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Applied math in geophysical fluids : partially trapped wave problems and mining plumes
(Massachusetts Institute of Technology, 2018)
The first portion of this work focuses on leaky modes in the atmospheric sciences. Leaky modes (related to quasi-modes, scattering resonances, and the singularity expansion method) are discrete, oscillatory and decaying ...
On the metric structure of random planar maps and SLE-decorated Liouville quantum gravity
(Massachusetts Institute of Technology, 2018)
A random planar map is a graph embedded in the sphere, viewed modulo orientation-preserving homeomorphisms. Random planar maps are the discrete analogues of random fractal surfaces called [gamma]-Liouville quantum gravity ...
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 ...
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 ...
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 ...