Georgia Institute of Technology College of Sciences

A fundamental problem in algebraic geometry is to construct compact moduli spaces parametrizing algebraic shapes. I will discuss a new approach to this problem in the case of Calabi--Yau pairs (X,D) for which D is ample. Such pairs arise from many well-studied algebraic varieties such as plane curves, K3 surfaces, and del Pezzo surfaces. In the case of Calabi-Yau pairs of dimension two, this approach outputs a projective moduli space on which the Hodge line bundle is ample. This is based on joint work with Yuchen Liu that builds on earlier work with Ascher, Bejleri, DeVleming, Inchiostro, Liu, and Wang.

In work in progress with Ryan Stees, we show that every closed, oriented 3-manifold can be obtained by 0-surgery on a link. Since the 0-surgery of a link can capture the data of many of the typical isotopy and concordance invariants of a link, particularly in the pairwise linking number 0 case, this result gives us a nice lens through which to study both 3-manifolds and links. However, 0-surgery on a link is certainly not a complete link invariant, and we also give multiple constructions for non-isotopic (and even non-concordant) links with homeomorphic 0-surgeries.

I will review a general framework for developing numerical methods working with non-parametrically defined surfaces for various problems. In this talk, I will focus on boundary integral equations. The main idea is to formulate appropriate extensions of a given problem defined on a surface to ones in the narrow band of the surface in the embedding space. The extensions are arranged so that the solutions to the extended problems are equivalent, in a strong sense, to the surface problems that we set out to solve. Such extension approaches allow us to analyze the well-posedness of the resulting system, develop, systematically and in a unified fashion, numerical schemes for treating a wide range of problems involving differential and integral operators, and deal with similar problems in which only point clouds sampling the surfaces are given. At the end of this talk, I will mention our work in developing multilevel neural network methods for inverting dense and large matrices that arise from boundary integral equations.

This talk provides an elementary introduction to skein algebras of surfaces, which serve as quantizations of $SL_n$-character varieties. For surfaces with boundary, we extend this framework to stated skein algebras, demonstrating how they provide simple and transparent geometric interpretations of various quantum group structures.

Specifically, we present a geometric realization of the dual canonical basis of $\mathscr{O}_q(\mathfrak{sl}_n)$ using skeins for $n=2$ and $n=3$. If time permits, we will also show how the skein algebra framework can be used to recover the Shapiro–Schrader embedding of the quantized enveloping algebra into a quantum torus algebra.

The quest of blowup solutions to the NSE remains a challenging topic. We will discuss a recent construction that illustrates a blowup phenomenon unexplored previously for the NSE. In particular, the constructed solutions start from smooth initial data and exhibit an instantaneous blowup saturating Type-I blowup rate at a finite time. Moreover, there are infinitely many such blowup solutions; and the non-uniqueness occurs in borderline spaces of known criteria which ensure uniqueness. This is joint work with Alexey Cheskidov and Stan Palasek.

I will present on recent work  - joint with John Green, Terence Harris, Kevin Ren, and Yumeng Ou -  towards proving lower bounds for the dimensions of Furstenberg sets of circles and sine curves in the plane.  A circular $(u,v)$-Furstenberg set is a set that contains a $u$-dimensional subset of each circle from a $v$-dimensional family of circles.  One can approach the circular Furstenberg problem by proving estimates for the number of incidences between families of $\delta$-disks and $\delta$-annuli that satisfy certain dimension conditions.  For different values of $u$ and $v$, we prove incidence estimates using local smoothing and using trilinear restriction estimates for the cone in $\mathbb{R}^3$.  As time permits, I will discuss work relevant to proving dimension estimates for Furstenberg sets of sine curves (which satisfy all of the bounds we prove for circular Furstenberg sets) and/or work for Furstenberg sets of curves that satisfy a more general cinematic curvature condition.

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.

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.

In the study of random growth models belonging to the Kardar--Parisi--Zhang (KPZ) universality class, a notably successful approach has been to analyze stationary initial conditions defined by the Busemann functions. Recently, this perspective has been extended to handle multiple asymptotic directions simultaneously, but the joint distribution of the Busemann process is more difficult to access, and many aspects of this process remain elusive. In particular, the remarkable independence property present in the exactly solvable setting fails when considering Busemann functions across different directions. In the corner growth model, also known as exponential last-passage percolation (LPP), we prove that, regardless of their different directions, Busemann functions along a down-right path are always negatively associated across each individual direction. In other words, increasing the value of Busemann functions in one direction tends to probabilistically decrease the values of neighboring ones. As an application, we obtain an exponential concentration inequality on the diffusive scale for Busemann functions along a down-right path, in the absence of independence. Joint work with Erik Bates.

I will give a brief overview of spectral theory of discrete Schrodinger-type operators on periodic graphs and discuss the recent result (joint with Matthew Faust) on the absence of flat bands for generic potentials.

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.