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Coherent sheaves on varieties arising in Springer theory, and category 0
(Massachusetts Institute of Technology, 2015)
In this thesis, we will study three topics related to Springer theory (specifically, the geometry of the exotic nilpotent cone, and two-block Springer fibers), and stability conditions for category 0. In the first chapter, ...
Parametrized higher category theory
(Massachusetts Institute of Technology, 2017)
We develop foundations for the category theory of [infinity]-categories parametrized by a base occategory. Our main contribution is a theory of parametrized homotopy limits and colimits, which recovers and extends the ...
Asymptotic description of the formation of black holes from short-pulse data
(Massachusetts Institute of Technology, 2020)
In this thesis we present partial progress towards the dynamic formation of black holes in the four-dimensional Einstein vacuum equations from Christodoulou's short-pulse ansatz. We identify natural scaling in a putative ...
Quantitative invertibility of random matrices : a combinatorial perspective
(Massachusetts Institute of Technology, 2020)
In this thesis, we develop a novel framework for investigating the lower tail behavior of the least singular value of random matrices - a subject which has been intensely studied in the past two decades. Our focus is on ...
Towards a functor between affine and finite Hecke categories in type A
(Massachusetts Institute of Technology, 2018)
In this thesis we construct a functor from the perfect subcategory of the coherent version of the affine Hecke category in type A to the finite constructible Hecke category, partly categorifying a certain natural homomorphism ...
Some results on principal series for GL(n,R).
(Massachusetts Institute of Technology, 1977)
Linear Algebra, Random Matrices and Lie Theory
(Massachusetts Institute of Technology, 2022-02)
This thesis bridges the gap between pure and applied mathematics. The first part of this thesis focuses on the theory and computation of various Lie groups. The classical Lie groups as well as the automorphism groups of ...
On long time dynamic and singularity formation of NLS
(Massachusetts Institute of Technology, 2017)
In this thesis, we investigate the long time behavior of focusing mass critical nonlinear Schrödinger equation (NLS). We will focus on the singularity formation and long time asymptotics. To be specific, there are two parts ...
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 ...
Homotopically nontrivial area contracting maps
(Massachusetts Institute of Technology, 2023-06)
In this thesis, we investigate the existence of area contracting maps in a given ho-motopy class. In particular, we give various generalizations of Guth’s h-principle for k-dilation of maps between spheres to other manifolds. ...