Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, September 24, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, September 24, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, September 24, 2019

#### Speeds of hereditary properties and mutual algebricity

###### Caroline Terry (U Chicago Math)

Abstract: A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property $H$, the speed of $H$ is the function which sends an integer $n$ to the number of distinct elements in $H$ with underlying set $\{1,...,n\}$. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete "jumps" in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollobás, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss. This is joint work with Chris Laskowski.

2:00 pm in 347 Altgeld Hall,Tuesday, September 24, 2019

#### On the potential theory of Markov processes with jump kernels decaying at the boundary

###### Zoran Vondracek (University of Zagreb)

Abstract: Consider a $\beta$-stable process in the Euclidean space $\mathbb{R}^d$, $0<\beta\le 2$, which is killed upon exiting an open subset $D$. The killed process is then subordinated via an independent $\gamma$-stable subordinator. The resulting process $Y^D$ is called a subordinate killed stable process. In two recent papers, it has been shown that the potential theory of this process exhibits some interesting features. The first one is the form of the jumping kernel which depends on the distance of points to the boundary in a novel way. The second and unexpected feature is the fact that for some values of the stability index $\gamma$, the boundary Harnack principle fails. In the first part of the talk, I will review these results. The second part of the talk will be devoted to ongoing work on potential theory of jump processes in open subset $D$ of $\mathbb{R}^d$ defined through their jumping kernels that depend not only on the distance between two points, but also on the distance of each point to the boundary $\partial D$ of the state space $D$. Joint work with Panki Kim and Renming Song

2:00 pm in 243 Altgeld Hall,Tuesday, September 24, 2019

#### Defective DP-colorings of sparse multigraphs

###### Fuhong Ma (Shangdong University)

Abstract: DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. We introduce and study defective DP-colorings of graphs and multigraphs with 2 colors. Each vertex $v$ of a multigraph $G$ has colors $\alpha(v)$ and $\beta(v)$ in its list. In an $(i,j)$-coloring of $G$, if $v$ is colored with $\alpha(v)$, then it can be incident to at most $i$ 'conflicting' edges, otherwise it can be incident to at most $j$ such edges. We concentrate on $(i,j)$-colorings of sparse multigraphs.

Let $f_{DP}(i,j,n)$ be the minimum number of edges that may have an $n$-vertex $(i,j)$-critical multigraph, that is, a multigraph $G$ that has no $(i,j)$-defective DP-coloring but whose every proper subgraph has such a coloring. For all $i$ and $j$, we find linear lower bounds on $f_{DP}(i,j,n)$ that are exact for infinitely many $n$.

3:00 pm in 243 Altgeld Hall,Tuesday, September 24, 2019

#### Motivic Chern classes and Iwahori invariants of principal series

###### Changjian Su (University of Toronto)

Abstract: Let G be a split reductive p-adic group. In the Iwahori invariants of an unramified principal series representation of G, there are two bases, one of which is the so-called Casselman basis. In this talk, we will prove conjectures of Bump, Nakasuji and Naruse about certain transition matrix coefficients between these two bases. The ingredients of the proof involve Maulik and Okounkov's stable envelopes and Brasselet--Schurmann--Yokura's motivic Chern classes for the complex Langlands dual groups. This is based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.

4:00 pm in 245 Altgeld Hall,Tuesday, September 24, 2019

#### The University of Illinois Math Community in Prison: Calculus, Coding, and Beyond

###### Simone Sisneros-Thiry   [email] (University of Illinois at Urbana-Champaign)

Abstract: From spring 2018 through spring 2019, a cohort of Education Justice Project (EJP) students enrolled in the University of Illinois calculus series (Math 115-231) at the Danville Correctional Center. The value of and need for math education in prisons has been keenly felt by EJP students. Our interest in mathematics and related fields has prompted these courses as a part of an increasingly robust math and engineering curriculum. Our presentation will introduce the history and current status of EJP math programming. It will focus on student motivation, interest, and background in mathematics, and will include reflections by students and instructors on approaches and outcomes.