- Coxeter groups are a large family of groups whose definition
is motivated by the spherical and euclidean reflection groups that
encode the symmetries of the regular polytopes and the euclidean
tilings at the heart of Lie theory, respectively.

Artin groups are another large family of groups whose definition is a geometrically motivated variation on the definition of Coxeter groups. The relationship between the two families is essentially the same as the relationship between the symmetric group and the braid group.

In this short series of talks, I will introduce the geometric spaces associated to Artin groups and describe what we know about them. The first talk is a quick overview of some key aspects of the theory of Coxeter groups leading to an explicit description of the spaces that are used to define Artin groups.

- For certain root posets, there exists a "doppelgĂ¤nger" minuscule poset---that is, both posets have the same number of linear extensions and the same number of plane partitions of height k. In these cases, there happens to also be a second minuscule poset whose top half is the root poset. These two facts are related.

We synthesize M. Haiman's rectification, H. Thomas and A. Yong's minuscule K-theoretic Schubert calculus techniques, and a remark made by R. Proctor to give a framework for combinatorial proofs of these poset coincidences.

This is joint work with Zachary Hamaker, Rebecca Patrias, and Oliver Pechenik.

- In combinatorics, the integer sequence known as the Catalan numbers lies at the center of several different counting problems. Aside from being the solution to many enumerative questions, these numbers also arise in seemingly disparate topics such as Temperley-Lieb algebras, hyperplane arrangements, and algorithmic questions for Coxeter and Artin groups. In this talk, we will discuss the basics of Catalan numbers and an appearance of their structure in the braid group.

- Some of you may remember last year I gave a talk about the space of lattices in the plane and the trefoil knot. This talk is not a sequel, but was inspired by that stuff - over the past year I have tracked down a couple of different objects whose "deformation spaces" are homeomorphic to the trefoil complement, including the space of certain cubic equations, SL(2,Z) invariant tilings of H^2 and (at most 3 element) subsets of the circle. My goal here is to explain some of the cool things that can be gained from taking these different perspectives: for example considering cubic equations will give us a nice description of the trefoil knot group and considering hyperbolic tilings will help us visualize the SL2 geometry of the complement.

Mostly, this talk is an excuse for me to show you guys some neat videos I have been producing in Mathematica related to all of this - so even if the trefoil knot is not your thing there should be enough pretty pictures to make it fun!

- Positroids are a well-behaved class of matroids, defined in terms of totally nonnegative cells in the real Grassmannain Gr(k,n). They have excellent topological and combinatorial properties, and are now beginning to show up across mathematics and physics: in noncommutative algebra, cluster algebras, solutions to the KP equation, and calculation of scattering amplitudes. I'll define positroids, talk about some of their basic properties, and show how they're linked to Wilson loop diagrams (one method of computing scattering amplitudes in N=4 SYM).