Sunday  Monday  Tuesday  Wednesday  Thursday  Friday  Saturday 
28 
29 Geometry and Combinatorics
Geometry and Combinatorics Speaker: (Western) "no seminar this week" Time: 15:30 Room: 
30 
31 Geometry and Topology
Geometry and Topology Speaker: Ralph Kaufmann (Purdue University) "Natural appearances of cubical structures" Time: 15:30 Room: Zoom Meeting ID: 958 6908 4555 Cubical structures appear naturally in several situations, sometimes clandestinely. In particular, we will consider such structures related to Feynman categories and discuss their origins. In this context, they appear for instance in coproduct structures, in a Wconstruction and relative Wconstructions given via pushforwards, which give natural cubical complexes that are homotopy equivalent to moduli spaces. 
1 Colloquium
Colloquium Speaker: Graham Denham (Western) "Lorentzian polynomials 1" Time: 15:30 Room: Via Zoom A complex polynomial in n variables satisfies the (Hurwitz) halfplane 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 BrÃƒÂ¤ndÃƒÂ©n
[Duke J. Math 2008] and leads to the notion of Lorentzian polynomials, recently introduced by BrÃƒÂ¤ndÃƒÂ©n
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 twopart 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.

2 
3 
4 
5 Geometry and Combinatorics
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 
6 
7 Geometry and Topology
Geometry and Topology Speaker: Rick Jardine (Western) "The UMAP algorithm, reimagined" Time: 15:30 Room: Zoom Meeting ID: 958 6908 4555 The HealyMcInnes 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 pseudometric spaces (epmetric 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. 
8 Colloquium
Colloquium Speaker: Graham Denham (Western) "Lorentzian polynomials 2" Time: 15:30 Room: Via Zoom A complex polynomial in n variables satisfies the (Hurwitz) halfplane 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 BrÃ¤ndÃ©n
[Duke J. Math 2008] and leads to the notion of Lorentzian polynomials, recently introduced by BrÃ¤ndÃ©n
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 twopart 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.

9 Algebra Seminar
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. 
10 
11 
12 
13 
14 Geometry and Topology
Geometry and Topology Speaker: Rune Haugseng (NTNU) "Homotopycoherent distributivity and the universal property of bispans" Time: 11:30 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 Gsets can be used to encode Tambara functors, which are the structure on $\pi_0$ of Gequivariant 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 ''monoidlike'' structures to ''ringlike'' 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. 
15 
16 Algebra Seminar
Algebra Seminar Speaker: Brett Nasserden (Waterloo) "Heights on Stacky Curves" Time: 14:30 Room: Zoom In a forthcoming work, Jordan Ellenberg, Matthew Satriano, and David ZureickBrown introduce a new theory of heights on algebraic stacks. This theory extends the classical theory of heights on algebraic varieties. Moreover, Ellenberg, Satriano, and ZureickBrown 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 Malleâ€™s conjecture for the group G; Malleâ€™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 anticanonical 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 ZureickBrown conjecture and discuss some open problems and ongoing investigations in this area.

17 
18 
19 
20 
21 Geometry and Topology
Geometry and Topology Speaker: Andres F. Fontalvo Orozco (ETH Zurich) "Traces and Link Invariants" Time: 11:30 Room: Zoom Meeting ID: 958 6908 4555 Ribbon categories can be used to produce invariants of framed links via the ReshetikhinTuraev 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. 
22 
23 
24 
25 
26 
27 
28 Geometry and Topology
Geometry and Topology Speaker: Martina Rovelli (University of Massachusetts Amherst) "$n$complicial sets as a model for $(\infty, n)$categories" Time: 15:30 Room: Zoom Meeting ID: 958 6908 4555 With the rising significance of $(\infty, n)$categories, it is important to have easytohandle 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. 
29 
30 
1 
