Category theory is the mathematical study of connections. While it is often used at a very abstract level to work on mathematical structures, recent work has shown that category theory can also be used to explain and model ecological food webs, chemical reactions, and data transformations. In this directed reading we will be focusing on the first of these, building up an understanding of category theory using the book Seven Sketches in Compositionality before leaping into more recent research to investigate how several longstanding questions of biodiversity have been answered using categorical methods.

(Prerequisites: Intro to Abstract Algebra (Math 3170))

Universal algebra studies properties of abstract algebraic structures. This is distinct from the related branch of mathematics, abstract algebra. Where abstract algebra studies algebraic structures and asks questions like "What are the properties of fields?" or "What is a representation theory of groups?", universal algebra treats the structures themselves as objects of study and asks questions such as "What are the properties of the class of all groups?" or "What is the subalgebra structure of the class of all bounded lattices?".

Book: A Course in Universal Algebra The Millenium Edition by S. Burris and H.P. Sankappanavar. This book is available for free online, a title search should easily find it.

(Prerequisites: Intro to Proofs and ideally familiarity with diagonalization arguments although not a strict requirement and can be covered in our readings.)

While the notion of an algorithm has existed since antiquity, modern computability theory began with David Hilbert's Entscheidungsproblem in 1928. Mathematicians spent the better part of the 1930s searching for the correct model of computation. In 1936, Turing introduced his model of computation and simultaneously dashed Hilbert's dream that all problems are computable. 

We will study the foundational notions of computability from the perspective of Turing machines. This includes the Church-Turing thesis, the halting problem (an example of a non-computable problem), the recursion theorem (which implies the existence of a self-referential program), oracle machines, computably enumerable sets, and more!

Depending on the interest of the student, we will then be well equipped to study other models of computation, complexity theory, formal language theory, and advanced topics in computability theory such as forcing, the arithmetic hierarchy, and the structure of the Turing degrees. 

Book: Turing Computability: Theory and Applications  by Robert I. Soare