In this thesis we study the $L^p$ Dirichlet problem for second order divergence-form operators having a BMO antisymmetric part. To be precise, for the divergence-form …
The Teichmüller space of a surface that admits a hyperbolic structure can be equipped with the Weil-Petersson metric. The Weil-Persson Ending Lamination is a candidate …
This dissertation is devoted to the study of inverse spectral problem of Hankel operators. It is well-known that spectral characteristics of a Hankel operator does …
Polygon iterations, which can be thought of as discrete dynamical systems on the space of polygons, provide an abundance of interesting discrete dynamical systems in …
This thesis consists of four chapters on the author's research in algebraic dynamics. The first chapter introduces the main results. The second chapter is an …
In the field of arithmetic dynamics, we study number theoretic aspects of discrete dynamical systems induced by rational maps on projective spaces. Among the rational …
We consider the one-dimensional nonlinear Schroedinger equation (NLS) with focusing point nonlinearity, which replaces the pure power nonlinearity in the standard NLS by inserting a …
We present a series of C^2-Morse functions on the Deligne-Mumford compactification M_{g,n} bar of the moduli space of genus g Riemann/hyperbolic surfaces with n punctures. …
The work in this thesis concerns three projects at the interface of statistical mechanics and cluster integrable systems. We describe each of these in the …
In this thesis, we study three families of moduli spaces: those of tropical curves, relative stable maps to the Riemann sphere, and weighted stable curves. …
This thesis exhibits two of the author's works: the first is about interpreting the derived equivalences of K3 surfaces through Cremona transformations, where we construct …
We study the one-dimensional Gross-Pitaevskii equation, a cubic defocusing non-linear Schrodinger equation with nonvanishing boundary conditions. In particular we linearize around the dark solitons, which …
Anonymous credentials allow a user to prove possession of a set of credentials, issued by some trusted issuer or issuers, authorizing access to a resource. …
Fractional-order partial differential equations (FPDEs) describe macroscopic properties of systems driven by Levy processes and, more generally, Continuous-Time Random Walks (CTRWs). CTRWs form a versatile …
Domain exchange transformations (DETs) are dynamical systems of piecewise translations defined on piecewise smooth Jordan domains. DETs are two-dimensional generalizations of interval exchange transformations which …
This dissertation discusses the ramification-theoretic behavior of Galois representations attached to dynamical systems over local fields, with applications to global fields. These arboreal representations are …
Chapter 1 presents joint work with Nikolaos Kapouleas and is concerned with constructions of new closed, embedded minimal surfaces in the round three sphere using …
We prove a weak factorization result on birational maps of Deligne-Mumford stacks, and deduce the following: Let $U \subset X$ be an open embedding of …
We develop a functor calculus for functors from FI, the category of finite sets and injections, to an arbitrary stable presentable infinity-category, which we call …
