Geometry and Combinatorics

Speaker: Priyavrat Deshpande (Chennai Mathematical Institute)

"A statistic on labeled threshold graphs: interpreting coefficients of the threshold characteristic polynomial"

Time: 09:30

Room: Zoom

Consider the collection of hyperplanes in $\mathbb{R}^n$ whose defining equations are of the form $x_i + x_j =0$. This arrangement is called the threshold arrangement since its regions are in bijection with labeled threshold graphs on $n$ vertices. Zaslavsky's theorem implies that the number of regions is the sum of coefficients of the characteristic polynomial of the arrangement. In this talk I will explain how to give a combinatorial meaning to these coefficients as the number of labeled threshold graphs with a certain property, thus answering a question posed by Stanley. This talk is based on joint work with Krishna Menon and Anurag Singh

Geometry and Topology

Speaker: Rick Jardine (Western)

"The UMAP algorithm, reimagined"

Time: 15:30

Room: Zoom Meeting ID: 958 6908 4555

The Healy-McInnes UMAP algorithm for a data set $X$ has been highly successful as a data science tool. The goal of this talk is to present a geometric version of this algorithm. The present approach requires Barr's generalized fuzzy sets with coefficients in the interval $[0,\infty]$, completeness properties of extended pseudo-metric spaces (ep-metric spaces), and manipulations of simplicial presheaves. We also use a geometric version of the cross entropy function for fuzzy sets with coefficients in $[0,\infty]$, which can be effectively bounded by distance.

Colloquium

Speaker: Graham Denham (Western)

"Lorentzian polynomials 2"

Time: 15:30

Room: Via Zoom

A complex polynomial in n variables satisfies the (Hurwitz) half-plane property if its value is nonzero when its inputs all have positive real part. This classical definition is the start of an interesting story that factors through the closely related theory of stable polynomials, due to Borcea and Brändén [Duke J. Math 2008] and leads to the notion of Lorentzian polynomials, recently introduced by Brändén and Huh [Annals of Math 2020]. Lorentzian polynomials also have an elementary definition, though subtle properties and close links to (statistical) negative dependence, matroid theory, and discrete convexity.

In this two-part Basic Notions seminar, Graham plans to spend a few minutes saying why he thinks this is interesting. Then together we will watch a recording of an introductory lecture that June Huh gave at the IAS in 2019. We will break in the middle, since the lecture is 90 minutes long. We will resume in the second week and conclude with some informal discussion.

Algebra Seminar

Speaker: Alexandru Buium (University of New Mexico)

"Arithmetic differential equations"

Time: 14:30

Room: Zoom

An arithmetic analogue of the theory of differential equations (both ordinary and partial) has been developed in recent years. It is based on replacing derivatives of functions with Fermat quotients of numbers and it led to a series of applications to Diophantine geometry. The talk will offer an overview of this development including recent joint work with L. Miler.