For any metric space, the metric gives deep insight into the space such as geometry and topology of the space. Rather than directly studying a specific space, one may wish to study a sequence of spaces with nice properties and then show that desired properties hold in the limit. To study such a sequence of…

The group of permutations is the canonical example of a finite Coxeter group, and each permutation can be represented visually by a noncrossing arc diagram. Each diagram encodes the canonical join representation of its permutation, and diagrams can be used to understand lattice congruences on the weak order of type A, equivalence relations that respect the…

For some nice subclass of braided monoidal categories, the associator and braiding data are described by an Abelian 3-cocycle . Eilenberg and MacLane demonstrated a surprisingly simple isomorphism between the groups of Abelian 3-cocycles with coefficients in  and quadratic forms with coefficients in , the trace map  by . Like many category theoretical results, this was proven at the level of…

Ed Witten's papers in 1988 drew out the connection between Jones Polynomials of Knots and Chern-Simons theory of 3-dimensional TQFT. Following his work, Reshetikhin and Turaev formulated invariants of 3-manifolds via colored framed knots leading to new TQFTs. In this talk, I will introduce Dehn Surgery on knots, framed knots colored with representations of Hopf…

In this talk, we begin by introducing the audience to some fundamentals of differential geometry of surfaces in ℝ3, focusing on curvature and the adequately named first and second fundamental forms. We will discuss what special parameterizations and properties give rise to isothermic surfaces. Equipped with these tools, we will look at the fundamental theorem for surfaces,…