Georgia Institute of Technology College of Sciences

Contact 3-manifolds arise organically as boundaries of symplectic 4-manifolds, so it’s natural to ask: Given a contact 3-manifold Y, does there exist a symplectic 4-manifold X filling Y in a compatible way? Stein fillability is one such notion of compatibility that can be explored via open books: representations of a 3-manifold by means of a surface with boundary and its self-diffeomorphism, called a monodromy. I will discuss joint work with Andy Wand in which we exhibit first known Stein-fillable contact manifolds whose supporting open books of genus one have non-positive monodromies. This settles the question of correspondence between Stein fillings and positive monodromies for open books of all genera. Our methods rely on a combination of results of J. Conway, Lecuona and Lisca, and some observations about lantern relations in the mapping class group of the twice-punctured torus.

Knots in contact manifolds are interesting objects to study. In this talk, I will focus on knots in overtwisted manifolds. There are two types of knots in an overtwisted manifold, one with overtwisted complement (known as loose) and one with tight complement (known as non-loose). Not very surprisingly, non-loose knots behave very mysteriously. They are interesting in their own right as we still do not understand them well. But also one might want to study them because surgery on them produces tight contact structures and understanding tight contact structures is a major problem in the contact world. I'll give a brief history on these knots and discuss some of their classification and structure problems and how these problems differ from the classification/ structure problems of knots in tight manifolds.
 

I will talk about the application of algebra and algebraic geometry to matroid theory. Baker and Bowler developed the notions of weak and strong matroids over tracts. Later, Baker and Lorscheid developed the notion of foundation of a matroid, which characterize the representability of the matroid. I will introduce a variety of topics under this theme. First, I will talk about a condition which is sufficient to guarantee that the notions of strong and weak matroids coincide. Next, I will describe a software program that computes all representations of matroids over a field, based on the theory of foundations. Finally, I will define a notion of rank for matrices over tracts in order to get uniform proofs of various results about ranks of matrices over fields.

Many recent breakthroughs in additive combinatorics, such as results relating to Roth’s theorem or inverse sum set theorems, utilize a combination of Fourier analytical and physical methods. Physical methods refer to results relating to the physical space, such as almost-periodicity results regarding convolutions. This thesis focuses on the properties of convolutions.

Given a group G and sets A ⊆ G, we study the properties of the convolution for sum sets and difference sets, 1A ∗1A and 1A ∗1−A. Given x ∈ Gn, we study the set image of its sum set and difference set. We break down the study of set images into two cases, when x is independent, and when x is an arithmetic progression. In both cases, we provide some convexity result for the set image of both the sum set and difference set. For the case of the arithmetic progression, we prove convexity by first showing a recurrence relation for the distribution of the convolution.

Finally, we prove a smoothness property regarding 4-fold convolutions 1A ∗1A ∗1A ∗1A. We then construct different examples to better understand possible bounds for the smoothness property in the case of 2-fold convolutions 1A ∗ 1A.

Committee

Prof. Ernie Croot, Advisor

Prof. Michael Lacey

Prof. Josephine Yu

Prof. Anton Leykin

Prof. Will Perkins