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 - 10:30

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 - 16:30

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 - 16:30

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 - 15:30

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.

Geometry and Topology

Speaker: Rune Haugseng (NTNU)

"Homotopy-coherent distributivity and the universal property of bispans"

Time: 11:30 - 12:30

Room: Zoom Meeting ID: 958 6908 4555

Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\infty$-)categories of spans (or correspondences). In some cases we have two pushforwards (an ''additive'' and a ''multiplicative'' one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). For example, commutative semirings can be described in terms of bispans of finite sets, while bispans in finite G-sets can be used to encode Tambara functors, which are the structure on $\pi_0$ of G-equivariant commutative ring spectra. Motivated by applications of the $\infty$-categorical upgrade of such descriptions to motivic and equivariant ring spectra, I will discuss the universal property of $(\infty, 2)$-categories of bispans. This gives a universal way to obtain functors from bispans, which amounts to upgrading ''monoid-like'' structures to ''ring-like'' ones. In the talk I will focus on the simplest case of bispans in finite sets, where this gives a new construction of the semiring structure on a symmetric monoidal $\infty$-category whose tensor product commutes with coproducts. This is joint work with Elden Elmanto.

Algebra Seminar

Speaker: Brett Nasserden (Waterloo)

"Heights on Stacky Curves"

Time: 14:30 - 15:30

Room: Zoom

In a forthcoming work, Jordan Ellenberg, Matthew Satriano, and David Zureick-Brown introduce a new theory of heights on algebraic stacks. This theory extends the classical theory of heights on algebraic varieties. Moreover, Ellenberg, Satriano, and Zureick-Brown have formulated a stacky version of the Manin conjecture which predicts the distribution of rational points on a suitable algebraic stack with respect to a suitable stacky height. This conjecture when applied to the classifying stack of a finite group G recovers a version of Malle’s conjecture for the group G; Malle’s conjecture predicts the asymptotic distribution of number fields of bounded discriminant with Galois group G.

I will give an introduction to this circle of ideas in the case of stacky curves. In this setting the theory is simpler and can often be explicitly described. In particular, one can explicitly describe the stacky height function associated to the anti-canonical bundle of a stacky projective line with chosen number of half points. With this explicit description in hand I will discuss some recent work with Stanely Xiao which verifies a new instance of the Ellenberg, Satriano, and Zureick-Brown conjecture and discuss some open problems and ongoing investigations in this area.

Geometry and Topology

Speaker: Andres F. Fontalvo Orozco (ETH Zurich)

"Traces and Link Invariants"

Time: 11:30 - 12:30

Room: Zoom Meeting ID: 958 6908 4555

Ribbon categories can be used to produce invariants of framed links via the Reshetikhin-Turaev construction. We will examine this construction and use it to motivate a generalization of the categorical trace known as the module trace. We will then examine a few examples, and basic properties. We will finish discussing partial results regarding existence and uniqueness.

Geometry and Topology

Speaker: Martina Rovelli (University of Massachusetts Amherst)

"$n$-complicial sets as a model for $(\infty, n)$-categories"

Time: 15:30 - 16:30

Room: Zoom Meeting ID: 958 6908 4555

With the rising significance of $(\infty, n)$-categories, it is important to have easy-to-handle models for those and understand them as much as possible. In this talk we will discuss how $n$-complicial sets provide a model for $(\infty, n)$-categories, and how one can recover strict $n$-categories through a suitable nerve construction. We will focus on $n = 2$, for which more results are available, but keep an eye towards the general case. Time permitting, we will also discuss a few recent research directions and work in progress about $n$-complicial sets.