## January 2019

### Simplicial Complexes, Configuration Spaces, and “Chromatic” Invariants (Andrew Cooper, NC State)

Given a space $X$, the configuration space $F(X,n)$ is the space of possible ways to place $n$ points on $X$, so that no two occupy the same position. But what if we allow some of the points to coincide? The natural way to encode the allowed coincidences is as a simplicial complex $S$. I will describe how the configuration space $M(S,X)$ obtained in this way gives rise to polynomial and homological invariants of $S$, how those invariants are related to…

Find out more »### The Roger-Yang Arc Algebra (Helen Wong, CMC)

Based on geometric considerations, J. Roger and T. Yang in 2014 defined a version of the Kauffman bracket skein algebra for punctured surfaces that includes arcs going from puncture to puncture. We'll provide a brief survey of known results about this arc algebra. In particular, I'd like to mention a recent algebraic result whose proof uses "generalized'' corner coordinates to describe arcs on a triangulated surface. This is joint work with Han-bom Moon.

Find out more »## February 2019

### A nonorientable version of the Milnor Conjecture (Cornelia A. Van Cott, USF)

In 1968, Milnor famously conjectured that the smooth 4-genus of the torus knot T(p,q) is given by (p-1)(q-1)/2. This conjecture was first verified by Kronheimer and Mrowka in 1993 and has received several other proofs since then. In this talk, we discuss a nonorientable analogue of this conjecture, first formulated by Josh Batson. We prove the conjecture for infinite families of of torus knots, using tools from knot Floer homology. We also connect the problem to the world of continued…

Find out more »### Applying Quantum Representations of Mapping Class Groups (Wade Bloomquist, UCSB)

One foundational pillar of low dimensional topology is the connection between link invariants and 3-manifold invariants. One generalization of this has been given by Reshetikhin and Turaev to a surgery theory for colored ribbon graphs. Then to complete the analogy rather than 3-manifold invariants we now have a 2+1 dimensional topology quantum field theory (TQFT). For this talk we will only be focusing on one corner of a TQFT, in particular the representations of mapping class groups which are afforded…

Find out more »## April 2019

### Geometry of quotient varieties and the algebra of conformal blocks (Han-Bom Moon Fordham University)

An important question in classical representation theory is when the tensor product of two irreducible representations has another representation as a factor. In this talk, I will introduce a quantum generalization of this question and explain how we may relate this question to geometry of quotients of certain complex manifolds. This is joint work with Sang-Bum Yoo.

Find out more »### Enhancements of the quandle coloring invariant for knots (Karina Cho, Harvey Mudd College)

Quandles are algebraic structures that play nicely with knots. The multiplicative structure of finite quandles gives us a way to "color" knot diagrams, and the number of such colorings for a given knot and quandle is called the quandle coloring invariant. We strengthen this invariant by examining the relationships between the colorings, which are given by endomorphisms. This can be visualized using a directed graph that we call the quandle coloring quiver. We will show that the quandle coloring quiver…

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