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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 ...
Self-shrinkers of mean curvature flow and harmonic map heat flow with rough boundary data
(Massachusetts Institute of Technology, 2011)
In this thesis, first, joint with Longzhi Lin, we establish estimates for the harmonic map heat flow from the unit circle into a closed manifold, and use it to construct sweepouts with the following good property: each ...
Sparse regularity and relative Szemerédi theorems
(Massachusetts Institute of Technology, 2015)
We extend various fundamental combinatorial theorems and techniques from the dense setting to the sparse setting. First, we consider Szemerédi regularity lemma, a fundamental tool in extremal combinatorics. The regularity ...
Effective Chabauty for symmetric powers of curves
(Massachusetts Institute of Technology, 2014)
Faltings' theorem states that curves of genus g > 2 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the upper bound on the number of rational ...
Index theorems and magnetic monopoles on asymptotically conic manifolds
(Massachusetts Institute of Technology, 2010)
In this thesis, I investigate the index of Callias type operators on asymptotically conic manifolds (also known as asymptotically locally Euclidean manifolds or scattering manifolds) and give an application to the moduli ...
Whittaker functions on metaplectic groups
(Massachusetts Institute of Technology, 2010)
The theory of Whittaker functions is of crucial importance in the classical study of automorphic forms on adele groups. Motivated by the appearance of Whittaker functions for covers of reductive groups in the theory of ...
Origami manifolds
(Massachusetts Institute of Technology, 2010)
An origami manifold is a manifold equipped with a closed 2-form which is symplectic everywhere except on a hypersurface, where it is a folded form whose kernel defines a circle fibration. In this thesis I explain how an ...
A free boundary problem inspired by a conjecture of De Giorgi
(Massachusetts Institute of Technology, 2012)
We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional I(u) = f lVul2 + V(U), where V(u) is the characteristic function of the interval (-1, 1). This functional ...
The Seiberg-Witten equations on manifolds with boundary
(Massachusetts Institute of Technology, 2011)
In this thesis, we undertake an in-depth study of the Seiberg-Witten equations on manifolds with boundary. We divide our study into three parts. In Part One, we study the Seiberg-Witten equations on a compact 3-manifold ...