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...
The BV Laplacian Δ, first introduced by Batalin and Vilkovisky, is a second-order differential operator that appears in the quantum master equation for quantizing gauge theories. The geometric framework for the BV formalism was later recognized by Schwarz as the setting of odd symplectic geometry and Khudaverdian showed that Δ...
In this thesis, we study the homological mirror symmetry for the Gross-Siebert program of local mirror symmetry. We construct a pair of mirror objects by lifting a tropical curve in the integral tropical manifold of the Gross-Siebert program. Furthermore, we evaluate the central charges of the mirror objects and show...
In this thesis we study the geometric limits under degree growth of Julia sets and filled Julia sets for complex polynomials with a unique critical point at $z = 0$. Specifically, for $c \in \mathbb{S}^1$, we are interested in the limit of the associated sequence of Julia sets $J(f_{n,c})$ in...
We prove the Rigidity Conjecture of Goette and Igusa, which states that, after rationalizing, there are no stable exotic smoothings of manifold bundles with closed even dimensional fibers. The key ingredients of the proof are fiberwise Poincaré–Hopf theorems generalizing earlier such results about the Becker–Gottlieb transfer. These theorems show how...
In this dissertation, we study and prove formulas of Hodge ideals for three types of isolated hypersurface singularities, including quasihomogeneous singularities, Newton non-degenerate singularities and analytically irreducible plane curve singularities.
For quasihomogeneous singularities, we provide
a formula of Hodge ideals of $D=\alpha Z$ which is a Q-divisor.
A direct consequence...
We study Hermitian metrics with constant Chern scalar curvature on a compact complex manifold. In the first part of the thesis, we prove a priori estimates for metrics of constant Chern scalar curvature on a compact complex manifold conditional on an upper bound on the entropy, extending a recent result...
The topic of this dissertation is the coherent-constructible correspondence (CCC). CCC is a version of homological mirror symmetry for toric varieties. It equates the derived category of coherent sheaves on a toric variety and the category of constructible sheaves on a torus whose singular support lies in the specific conical...
We study plurisubharmonic functions and their applications to K\"ahler geometry. We begin by studying regularity of envelopes of plurisubharmonic functions, particularly when the reference form is degenerate. This is then applied to show regularity of geodesic of K\"ahler metrics on singular varieties, as well as regularity of certain geodesic rays....
In this thesis, we study applications of the theory of perverse sheaves and their enhancements to problems in birational geometry. In the first application, we give positive results towards a conjecture of Batyrev about the nonnegativity of stringy Hodge numbers. In particular, we prove the nonnegativity of $(p,1)$-stringy Hodge numbers...
The variation of entropy in a family of dynamical systems is a natural indication of the bifurcations that the family undergoes. In the context of one-dimensional dynamics, Milnor's monotonicity of entropy conjecture (now a theorem of Bruin and van Strien) asserts that for polynomial interval maps with real critical points...
In this dissertation we study the connections over principal bundles in dimension four with bounded Yang-Mills energy, and present a new result on the existence a global Coulomb gauge with estimate in optimal space. To be precise, let $A$ be a $W^{1,2}$-connection on a principal $\text{SU}(2)$-bundle $P$ over a smooth...
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...
This thesis provides an introduction to decay rates for the damped wave equation on compact manifolds. It also gives a proof of a sharp decay rate for solutions to the damped wave equation on the torus with damping of a particular polynomial form. Finally it gives a proof of a...
In this thesis, we study the geometry of planar shapes and their harmonic caps. Specifically, given a compact continuum $P$, we are interested in constructing a planar cap $\hat P$ such that $P$ and $\hat P$ can be glued together along their boundary to form a topological sphere with prescribed...
We study modular forms, Jacobi forms, and hermitian formal Fourier-Jacobi series over imaginary quadratic fields. In the first section, we prove that the ring of classical Jacobi forms of a fixed genus g, varying index m and weight k is generated by theta functions. From this result we show that...
Arising from the study of multiple ergodic averages, nilsequences and multiple correlation sequences lie at the crossroads of ergodic theory, combinatorics and number theory. We study these types of sequences along various subsequences of integers, and provide applications to ergodic theory and harmonic analysis. Our first result involves multiple correlation...
We address the problem of efficient maintenance of the answer to a new type of query: Continuous Maximizing Range-Sum (Co-MaxRS) for moving objects trajectories. The traditional static/spatial MaxRS problem finds a location for placing the centroid of a given (axes-parallel) rectangle $R$ so that the sum of the weights of...
In this paper, we study the basic locus in the fiber at $p$ of a certain unitary Shimura variety with a certain parahoric level structure. The basic locus $\widehat{\CM^{ss}}$ is uniformized by a formal scheme $\CN$ which is called Rapoport-Zink space. We show that the irreducible components of the induced...
The Picard group is an important invariant of the $K(n)$-local category. If the prime $p$ is relatively large compared to the height $n$, the Picard group of the $K(n)$-local category is purely algebraic. In \cref{chapter:finitetype}, we describe the necessary and sufficient numerical condition when an element $X$ in the Picard...