Algebraic complexity theory is the study of the computational difficulty of infinite families of polynomials -- a generalization of the computational complexity of decision problems. In this theory, there are analogues of the usual complexity classes P and NP, as well as different NP-complete problems. The main aim is to find complexity lower bounds for certain specific families, such as the families for matrix multiplication and the permanent family. There are different approaches using algebraic geometry and representation theory to attack such lower bound problems. This talk will be a brief introduction to this area and its central problems.

The existence and uniqueness of Gevrey regularity solution for a class of nonlinear
bistable gradient flows, with the energy decomposed into purely convex and concave parts,
such as epitaxial thin film growth and square phase field crystal models, are discussed in this talk.
The polynomial pattern of the nonlinear terms in the chemical potential enables one to derive a
local in time solution with Gevrey regularity, with the existence time interval length dependent
on certain functional norms of the initial data. Moreover, a detailed Sobolev estimate for the gradient
equations results in a uniform in time bound, which in turn establishes a global in
time solution with Gevrey regularity. An extension to a system of gradient flows,
such as the three-component Cahn-Hilliard equations, is also addressed in this talk.

Motivated from his p-adic study of the variation of the zeta function as
the variety moves through a family, Dwork conjectured that a new type of
L-function, the so-called unit root L-function, was always p-adic
meromorphic. In the late 1990s, Wan proved this using the theory of
sigma-modules, demonstrating that unit root L-functions have structure.
Little more is known.

This talk is concerned with unit root L-functions coming from families of
exponential sums. In this case, we demonstrate that Wan's theory may be
used to extend Dwork's theory -- including p-adic cohomology -- to these
L-functions. To illustrate the technique, the unit root L-function of the
Kloosterman family is studied in depth.

The purpose of this thesis is to use the tools of Inner Model Theory to the study of notions relative to generic embeddings induced by ideals. We seek to apply the Core Model Induction technique to obtain lower bounds in consistency strength for a specific stationary catching principle called StatCatch*(I), related to the saturation of an ideal I of omega_2. This principle involves the central notion of self-genericity in its formulation, introduced by Foreman, Magidor and Shelah. In particular, we show that assuming StatCatch*(I) (plus some additional hypothesis in the universe), we can obtain, for every finite n, an inner model with n Woodin Cardinals.

We will explain a systematic way to study a type of curve flows in R^{n+1, n}, whose geometric invariants are solutions of some integrable systems. In detail, we will construct a hierarchy of isotropic curve flows in R^{2, 1}, construct explicit solutions from Backlund transformation, and study its Hamiltonian formulation.