Events

Title: Crystal invariant theory and geometric RSK

Abstract:

The original problem of classical invariant theory was to describe the invariants of SL_m acting on a polynomial ring in an m \times n matrix of variables. One way to solve this problem is to consider the polynomial ring as a GL_m \times GL_n representation, and decompose this representation into its irreducible components.

Title: Laplacian States

Abstract:

It is customary to assume that the initial state of a continuous quantum walk on a graph $X$ is a vertex. However the Laplacian matrix of a graph with vertex set $V(X)$ is positive semidefinite, and can be scaled to produce a density matrix, and so provides an initial state for a walk on $X$.

Title: Enumerating Matroids and Linear Spaces

Abstract:

We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the puzzle for enumerating fixed-rank matroids at this level of accuracy: there are exact formulas for enumeration of rank-1 and rank-2 matroids, and it was recently proved by van der Hofstad, Pendavingh, and van der Pol that for constant $r\ge 4$ there are $(e^{1-r}n+o(n))^{n^{r-1}/r!}$ rank-$r$ matroids on a ground set of size $n$.

Title: Guessing with little data

Abstract:

A popular and powerful technique in experimental mathematics takes as input the first few terms of an infinite sequence and returns plausible candidates for recurrence equations that the sequence may satisfy. In a way, the search for such candidates is a generalization of polynomial interpolation. For polynomial interpolation, it is well known and easy to see that d+1 sample points are needed in order to recover a polynomial of degree d. Similarly, it turns out that (r+1)*(d+2) consecutive terms of a sequence are needed in order to detect a linear recurrence of order r with polynomial coefficients of degree at most d.

Title: Minimal relations for an algebra inspired by algebraic graph theory

Abstract:

The balanced algebra has two generators, R and L, and its defining relations are that any pair of balanced words commutes. For example, RL and LR are balanced (contain the same number of both generators), so in the balanced algebra, (RL)(LR)=(LR)(RL).

Title: A Matching Theoretic Flat Wall Theorem

Abstract:

One of the key theorems in Graph Minors is the Flat Wall Theorem which asserts the existence of a large wall under certain conditions. In this talk, we discuss about graphs with perfect matchings and their relationship with digraphs. Our main focus is on a matching theoretic analogue of the Flat Wall Theorem for bipartite graphs excluding a fixed matching minor.

Title: Random Self-reducibility of Ideal-SVP via Arakelov Random Walks

Abstract:

Fixing a number field, the space of all ideal lattices, up to isometry, is naturally an Abelian group, called the *Arakelov class group*. This fact, well known to number theorists, has so far not been explicitly used in the literature on lattice-based cryptography. Remarkably, the Arakelov class group is a combination of two groups that have already led to significant cryptanalytic advances: the class group and the unit torus.

Title: Springer fibers and the Delta Conjecture at t=0

Abstract:

Springer fibers are a family of varieties that have remarkable connections to combinatorics and representation theory. Springer used them to geometrically construct all of the irreducible representations of the symmetric group (Specht modules). Moreover, they give a geometric meaning to Hall-Littlewood symmetric functions.

Title: Traveling Salesman Problems: Approximation Algorithms and Black-Box Reductions

Abstract:

We survey the recent progress on approximation algorithms and integrality ratios for variants of the traveling salesman problem, with a focus on black-box reductions from one problem to another. In particular, we explain recent results for the Path TSP and the Capacitated Vehicle Routing Problem, which are joint works with Vera Traub and Rico Zenklusen and with Jannis Blauth and Vera Traub.

Title: Hadamard diagonalizable graphs of small order

Abstract:

A graph whose Laplacian matrix has a full set of eigenvectors with entries in {1,-1} is said to be Hadamard diagonalizable (i.e. there exists a Hadamard matrix which diagonalizes the Laplacian matrix). We demonstrate that the only diagonalizable graphs on n=8k+4 vertices are K_n and K_{n/2,n/2}.