Mathematics

I will begin by introducing some of the most basic combinatorial objects - partitions. It turns out that their generating function is a prototypical example of a modular form. These are objects with infinite symmetry, in turn giving them extraordinary properties. I'll then talk about the asymptotic behaviour of various modular-type objects arising from combinatorics and topology using the Circle Method of Hardy-Ramanujan and Wright, as well as one can even obtain exact formulae. In particular, I'll highlight the asymptotic (non)-equidistribution properties of Betti numbers of various Hilbert schemes as well as t-hooks in partitions. This talk will include various works with configurations of my collaborators Kathrin Bringmann, Giulia Cesana, William Craig, Daniel Johnston, Ken Ono, and Aleksander Simonič.

Speaker Biography:

Joshua Males received his MMath (masters + bachelors) degree from Durham University, UK under the supervision of Jens Funke, before taking a year sabbatical in Durham. In late 2017 he joined Kathrin Bringmann's number theory group at the University of Cologne, Germany, where he earned his PhD in May 2021. Since August 2021, Joshua has been a PIMS postdoctoral fellow at the University of Manitoba, working under his mentor Siddarth Sankaran. His research focuses on modular forms and their use in number theory and beyond, with connections to combinatorics, topology, and arithmetic geometry. At the time of writing, Joshua has 8 published articles (4 solo author) and 6 preprints (1 solo author) as well as 3 more articles in the latter stages of preparation.

Estimating the COVID-19 infection fatality rate (IFR) has proven to be challenging, since data on deaths and data on the number of infections are subject to various biases. I will describe some joint work with Harlan Campbell and others on both methodological and applied aspects of meeting this challenge, in a meta-analytic framework of combining data from different populations. I will start with the easier case when the infection data are obtained via random sampling. Then I will discuss drawing in additional infection data obtained in decidedly non-random manner.

How far inside a domain does a flux of Brownian particles perturb a background concentration when particles can escape through a neighboring window? What motivates this question is the dynamics of ions entering and exiting nanoregions of excitable cells through ionic membrane channels. Here this is explored using a simple diffusion model consisting of the Laplace's equation in a domain whose boundary is everywhere reflective except for a collection of narrow circular windows, where either flux or absorbing boundary conditions are prescribed. We derive asymptotic formulas revealing the role of the influx amplitude, the diffusion properties, and the geometry, on the concentration difference. Lastly, a length scale to estimate how deep inside a domain a local diffusion current can spread is introduced. This is joint work with David Holcman at ENS.

Knot Floer homology is a package of widely-used knot invariants constructed via pseudo-holomorphic curves. In this talk, we will restrict our attention to the knot Floer homology of a class of knots called satellite knots; understanding these invariants figure prominently in studying 4-dimensional questions in knot theory, such as analyzing surfaces bounded by knots in 4-manifolds. However, previous methods of computing these invariants are rather involved. In this talk, I will present a new and more effective way to compute the knot Floer homology of satellite knots; our approach is built on the immersed-curve technique introduced by Hanselman-Rasmussen-Watson in bordered Heegaard Floer homology. This talk is based on joint work in progress with Jonathan Hanselman.

Speaker Biography:

Wenzhao Chen obtained his Ph.D. at Michigan State University in 2019, where he studied Heegaard Floer homology and low dimensional topology under the supervision of Dr. Matt Hedden. He was a postdoc in the Max Planck Institue for Mathematics in Bonn from 2019 to 2021. Currently, He is a PIMS Postdoctoral Fellow at the University of British Columbia. He is working with Dr. Liam Watson in low-dimensional topology.

Let \(G\) be a finite group acting transitively on \(X\). We say \(g,h \in G\) are intersecting if \(gh^{-1}\) fixes a point in \(X\). A subset \(S\) of \(G\) is said to be an intersecting set if every pair of elements in \(S\) intersect. Cosets of point stabilizers are canonical examples of intersecting sets. The group action version of the classical Erdos-Ko-Rado problem asks about the size and characterization of intersecting sets of maximum possible size. A group action is said to satisfy the EKR property if the size of every intersecting set is bounded above by the size of a point stabilizer. A group action is said to satisfy the strict-EKR property if every maximum intersecting set is a coset of a point stabilizer. It is an active line of research to find group actions satisfying these properties. It was shown that all \(2\)-transitive satisfy the EKR property. While some \(2\)-transitive groups satisfy the strict-EKR property, not all of them do. However a recent result shows that all \(2\)-transitive groups satisfy the slightly weaker "EKR-module property"(EKRM), that is, the characteristic vector of a maximum intersecting set is a linear span of characteristic vectors of cosets of point stabilizers. We will discuss about a few more infinite classes of group actions that satisfy the EKRM property. I will also provide a few non-examples and a characterization of the EKRM property using characters of \(G\) .