In 1984, knot theory was revolutionized with the discovery of the Jones polynomial. Fifteen years later, with several questions about it still unanswered, the polynomial was categorified into what is presently known as Khovanov homology. Parts of the Przytycki-Sazdanovic braid conjecture state...

Classical knot theory is the study of simple closed curves in three dimensional space. The dissertation studies algebraic structures related to and motivated by knot theory. Since the first half of 1990, several homology theories have been defined for quandles and, more generally, for shelves...

In this paper, we investigate pattern formation in a
two-phase system on a two-dimensional manifold by numerically computing the
minimizers of a Cahn-Hilliard-like model for micro-phase separation of
diblock copolymers. The total energy of the system includes a short-range
term - a Landau...

We prove that if Q is a finite quasigroup quandle, then
|Q| annihilates the torsion of its homology. It is a classical result in
reduced homology of finite groups that the order of a group annihilates its
homology. From the very beginning of the rack homology (between 1990 and
1995) the...

In this thesis, we study global behavior of solutions to various nonlinear dispersive partial differential equations, which are important in real-world applications such as nonlinear wave equation (NLW) and nonlinear Klein-Gordon equation (NLKG), relevant to wave propagation and turbulence;...

This dissertation consists of four parts. In the first part we prove that two different types of set-theoretic Yang-Baxter homology theories lead to the same homology. In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang-Baxter operators (we will call it the...

Energy-driven pattern formation induced by competing short and long range interaction is common in many biological and physical systems. We report on our work through two models. The sharp interface model is a nonlocal and non-convex geometric variational problem. The admissible class of the...

Using the notions and methods of computability theory, we study effectiveproperties of computable magmas. A magma is an algebraic structure with asingle binary operation that is not necessarily associative nor commutative.We define a magma to be computable if it is finite or if its domain can...

This dissertation concerns the mathematical analysis of a mathematical model for priceformation. We take a large number of rational buyers and vendors in the market who aretrading the same good into consideration. Each buyer or vendor will choose his optimalstrategy to buy or sell the good. Since...

In this thesis, we investigate pattern formation in a two-phase system based on the Otha-Kawasaki model for diblock copolymers. In the Ohta-Kawasaki model, the total energy of the systemincludes a short-range term - a Landau free energy and a long-range term - the Otha-Kawasakifunctional. The...