(prerequisites: none)

How do we describe the motion of the planets? At any single moment each planet has a mass, a position, a velocity and an acceleration. But then, due to the position of other planets and the laws of gravity, each planet’s position, velocity and acceleration will change. Then because of that change, how gravity acts upon this new “state” of the planets will change. And so on.

The theory of dynamical systems offers a tool for describing the behavior of complex evolving systems. It has applications in planetary motion, biological systems, particle motion, flow of water and even data storage.

This project will provide an introduction to the tools of dynamical systems. To do this we will first cover the basics of topology. We will then explore examples of dynamical systems which arise in various areas of mathematics.

(Prerequisites: Some familiarity with proofs)

In this directed reading, the student will learn about the connections of syntax and semantics in logic. In particular, they will study soundness and completeness results which will exploit the connection of formal proofs in a deductive system with equational truths in respective algebras. The main focus will be the propositional logic and it's algebraic counterpart Boolean algebras. However, depending on the pace and the understanding, I would like to tackle other classical logics such as modal and intuitionistic logics. 

(Prerequesits: Linear Algebra encouraged, but a basic understanding of matrices and vectors will suffice. )

Knots, Quandles, and Algebraic Structure 

Knots are some of the most universal objects in human art and aesthetics, and can be found in cultures throughout the world. They also show up everywhere in nature, from protein folding to quantum mechanics. What mathematics are hiding within knots, and how can we build a theory to characterize them? What does it mean to turn a geometric object into an algebraic structure? 

Knot theory provides a way to formalize and characterize knots, leading to powerful invariants and beautiful classification theorems. Knot theory brings together concepts across many fields in mathematics, from abstract algebra to algebraic topology. In particular, a modern technique uses algebraic objects known as quandles, which have a self-distributive axiom in place of associativity. 

This project will introduce the basics of knot theory and abstract algebra in a hands-on way. Knot properties will be shown using pictures and diagrams, and then algebraic structures will be developed through concrete examples on matrices. The goal will be a motivation of what an algebraic structure is and how to build one, rather than starting from axioms. With these tools, we will cover the basic theory of quandles and how they can be used with knots. Depending on how quickly we move, there could be time to talk about tools in algebraic topology applied to knots, at a very introductory level.