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Faster algorithms for convex and combinatorial optimization
(Massachusetts Institute of Technology, 2016)
In this thesis, we revisit three algorithmic techniques: sparsification, cutting and collapsing. We use them to obtain the following results on convex and combinatorial optimization: --Linear Programming: We obtain the ...
Revealing and analyzing imperceptible deviations in images and videos
(Massachusetts Institute of Technology, 2016)
The world is filled with objects that appear to follow some perfect model. A sleeping baby might look still and a house's roof .should be straight. However, both the baby and the roof can deviate subtly from their ideal ...
Parabolic Springer resolution
(Massachusetts Institute of Technology, 2016)
Let G be a reductive group over a field k = k. Let P be a parabolic subgroup. We construct a functor Groupoid ... is a connected space, which induces an action of generalizing a classical result. It is also a part of a ...
Fields of rationality of cuspidal automorphic representations
(Massachusetts Institute of Technology, 2016)
This thesis examines questions related to the growth of fields of rationality of cuspidal automorphic representations in families. Specifically, if F is a family of cuspidal automorphic representations with fixed central ...
Integral geometry, Hamiltonian dynamics, and Markov Chain Monte Carlo
(Massachusetts Institute of Technology, 2016)
This thesis presents applications of differential geometry and graph theory to the design and analysis of Markov chain Monte Carlo (MCMC) algorithms. MCMC algorithms are used to generate samples from an arbitrary probability ...
The Morse index of mean curvature flow self-shrinkers
(Massachusetts Institute of Technology, 2016)
In this thesis, we will introduce a notion of index of shrinkers of the mean curvature flow. We will then prove a gap theorem for the index of rotationally symmetric immersed shrinkers in R3, namely, that such shrinkers ...
On folding and unfolding with linkages and origami
(Massachusetts Institute of Technology, 2016)
We revisit foundational questions in the kinetic theory of linkages and origami, investigating their folding/unfolding behaviors and the computational complexity thereof. In Chapter 2, we exactly settle the complexity of ...
Mirror symmetry and the K theory of a p-adic group
(Massachusetts Institute of Technology, 2016)
Let G be a split, semisimple p-adic group. We construct a derived localization functor Loc : ... from the compactified category of [BK2] associated to G to the category of equivariant sheaves on the Bruhat-Tits building ...
Sculpting representations for deep learning
(Massachusetts Institute of Technology, 2016)
In machine learning, the choice of space in which to represent our data is of vital importance to their effective and efficient analysis. In this thesis, we develop approaches to address a number of problems in representation ...
Self-shrinkers and translating solitons of mean curvature flow
(Massachusetts Institute of Technology, 2016)
We study singularity models of mean curvature flow ("MCF") and their generalizations. In the first part, we focus on rigidity and curvature estimates for self-shrinkers. We give a rigidity theorem proving that any self-shrinker ...