecture that combines contact geometry with the novel theory of bridge trisections of knotted surfaces. Notably, the proof completely avoids any gauge theory or pseudoholomorphic curve techniques.

Bring a laptop with Macaulay 2 installed.