Georgia Institute of Technology College of Sciences

A fundamental result in 3-dimensional contact topology due to Ding-Geiges tells us that any contact 3-manifold can be obtained via doing a surgery on a Legendrian link in the standard contact 3-sphere. So it's natural to ask how simple or complicated a surgery diagram could be for a given contact manifold? Contact surgery number is a measure of  this complexity. In this talk, I will discuss this notion of complexity along with some examples. This is joint work with Marc Kegel.

Algebraic geometry is the study of shapes defined by polynomial equations called algebraic varieties. One natural approach to study them is to construct a moduli space, which is a space parameterizing such shapes of a given type (e.g. algebraic curves). After surveying this topic, I will focus on the problem of constructing moduli spaces parametrizing Fano varieties, which are a class of positively curved complex manifolds that form one of the three main building blocks of varieties in algebraic geometry. While algebraic geometers once considered this problem intractable due to various pathologies that occur, it has recently been solved using K-stability, which is an algebraic definition introduced by differential geometers to characterize when a Fano variety admits a Kähler-Einstein metric.

In this dissertation we study a variety of graph-theoretic problems lying at the intersection of mathematics, computer science, and statistics. This work consists of three parts, all of which use probabilistic techniques. 

In Part 1, we consider structurally constrained graphs and hypergraphs. We examine a celebrated conjecture of Alon, Krivelevich, and Sudakov regarding vertex coloring. Our results provide improved bounds in all known cases for which the conjecture holds. We introduce a generalized notion of local sparsity and study the independence and chromatic numbers of graphs satisfying this property. We also consider multipartite hypergraphs, a natural extension of bipartite graphs. We show how certain probabilistic techniques for problems on bipartite graphs can be adapted to multipartite hypergraphs, and are therefore able to extend and generalize a number of results.

In Part 2, we investigate edge coloring from an algorithmic standpoint. We focus on multigraphs of bounded maximum degree, i.e., $\Delta(G) = O(1)$. Following the so-called augmenting subgraph approach, we design deterministic and randomized algorithms using a near-optimal number of colors in the sequential setting as well as in the LOCAL model of distributed computing. Additionally, we study list-edge-coloring for list assignments satisfying certain local constraints, and describe a polynomial-time algorithm to compute such a coloring.

Finally, in Part 3, we explore a number of statistical inference problems in random hypergraph models. Specifically, we consider the statistical-computational gap for finding large independent sets in sparse random hypergraphs, and the computational threshold for the detection of planted dense subhypergraphs (a generalization of the classical planted clique problem). We explore the power and limitations of low-degree polynomial algorithms, a powerful class of algorithms which includes the class of local algorithms as well as approximate message passing and power iteration.

This presentation introduces a methodology for generating computer-assisted proofs (CAPs) aimed at establishing the existence of solutions for nonlinear differential equations featuring non-polynomial analytic nonlinearities. Our approach combines the Fast Fourier Transform (FFT) algorithm with interval arithmetic and a Newton-Kantorovich argument to effectively construct CAPs. A key highlight is the rigorous management of Fourier coefficients of the nonlinear term Fourier series, achieved through insights from complex analysis and the Discrete Poisson Summation Formula. We demonstrate the effectiveness of our method through two illustrative examples: firstly, proving the existence of periodic orbits in the Mackey-Glass (delay) equation, and secondly, establishing the existence of periodic localized traveling waves in the two-dimensional suspension bridge equation.

This is joint work with Jan Bouwe van den Berg (VU Amsterdam, The Netherlands), Maxime Breden (École Polytechnique, France) and Jason D. Mireles James (Florida Atlantic University, USA)