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Computational limitations for small depth circuits
(Massachusetts Institute of Technology, 1986)
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 ...
Shortest paths, Markov chains, matrix scaling and beyond : improved algorithms through the lens of continuous optimization
(Massachusetts Institute of Technology, 2017)
In this thesis, we build connections between classic methods from convex optimization and the modern toolkit from the fast Laplacian solver literature, in order to make progress on a number of fundamental algorithmic ...
Getting a handle on contact manifolds
(Massachusetts Institute of Technology, 2019)
In this thesis, we develop the details of a surgery theory for contact manifolds of arbitrary dimension via convex structures, extending the 3-dimensional theory developed by Giroux. The theory is analogous to that of ...
Semi-algebraic graphs and hypergraphs in incidence geometry
(Massachusetts Institute of Technology, 2019)
A (hyper)graph is semi-algebraic if its vertices are points in some Euclidean spaces and the (hyper)edge relation is defined by a finite set of polynomial inequalities. Semi-algebraic (hyper)graphs have been studied ...
Distinguishing open symplectic mapping tori via their wrapped Fukaya categories
(Massachusetts Institute of Technology, 2019)
The main goal of this thesis is to use homological methods as a step towards the classification of symplectic mapping tori. More precisely, we exploit the dynamics of wrapped Fukaya categories to distinguish an open version ...
Algebraic geometry and representation theory in the Verlinde category
(Massachusetts Institute of Technology, 2019)
This thesis studies algebraic geometry and the representation theory of group schemes in the setting of symmetric tensor categories over algebraically closed fields of positive characteristic. A specific focus is paid to ...
The geometry and dynamics of twisted quadratic differentials
(Massachusetts Institute of Technology, 2019)
This thesis examines twisted quadratic differentials, also known as dilation surfaces. These are variants of translation surfaces, their more well-studied counterpart. In this work, we study questions about the realizability ...
Unstable operations in the Bousfield-Kan spectral sequence for simplicial commutative FF₂-algebras
(Massachusetts Institute of Technology, 2015)
In this thesis we study the Bousfield-Kan spectral sequence (BKSS) in the Quillen model category sCom of simplicial commutative FF₂ -algebras. We develop a theory of unstable operations for this BKSS and relate these ...