Georgia Institute of Technology College of Sciences

Skein lasagna modules are smooth 4-manifold invariants constructed from functorial link homology theories. These invariants are capable of detecting exotic phenomena in dimension 4. Wall-type stabilization questions ask about the behavior of exotic smooth structures under various topological operations. In studying applications of these invariants to Wall-type stabilization problems, we construct an isomorphism between the skein lasagna invariant of a pair of the form (S^2xD^2, L), where L is a link in the boundary S^1xS^2, and the Rozansky-Willis homology of L in S^1xS^2 up to an extra tensor factor. In this talk, we will describe both invariants, describe their relationship, and discuss relevant properties. We will then briefly sketch how the properties of skein lasagna modules are used to establish functoriality for Rozansky-Willis homology and how to upgrade the theory to a new 4-manifold invariant.

How large of a random subset $D \subset \mathbb{F}_p^n$ does one need to almost surely intersect zero sets cut out by at most $s$ polynomials each of degree at most $k$? We determine the sharp threshold for this problem for all fixed $s$ and $k$. A corollary is that there exists a dense subset $A \subset \mathbb{F}_p^n$ free of k-term arithmetic progressions with common difference in a sufficiently small $D$, improving the lower bound for what is known as Szemerédi’s theorem with random differences. Our bound is the first to capture dependence of $|D|$ on $|A|$ in the finite field setting, giving better dependence than what is known in the integers. Based on joint work with Daniel Altman.

In this dissertation, we use methods of probabilistic combinatorics to work towards a conjecture of Alon and Kim about coloring the edges of k-uniform t-simple hypergraphs. A hypergraph is k-uniform if every edge contains exactly k vertices, and t-simple if any two edges intersect in at most t vertices. The chromatic index of a hypergraph H is the smallest integer N such that one can properly color the edges of H with N colors.

In 1997, Alon and Kim conjectured that if H is a k-uniform t-simple hypergraph with maximum degree D sufficiently large, then the chromatic index \chi'(H) is upper bounded by (t-1+1/t+\epsilon)D. Using probabilistic techniques and a nibble coloring method, we prove a general coloring theorem stating that a k-uniform t-simple hypergraph H with large maximum degree D has chromatic index at most (b+\epsilon)kD, where b is a particular parameter derived from local structural information about H.

We use structural techniques to prove sharp upper bounds on b in the 3-uniform 2-simple, and 3-uniform 3-simple cases. In particular, we deduce as a corollary that for sufficiently large D, every 3-uniform 2-simple and 3-simple hypergraph of maximum degree at most D has chromatic index at most 2.358D and 2.679D, respectively. We also prove that for sufficiently large D, every 3-uniform 2-simple hypergraph has fractional chromatic index at most 2D.

I will present some recent work with Debmalya Basak and Alexandru Zaharescu on potential improvements to the Siegel—Walfisz upper bound on the greatest real zero of a Dirichlet $L$-function.