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Functional and cross-trait genetic architecture of common diseases and complex traits
(Massachusetts Institute of Technology, 2017)
In this thesis, I introduce new methods for learning about diseases and traits from genetic data. First, I introduce a method for partitioning heritability by functional annotation from genome-wide association summary ...
Negative answers to some positivity questions
(Massachusetts Institute of Technology, 2014)
We construct counterexamples to a number of questions related to positivity properties of line bundles on algebraic varieties. The examples are based on studying the geometry of varieties that admit pseudo automorphisms ...
Bounds on extremal functions of forbidden patterns
(Massachusetts Institute of Technology, 2015)
Extremal functions of forbidden sequences and 0 - 1 matrices have applications to many problems in discrete geometry and enumerative combinatorics. We present a new computational method for deriving upper bounds on extremal ...
Gaussian free field, Schramm-Loewner evolution and Liouville quantum gravity
(Massachusetts Institute of Technology, 2017)
Consider an instance h of the Gaussian free field on a simply connected domain ... We study several properties of the level lines: continuity, monotonicity, reversibility and target-independence ... In the second part, we ...
Mayer-Vietoris property for relative symplectic cohomology
(Massachusetts Institute of Technology, 2018)
In this thesis, I construct and investigate the properties of a Floer theoretic invariant called relative symplectic cohomology. The construction is based on Hamiltonian Floer theory. It assigns a module over the Novikov ...
Combinatorics of acyclic orientations of graphs : algebra, geometry and probability
(Massachusetts Institute of Technology, 2015)
This thesis studies aspects of the set of acyclic orientations of a simple undirected graph. Acyclic orientations of a graph may be readily obtained from bijective labellings of its vertex-set with a totally ordered set, ...
On planar rational cuspidal curves
(Massachusetts Institute of Technology, 2014)
This thesis studies rational curves in the complex projective plane that are homeomorphic to their normalizations. We derive some combinatorial constraints on such curves from a result of Borodzik-Livingston in Heegaard-Floer ...
Closed quasigeodesics, escaping from polygons, and conflict-free graph coloring
(Massachusetts Institute of Technology, 2018)
Closed quasigeodesics. A closed quasigeodesic on the surface of a polyhedron is a loop which can everywhere locally be unfolded to a straight line: thus, it's straight on faces, uniquely determined on edges, and has as ...
Symplectic cohomology of contractible surfaces
(Massachusetts Institute of Technology, 2014)
In 2004, Seidel and Smith proved that the Liouville manifold associated to Ramanujams surface contains a Lagrangian torus which is not displaceable by Hamiltonian isotopy, and hence that higher products of this manifold ...
Economic behavior from an evolutionary perspective
(Massachusetts Institute of Technology, 2015)
The conflict between rational models of economic behavior and their systematic deviations, often referred to as behavioral economics, is one of the most hotly debated issues in social sciences. This thesis reconciles the ...