This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) ...

In this thesis we study the self-Floer theory of a monotone Lagrangian submanifold $L$ of a closed symplectic manifold $X$ in the presence of various kinds of symmetry. First we consider the group $\mathrm{Symp}(X, L)$ ...

Let G be a semisimple linear algebraic group over an algebraically closed field of prime characteristic. In this thesis we outline the theory of such groups and their cohomology. We then concentrate on algebraic groups in ...

Smooth manifolds have been always understood intuitively as spaces that are infinitesimally linear at each point, and thus infinitesimally affine when forgetting about the base point. The aim of this thesis is to develop ...

This thesis presents a characterization of those categories with weak factorization systems that can interpret the theory of intensional dependent type theory with Σ, Π, and identity types. We use display map categories ...

The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\mathbb{C}$, ...

This thesis studies the geometry of objects from 2-dimensional statistical physics in the continuum. Chapter 1 is an introduction to Schramm-Loewner evolutions (SLE). SLEs are the canonical family of non-self-intersecting, ...

The purpose of this thesis is to define a “local” version of Ozsváth and Szabó’s Heegaard Floer homology HFL^ for links in the 3-sphere, i.e. a Heegaard Floer homology HFT^ for tangles in the 3-ball. The decategorification ...

I investigate the $K_2$ groups of the quotients of Fermat curves given in projective coordinates by the equation $F_n:X^n+Y^n=Z^n$. On any quotient where the number of known elements is equal to the rank predicted by ...

Floer theory is a rich collection of tools for studying symplectic manifolds and their Lagrangian submanifolds with the help of holomorphic curves. Its origins lie in estimating the numbers of equilibria in Hamiltonian ...

Spectral methods have become increasingly popular in designing fast algorithms for modern highdimensional datasets. This thesis looks at several problems in which spectral methods play a central role. In some cases, we ...

Many of the classes of objects studied in geometry are defined by first choosing a class of nice spaces and then allowing oneself to glue these local models together to construct more general spaces. The most well-known ...

We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion ...

We consider a short time existence problem motivated by a conjecture of Joyce in [8]. Specifically we prove that given any compact Lagrangian L ⊂ C^n with a finite number of singularities, each asymptotic to a pair of ...