Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

The (type A) Hecke algebra H_n(q) is an n!-dimensional q-analog of the symmetric group.  A related trace space of certain functions on H_n(q) has dimension equal to the number of integer partitions of n. If we could evaluate all functions belonging to some basis of the trace space on all elements of some basis of H_n(q), then by linearity we could evaluate em all traces on all elements of H_n(q).  Unfortunately there is no simple published formula which accomplishes this. We will consider a basis of H_n(q) which is related to structures called wiring diagrams, and a combinatorial rule for evaluating one trace basis on all elements of this wiring diagram basis. This result, the first of its kind, is joint work with Justin Lambright and Ryan Kaliszewski. 

This is Lecture 3 of a series of 3 lectures. See the abstract on Tuesday's ACO colloquium of this week.(Please note that this lecture will be 80 minutes' long.)

A tight k-uniform \ell-cycle, denoted by TC_\ell^k, is a k-uniform hypergraph whose vertex set is v_0, ..., v_{\ell-1}, and the edges are all the k-tuples {v_i, v_{i+1}, \cdots, v_{i+k-1}}, with subscripts modulo \ell. Motivated by a classic result in graph theory that every n-vertex cycle-free graph has at most n-1 edges, Sos and, independently, Verstraete asked whether for every integer k, a k-uniform n-vertex hypergraph without any tight k-uniform cycles has at most \binom{n-1}{k-1} edges. In this talk I will present a construction giving negative answer to this question, and discuss some related problems. Joint work with Jie Ma.

For a fixed graph $G$, let $\mathcal{L}_G$ denote the family of Lipschitz functions $f:V(G) \rightarrow \mathbb{R}$ such that $0 = \sum_u f(u)$. The \emph{spread} of $G$ is denoted $c(G) := \frac{1}{|V(G)|} \max_{f \in \mathcal{L}_G} \sum_u f(u)^2$ and the subgaussian constant is $e^{\sigma_G^2} := \sup_{t > 0} \max_{f \in \mathcal{L}_G} \left( \frac{1}{|V(G)|} \sum_u e^{t f(u)} \right)^{2/t^2}$. Motivation of these parameters comes from their relationship with the isoperimetric number of a graph (given a number $t$, find a set $W \subset V(G)$ such that $2|W| \geq |V(G)|$ that minimizes $i(G,t) := |\{u : d(u, W) \leq t \}|$). While the connection to the isoperimetric number is interesting, the spread and subgaussian constant have not been any easier to understand. In this talk, we will present results that describe the functions $f$ achieving the optimal values. As a corollary to these results, we will resolve two conjectures (one false, one true) about these parameters. The conjectures that we resolve are the following. We denote the Cartesian product of $G$ with itself $d$ times as $G^d$. Alon, Boppana, and Spencer proved that the set $\{u: f(u) < k\}$ for extremal function $f$ for the spread of $G^d$ gives a value that is asymptotically close to the isoperimetric number when $d, t$ grow at specific rates and $k=0$; and they conjectured that the value is exactly correct for large $d$ and $k,t$ in ``appropriate ranges.'' The conjecture was proven true for hypercubes by Harper and the discrete torus of even order by Bollob\'{a}s and Leader. Bobkov, Houdr\'{e}, and Tetali constructed a function over a cycle that they conjectured to be optimal for the subgaussian constant, and it was proven correct for cycles of even length by Sammer and Tetali. This work appears in the manuscript https://arxiv.org/abs/1705.09725 .