Events

Title: Nash-Williams Orientation for Infinite Graphs

Abstract: Nash-Williams proved in 1960 that an edge-connectivity of 2k is sufficient for a finite graph to admit a k-arc-connected orientation and conjectured that the same holds for infinite graphs. We show that the conjecture is true for locally finite graphs with countably many ends.

This is joint work with Max Pitz and Marcel Koloschin.

Title: Online edge colouring

Abstract: Vizing's Theorem provides an algorithm that edge colors any graph of maximum degree Δ can be edge-colored using Δ+1 colors, which is necessary for some graphs, and at most one higher than necessary for any graph. In online settings, the trivial greedy algorithm requires 2Δ-1 colors, and Bar-Noy, Motwani and Naor in the early 90s showed that this is best possible, at least in the low-degree regime.

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: Combinatorial Action in Causal Set Quantum Gravity

Abstract: In this talk, I will first provide a brief overview of causal set theory, an approach to quantum gravity. This theory proposes that spacetime is fundamentally characterized by a partially ordered set (poset), in which the partial order represents causal relations and the number of elements signifies the volume of a spacetime manifold region. I will then discuss how efforts to find a discrete counterpart of the d'Alembertian operator on a poset led to the formulation of the causal set action S_BDG. This action is defined as a linear combination of the counts of various order intervals. Further analysis has shown that while KR posets are predominant in the number of posets of size n, the quantum dynamics imposed by S_BDG suppresses them for large n. Finally, I will propose a method to derive the combinatorial analogue of Einstein's field equations on posets.

Title: Regret bounds for FTRL and Mirror Descent

Abstract: In the previous talk, I introduced FTRL and Mirror Descent and showed how they generalize two well-known algorithms of Multiplicative Weights and Gradient Descent. In this talk, I will show that FTRL and Mirror Descent are in fact equivalent in the sense that they produce the same sequence of predictions. Moreover, I will go over some regret bounds for these algorithms, that will generalize the regret bounds we get for multiplicative weights and gradient descent.

Title: Laplacian State Transfer

Abstract: Let $X$ be a graph and let $E_1,\ldots,E_d$ be the spectral idempotents of its adjacency matrix. If $a$ and $b$ are vertices in $X$, they are \textsl{strongly cospectral} if $E_re_ae_a^TE_r = E_re_be_b^T$ for each $r$. This is a relation between the two density matrices $e_aa_a^T$ and $e_be_b^T$, and is a necessary condition for state transfer between pure states.

If $L$ is the Laplacian of a graph $X$ with $m$ edges, the matrix $(1/2m)L$ is positive semidefinite with trace one, thus it is a density matrix. We call it a \textsl{Laplacian state}. It is pure only if $X$ is an edge. We have been investigating transfer between Laplacian states in continuous quantum walks. We have extended the definition of strongly cospectral to this case, have obtained a number of results are about various forms of state transfer. My talk will be a report on this.

(This is joint work with Ada Chan, Qiuting Chen, Wanting Sun and Xiaohong Zhang.)