Cascade Lectures in Combinatorics 2025
Topic
This is a series of one day conferences being organized in the Pacific Northwest region, with funding from the National Science Foundation (NSF), the Pacific Institute for the Mathematical Sciences (PIMS) and the Milliman Fund. These take place twice a year, with each meeting on a Saturday. They are aimed at graduate students, postdocs, and faculty, bringing in four or five expert speakers for each meeting and aiming to create a sense of community with friendly and frequent meetings. We alternate between in-person meetings each spring and online meetings each fall.
Details
Conference Schedule:
9:15-10am: Welcome/light breakfast
10-11am: Isabella Novik, talk slides
11-11:30am: Coffee break
11:30am-12:30pm: Petter Brändén, talk slides
12:30-2:30pm: Lunch
2:30-3:30pm: Alexander Woo, talk slides
3:30-4pm: Coffee break
4-5pm: Lauren Williams, talk slides
5:30pm: Group dinner at Big Time Brewery
Talk titles and abstracts:
Petter Brändén: Totally nonnegative matrices, chain enumeration and zeros of polynomials
Abstract: We prove a general theorem that relates totally nonnegative matrices to chain enumeration in partially ordered sets, and f-vectors of simplicial complexes and posets. It is used to develop a general theory for chain enumeration in posets and zeros chain polynomials. The results obtained extend and unify results of the speaker, Brenti, Welker and Athanasiadis. In the process we define a notion of h-vectors for a large class of posets which generalize the notions of h-vectors associated to simplicial and cubical complexes. We also use the methods developed to answer an open problem posed by Forgács and Tran on the real-rootedness of polynomials arising from certain bivariate rational functions. This is joint work with Leonardo Saud Maia Leite.
Isabella Novik: Transversal numbers of polytopes, spheres, and pure simplicial complexes
Abstract: A transversal of a uniform hypergraph H is a subset of the vertex set that intersects all edges of H. The transversal number T(H) of H is the minimum cardinality of a transversal of H. In this talk, I will discuss the transversal numbers of hypergraphs arising from pure simplicial complexes, simplicial polytopes, and simplicial spheres. In particular, I will discuss some new upper and lower bounds, including a few constructions of complexes with relatively large transversal numbers. Many of these constructions are closely related to cyclic polytopes. Joint work with Hailun Zheng.
Lauren Williams: Cyclic partial orders, Parke-Taylor polytopes, and the magic number conjecture for the amplituhedron
Abstract: The amplituhedron is a geometric object introduced by physicists to compute scattering amplitudes, certain probabilities that describe what happens when particles with given momenta collide. I'll give a gentle introduction to the amplituhedron and the magic number conjecture, which says that the cardinality of a tiling of the amplituhedron is the number of plane partitions which fit inside a particular box. (This is a generalization of the fact that triangulations of even-dimensional cyclic polytopes have the same size.) I will also introduce some interesting lattice polytopes called Parke-Taylor polytopes, as well as cyclic partial orders and circular extensions, which are cyclic analogues of the notion of partial order and linear extension. Finally I'll explain how these polytopes and cyclic partial orders are related to the magic number conjecture. Based on joint work with Matteo Parisi, Melissa Sherman-Bennett, and Ran Tessler.
Alex Woo: Towards Hessenberg-Schubert calculus
Abstract: Schubert calculus is the study of the classes of the Schubert varieties in the (equivariant) cohomology of the flag variety. In particular, there is a model of this (equivariant) cohomology ring where classes are labelings of the vertices of the Bruhat graph (the Cayley graph of the symmetric group as generated by all transpositions) by polynomials that satisfy restrictions coming from the edges. Classes of the Schubert varieties are given by Billey's formula. Motivated in part by the Stanley-Stembridge conjecture (recently proven by Hikita), I will discuss strategies for finding the classes for analogues of Schubert varieties in Hessenberg varieties, under a model where classes are labelings of a subgraph of the Bruhat graph. I will give explicit results for the Hessenberg variety of a path graph. This talk is based on joint work with Erik Insko and Martha Precup.
Registration: Participants should register by Thursday, March 6 at noon. To register, simply email plhersh@uoregon.edu letting her know you intend to participate. Thanks!
Colloquium on Friday: Lauren Williams will also give a math department colloquium at the University of Washington at 3:30pm on Friday, March 7 in ECE 125. Title to be announced. Conference participants are all welcome.
Additional Information
Visit the main event page for more information.