- Coxeter groups are a family of nicely behaved groups that
can be viewed as being geometrically generated by reflections and
Artin groups are another family of groups dervied from them. The
motivating example is the following. The symmetric group is a Coxeter
group and the braid group is the corresponding Artin group. In this
talk I will attempt to describe exactly which Artin groups we
understand and which ones we do not. More explicitly, my criterion
for "understanding a group" is whether or not we can solve its word
problem.

(This is the second talk in a series but it should be understandable even if you did not see the first talk that I gave last quarter.)

- The Robinson-Schensted correspondence is a combinatorial bijection between permutations of n elements and pairs of standard Young tableaux of the same shape with n boxes. It was proved by Spaltenstein and Steinberg that this bijection has a geometric interpretation in terms of the variety of complete flags in an n-dimensional vector space V. The correspondence can be generalized: for example we can consider signed permutations and pairs of standard bitableaux. I will explain how this generalization naturally appears geometrically when considering flags in Kato’s “exotic” version of symplectic space, although the combinatorial side is not completely understood yet. This is joint work with Vinoth Nandakumar and Neil Saunders.

- Tesler matrices are upper triangular matrices with nonnegative integer entries with certain restrictions on the sums of their rows and columns. Glenn Tesler studied these matrices in the 1990s and in 2011 Jim Haglund rediscovered them in his study of diagonal harmonics. We investigate a polytope whose integer points are the Tesler matrices. It turns out that this polytope is a flow polytope of the complete graph thus relating its lattice points to vector partition functions. We study the face structure of this polytope and show that it is a simple polytope. We show its h-vector is given by Mahonian numbers and its volume is a product of consecutive Catalan numbers and the number of Young tableaux of staircase shape.

- This talk will be a brief survey to a subfield of combinatorics called bijective combinatorics. The idea is simple: if two finite sets have the same size, then there should a bijection between them. An interesting quest is then how to find such a bijection. This can be really fun to try yourself as it does not require excessive machinery to get started. If time permits, I will discuss a new result on bijective study of the so-called basketball walks.

- In the first talk in this series I explained the basic spaces used to define and study Artin groups. In the second talk I described exactly which of these groups are well understood and which remain mysterious. In this third talk, I will describe what I view as a few tractable avenues to explore in the near future.

- The RSK correspondence is a many-facetted jewel of algebraic combinatorics. In one variation, it provides a bijection between permutations of n and pairs of standard Young tableaux with the same shape, which is a partition of n. In another (more general) version, it provides a bijection between fillings of a partition lambda by arbitrary non-negative integers and fillings of the same shape lambda by non-negative integers which weakly increase along rows and down columns. I will discuss a connection between RSK and the representation theory of type A quivers (i.e., directed graphs obtained by orienting a path graph). On the one hand, this provides yet another perspective on classical results. At the same time, it solves some questions of independent representation-theoretic interest. I will not assume familiarity with either RSK or with quiver representations. This is joint work with Al Garver and Becky Patrias.

- While it is not usually stated this way, Thales' Theorem says that if L is a line segment in the Euclidean plane, the set of points at which L subtends a right angle is the circle with diameter L. But what happens when we change the angle? What happens when we change L to some other subset of the plane? What happens in higher dimensions? In this talk, I will take this idea and run as fast as I can with it in as many directions as I can; my path won't be a space-filling curve, but I will encounter ideas from a variety of mathematical subject areas.

For context, I am hoping to submit a write-up of my intentionally incomplete results to a journal with an audience of undergraduate students/advisers looking for research directions. As such, the talk will (in theory) be accessible to undergraduates. Also, I am out of my depth in most of the subject areas I am going to talk about, so I am looking for feedback wherever possible.