The holomorphic sigma-model is a field theory that exists in any complex dimension that describes the moduli space of holomorphic maps from one complex manifold to another. We introduce the general notion of a holomorphic field theory, which is one that is sensitive to the underlying complex structure of the...
We prove a uniform scalar curvature bound for solutions of the conical Kahler-Ricci flow when the twisted canonical bundle is semiample and the cone divisor is obtained from the associated Iitaka-Kodaira fibration. In the course of the proof we establish uniform bounds for the potential of the metric and its...
This work is concerned with the Laudau-Ginzburg $A$-model, or the Fukaya-Seidel category, associated with a Laurent polynomial $f: (\C^*)^n \ o \C$. We use constructible sheaves on a real $n$-dimensional torus to describe the Lagrangian thimbles associated to $f$. Then we discuss the application to Homological Mirror Symmetry for smooth...
In this thesis, we consider the spherical spin glass models at zero temperature. We first determine the structureof the Parisi measure at zero temperature for the spherical p+s spin glass model. We then
consider the spherical mixed p-spin model and show that for the spherical spin models with
n components,...
We introduce empirical measures to study the $L^2$ norms of restrictions of quantum integrable eigenfunctions to the unique rotationally invariant geodesic $H$ on a convex surface of revolution. The weak* limit of these measures describes the dependence of their size on $H$ in terms of the angular momentum. The limit...
In this paper we begin the study of the (dual) Steenrod algebra of the motivic Witt cohomology spectrum H_Wℤ by determining the algebra structure of H_Wℤ∗∗H_Wℤ over fields k of characteristic not 2 which are extensions of fields F with K^M_2(F)/2=0. For example, this includes all fields of odd characteristic,...
This dissertation provides an introduction to the diffraction and scattering theory for the Aharonov--Bohm Hamiltonian with one or multiple poles on $\mathbf{R}^2$. It shows the propagation of diffractive singularities of the wave equation with the magnetic Hamiltonians with singular vector potential, which is related to the so-called Aharonov--Bohm effect. Based...
This thesis is composed of two major parts. The first major part includes the first two chapters. In Chapter 1, we go over the history and background of spin glass theory and introduce the Sherrington-Kirkpatrick (SK) model and the Ghatak-Sherrington (GS) model. In Chapter 2, we introduce the Thouless-Anderson-Palmer (TAP)...
We prove the uniqueness of equilibrium states for certain potentials satisfying the Bowen property for two flows related to geodesic flows on surfaces with sufficient hyperbolicity. Our first result is the uniqueness of equilibrium states for Hölder continuous potentials and the geometric potential for products of geodesic flows of rank...
We study the symmetry groups with respect to various equivalence relations defined on subshifts, and more generally, on Cantor systems. Two basic notions of equivalence for dynamical systems are conjugacy and flow equivalence. In this dissertation, we focus on the well-studied automorphism group, which is the group of self-conjugacies, and...