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Computability of rational points on curves over function fields in characteristic p
(Massachusetts Institute of Technology, 2020)
The motivating problem of this thesis is that of explicitly computing the K-rational points of a regular nonsmooth curve X over a αnitely generated αeld K of characteristic p. We start with an in-depth study of such curves ...
Arboreal representations, sectional monodromy groups, and abelian varieties over finite fields
(Massachusetts Institute of Technology, 2020)
This thesis consists of three independent parts. The first part studies arboreal representations of Galois groups - an arithmetic dynamics analogue of Tate modules - and proves some large image results, in particular ...
Contributions to sutured monopole and sutured instanton Floer homology theories
(Massachusetts Institute of Technology, 2020)
In this thesis, we present the development of some aspects of sutured monopole and sutured instanton Floer homology theories. Sutured monopole and instanton Floer homologies were introduced by Kronheimer and Mrowka. They ...
Unipotent representations of real reductive groups
(Massachusetts Institute of Technology, 2020)
Let G be a real reductive group and let Ĝ be the set of irreducible unitary representations of G. The determination of Ĝ (for arbitrary G) is one of the fundamental unsolved problems in representation theory. In the early ...
Explicit division and torsion points on superelliptic Curves and jacobians
(Massachusetts Institute of Technology, 2020)
In this thesis, I study two problems in the arithmetic of superelliptic curves. By a superelliptic curve, I mean the smooth projective model of the affine plane curve y[superscript n] = f(x) where f(x) is separable, n is ...
Building and using robust representations in image classification
(Massachusetts Institute of Technology, 2020)
One of the major appeals of the deep learning paradigm is the ability to learn high-level feature representations of complex data. These learned representations obviate manual data pre-processing, and are versatile enough ...
Semi-infinite Homology of Floer spaces
(Massachusetts Institute of Technology, 2020)
This dissertation presents a framework for defining Floer homology of infinite-dimensional spaces with a functional. This approach is meant to generalize the traditional constructions of Floer homologies which mimic the ...
The method of moments in convolved random matrix models and discrete analogues
(Massachusetts Institute of Technology, 2020)
We study the global and local asymptotics of Macdonald processes, its degenerations, and related models using the method of difference operators. We focus on three applications. First, we consider random plane partitions ...
Point processes of representation theoretic origin
(Massachusetts Institute of Technology, 2019)
There are two parts to this thesis. In the first part we compute the correlation functions of the 4-parameter family of BC type Z-measures. The result is given explicitly in terms of Gauss's hypergeometric function. The ...
The arrival time for mean curvature flow on a convex domain
(Massachusetts Institute of Technology, 2019)
We give asymptotics for the level set equation for mean curvature flow on a convex domain near the point where it attains a maximum. It was shown by Natasa Sesum that solutions are not necessarily C³, and we recover this ...