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...

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...

Motivated by knot theory, we introduce a homology theory for small categories with functor coefficients. Under this general framework, different familiar homology theories such as group homology, chromatic homology, poset homology, and Khovanov homology can be realized as homology of small...

In this thesis we lay the foundation for rational degree d as an element of Z[1/p] by using perfectoid analogue of projective space, and consider power series instead of polynomials. We start the groundwork by proving Weierstrass theorems for perfectoid spaceswhich are analogues of standard...

Categorification lifts numbers to vector spaces and vector spaces to categories. A prime example is turning Euler characteristic of a topological space into its homology groups. Important examples include various link homology groups which lift polynomial invariants of knots. For instance,...