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Fourier decoupling for convex sequences
(Massachusetts Institute of Technology, 2023-06)
We study the decoupling theory for functions on R with Fourier transform sup- ported in a neighborhood of a convex sequence [formula], where [formula] and 𝑔 : [0, 1] → R is a 𝐶² function satisfying 𝑔′(𝑥) > 0, 𝑔′′(𝑥) ...
Higher-order Fourier analysis with applications to additive combinatorics and theoretical computer science
(Massachusetts Institute of Technology, 2022-05)
Fourier analysis has been used for over one hundred years as a tool to study certain additive patterns. For example, Vinogradov used Fourier-analytic techniques (known in this context as the Hardy-Littlewood circle method) ...
Integrability in random conformal geometry
(Massachusetts Institute of Technology, 2022-05)
Liouville quantum gravity (LQG) is a random surface arising as the scaling limit of random planar maps. Schramm-Loewner evolution (SLE) is a random planar curve describing the scaling limits of interfaces in many statistical ...
Symmetric structures in the weak and strong Bruhat orders
(Massachusetts Institute of Technology, 2022-05)
The weak and strong Bruhat orders are classical and rich combinatorial objects, with connections to Schubert calculus, Lie algebras, hyperplane arrangements, sorting networks and so on. In this thesis, we study various new ...
K-stability of Log Fano Cone Singularities
(Massachusetts Institute of Technology, 2022-05)
In this thesis, we define the 𝛿-invariant for log Fano cone singularities, and show that the necessary and sufficient condition for K-semistability is 𝛿 ≥ 1. This generalizes the result of C. Li and K. Fujita. We also ...
Extremal semi-modular functions and combinatorial geometries
(Massachusetts Institute of Technology, 1975)
Towards characterizing morphims between high dimensional hypersurfaces
(Massachusetts Institute of Technology, 2003)
This thesis is organized into two papers. All results are proven over an algebraically closed field of characteristic zero. Paper 1 concerns morphisms between hypersurfaces in Pn, n =/> 4. We show that if the two hypersurfaces ...
Smooth K-theory and locally convex algebras
(Massachusetts Institute of Technology, 2003)
In this thesis, we improve the loop linearization process from the classical article of Atiyah and Bott on Bott periodicity. The linearization process is made explicit in terms of formulae for smooth loops. Using this ...
The instability of time-dependent jets
(Massachusetts Institute of Technology, 2002)
The central focus of my thesis is to study the instability jets of various complexity by analyzing the linear and nonlinear dynamics. We applied this methodology to four different situations in order to learn the following. ...
Geometric approaches to computing Kostka numbers and Littlewood-Richardson coefficients
(Massachusetts Institute of Technology, 2004)
Using tools from combinatorics, convex geometry and symplectic geometry, we study the behavior of the Kostka numbers and Littlewood-Richardson coefficients (the type A weight multiplicities and Clebsch-Gordan coefficients). ...