(Prerequisites: Formally, none; however, some familiarity with abstract algebra (groups) would be helpful)

Topology is the study of geometric properties of objects that are stable under certain kinds of transformations (e.g. the coffee mug and the donut are comparable in some sense).  Algebraic topology leverages certain connections with algebraic objects to study the topologies of certain spaces.

Topics to be studied include, but are not limited to: fundamental groups, covering spaces, Van Kampen's theorem, and homology.

(Prerequisites: Techniques of Proofs required, would recommend some interest in abstract algebra, such as familiarity with a group and modular arithmetic)

The foundations of 20th century mathematics are built on ZFC set theory. Naïvely, a set is a bucket you can put things into that cannot contain itself. This has generally been quite effective, and one can reasonably say mathematicians have accomplished a lot in the past century. However, there are conceptual elements to type theory that don't seem to line up with how mathematicians think. There are infinitely many groups with two elements, although since they are all isomorphic, we like to think of there only being one group of order 2, Z2. 

Type theory replaces sets with types, which always track what kind of data they contain. In homotopy type theory (HoTT), types become objects with a geometric flavor. Isomorphisms become paths linking together objects that we would want to consider "the same", and equality becomes a hierarchy of shapes that glue together things with "sameness" in possibly different ways. Homotopy type theory offers an alternative to foundations in mathematics that captures all the dynamic relationships between things.

In this project we will cover the basics of dependent type theory, and work towards understanding the univalent axiom of HoTT, building basic structures such as the natural numbers from the ground up. We will also learn how to formalize proofs as computer code. HoTT has found use in formalizing proofs for computers, and I see value in learning how this process works, rather than it being a mysterious black box.

Book: Introduction to Homotopy Type Theory by Egbert Rijke 

(prerequisites: Calc 3 is required; linear algebra would be nice but is not required; probability and/or real analysis is highly recommended)

How do you make a password that you can remember that machines can't crack? What on earth is entropy and why does nobody seem to agree? What is a jpeg really doing when it says it is "compressing" the data? How is it that sometimes when a cd is scratched it skips but sometimes it reads perfectly fine? All these questions are related to a field called information theory, which we will explore.