Mark Ellingham , Vanderbilt University.
Title: Maximum genus directed embeddings of digraphs
Abstract:
In topological graph theory we often want to find embeddings of a given connected graph with minimum genus, so that the underlying compact surface of the embedding is as simple as possible. If we restrict ourselves to cellular embeddings, where all faces are homeomorphic to disks, then it is also of interest to find embeddings with maximum genus. For undirected graphs this is a very well-solved problem. For digraphs we can consider directed embeddings, where each face is bounded by a directed walk in the digraph. The maximum genus problem for digraphs is related to self-assembly problems for models of graphs built from DNA or polypeptides. Previous work by other people determined the maximum genus for the very special case of regular tournaments, and in some cases of directed 4-regular graphs the maximum genus can be found using an algorithm for the representable delta-matroid parity problem. We describe some recent work, joint with Joanna Ellis-Monaghan of the University of Amsterdam, where we have solved the maximum directed genus problem in some reasonably general situations.
Olya Mandelshtam, University of Waterloo.
Title: A new compact formula for symmetric Macdonald polynomials via interacting particles
Abstract:
I will discuss recently discovered connections between one-dimensional interacting particle models—the ASEP and the TAZRP—and Macdonald polynomials, and the combinatorial objects (multiline queues) that make these connections explicit. Based on these objects, a tableau formula was found for the modified Macdonald polynomial in terms of the so-called `queue inversion' statistic, which naturally corresponds to the dynamics of the TAZRP. We present a new tableau formula for the symmetric Macdonald polynomials, using the same queue inversion statistic on non-attacking tableaux. Central to our approach is a probabilistic bijection on non-attacking fillings that enables swapping of entries between columns.
Vic Reiner University of Minnesota
Title: Koszulity and Stirling Representations
Abstract:
(based on arXiv:2404.10858, with Ayah Almousa and Sheila Sundaram)
Supersolvable hyperplane arrangements and matroids are known to give rise to certain Koszul algebras, namely their Orlik-Solomon algebras and graded Varchenko-Gel'fand algebras. We explore how this interacts with group actions, particularly for the braid arrangement and the action of the symmetric group, where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality.
Lutz Warnke , University of california San Diego
Title: Phase Transition in Random Graphs
Abstract:
The percolation phase transition is one of the most interesting and striking features of random graphs. As the edge-to-vertex ratio surpasses a critical threshold, the graph’s connectivity changes dramatically: from only small components, to a structure dominated by a single `giant' component alongside smaller components.
In this talk, we discuss how mathematical tools and ideas from different areas (including from partial differential equations, differential equations, generating functions, and branching processes) can be combined to analyze the phase transition in both classical and modern random graph models.
Shira Zerbib,Iowa State University
Title: Using the KKM theorem
Abstract:
The KKM theorem, due to Knaster, Kuratowski, and Mazurkiewicz in 1929, is a fundamental result in fixed-point theory, which has seen numerous extensions and applications. In this talk we survey old and recent generalizations of the KKM theorem and their applications in the areas of piercing numbers, mass partition, fair division, and matching theory. We also discuss new results utilizing KKM-type theorems, and related open problems. Joint with Daniel McGinnis.