Events

Title: Eigenvalues of high dimensional Laplacian operators

Abstract: A simplicial complex is a topological space built by gluing together simple building blocks (such as vertices, edges, triangles and their higher dimensional counterparts). Alternatively, we can define a simplicial complex combinatorially, as a family of finite sets that is closed under inclusion. In 1944, Eckmann introduced a class of high dimensional Laplacian operators acting on a simplicial complex. These operators generalize the Laplacian matrix of a graph (which can be seen as a 0-dimensional Laplacian), and are strongly related to the topology of the complex (and in particular, to its homology groups).

Title: The non-backtracking matrix

Abstract: A non-backtracking walk in a graph is any traversal of the vertices of a graph such that no edge is immediately repeated. The non-backtracking matrix of a graph is indexed by the directed edges of the graph, and encodes if two edges can be traversed in succession. Since this matrix is not symmetric, the question of when the matrix is diagonalizable is of interest to those who study it. Equivalently, such graphs have a non-trivial Jordan block. In this talk, I will present an overview of the non-backtracking matrix, including its history and applications. Finally, I will present recent results about graphs with non-trivial Jordan blocks for the non-backtracking matrix from joint work with Kristin Heysse, Kate Lorenzen, and Xinyu Wu.

Title: Pretty Good Fractional Revival via Magnetic Fields - theory and examples

Abstract: I will discuss the notion of PGFR relative to a given subset of nodes of a graph. This is a generalization of the more standard (pretty good) fractional revival between 2 nodes. In the process, I will introduce the proper generalization of cospectrality to the fractional setting, and give the appropriate extensions of methods already in use for the 2-vertex case. These include the Kronecker condition and the field-trace method. I will conclude by giving various families of examples of PGFR in the presence of (transcendental) magnetic fields.

Title: A new shifted Littlewood-Richardson rule

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.

Abstract: As Littlewood-Richardson rules compute linear representation theory of symmetric groups and cohomology of ordinary Grassmannians, shifted Littlewood-Richardson rules compute analogous projective representation theory of symmetric groups and cohomology of orthogonal Grassmannians. The first shifted Littlewood-Richardson rule is due to Stembridge (1989), building on a natural generalization by Sagan and Worley (1979/1984) of the jeu de taquin algorithm to shifted Young tableaux. We give a new shifted Littlewood-Richardson rule that requires consideration of fewer tableaux than Stembridge's rule and appears to involve an easier check on each. Our rule derives from applying old ideas of Lascoux and Schützenberger (1981) to the study of Haiman's mixed insertion (1989) and Serrano's shifted plactic monoid (2010). (Joint work with Oliver Pechenik).

Title: Follow the Regularized Leader and Mirror Descent

Abstract: In previous talks, we have seen how the multiplicative weights method and gradient descent solve the regret minimization problem. In this talk we will go over a meta-algorithm called Follow the Regularized Leader (FTRL). We will show how FTRL generalizes both multiplicative weights and gradient descent. We will also talk about the Mirror Descent meta-algorithm, and show its equivalence with FTRL.

Title: Biased cliques

Abstract: A biased clique is a collection of cycles in a complete graph G so that no theta of G contains exactly two cycles in the collection. They have interesting connections to both matroids and groups; I will give a survey of some results on these objects.

Title: Subconstituents of Drackns 

Abstract: For a distance-regular graph X and an arbitrary vertex v, we often find interesting structure in the subgraph of X induced on vertices at distance 2 from v.

For example:

Any strongly-regular graph with parameters (n,k,a,k/2) can be found at distance 2 from a vertex in a distance-regular graph of diameter 3.

Certain distance-regular graphs of diameter 3 can be found at distance 2 from a vertex in a Moore graph of girth 5.

These (and more) examples are known as second-subconstituents, and they can be studied using the Terwilliger (or subconstituent) algebra of X. I will discuss this theory in the case that X is a distance-regular antipodal cover of a complete graph (drackn). This setting generalizes the first example and includes the second.

I will describe some general techniques for studying the Terwilliger algebras of drackns and then restrict to drackns without triangles. In this setting I will explain how to compute the spectrum of the second-subconstituent of any triangle-free drackn, except possibly the second-subconstituent OF a second-subconstituent of a Moore graph of valency 57.

Title: Odd cycle transversal in P5-free graphs

Abstract: Odd cycle transversal is a fun computational problem, somewhere between colouring and independent set: we are (equivalently) looking for a bipartite induced subgraph of maximum weight. As one might expect, this is NP-hard; I will tell you how to solve this problem in polynomial time in P5-free graphs (and more). Joint work with Cece Henderson, Evelyne Smith-Roberge, and Rebecca Whitman.

Note: I am COVID-cautious and will bring masks for those willing to wear them. 

Title: More Martin and c2 details.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.  Please also come to the pre-seminar to discuss what sorts of snacks and refreshments we might want going forward.

Abstract:  I'm going to tell you more of the details behind my Martin polynomial work last year with Erik Panzer which led to the proof of the c2 completion conjecture. I will actually describe the key bijection that proves it all, say some things about the permanent invariant, and cover other details that there hadn't been time for in the colloquium level presentation of the result.

Title: A O(log log (rank) ) - competitive algorithm for the matroid secretary problem

Abstract: In the Matroid Secretary problem the weighted elements of a matroid arrive one by one in a uniformly random order where an online algorithm observes the value of the element and must make an irrevocable decision of whether or not to include the element in its solution before the arrival of the next element. The goal is to maximize the total value of the chosen elements under the condition that they must constitute an independent set. Other than knowing the cardinality of the ground set and having access to an independence oracle, the algorithm has no further information about the matroid.