Department of

# Mathematics

Seminar Calendar
for events the week of Sunday, September 24, 2017.

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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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Monday, September 25, 2017

3:00 pm in 141 Altgeld Hall,Monday, September 25, 2017

#### An Invitation to Motivic Homotopy Theory

###### Daniel Carmody   [email] (UIUC)

Abstract: In this talk I’ll introduce some of the basic constructions in motivic homotopy theory while trying to give motivations for some of the more complex definitions. This will be largely based on Dan Dugger’s $Universal$ $Homotopy$ $Theories$ paper.

4:00 pm in 245 Altgeld Hall,Monday, September 25, 2017

#### Rigidity in orbit equivalence via cost

###### Anush Tserunyan   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: The talk is tailored around the following basic question: Let $\mathbb{F}_n$↷$[0,1]$ and $\mathbb{F}_m$↷$[0,1]$ be free actions of the free groups on $n$ and $m$ generators and assume that these actions preserve the Lebesgue measure and are ergodic (a bunch of words you're not supposed to know—ignore). If these actions produce the same orbits (i.e. their orbit equivalence relations are equal), must it be that $n = m$? This is an instance of the more general question: how much of the group is "remembered" by the orbit equivalence relations of its free actions? The phenomenon of a weaker notion "remembering" a stronger one is referred to as rigidity. We will describe the answer to the initial question (due to D. Gaboriau) by introducing an invariant called cost, which is tied to measurable graphs and combinatorics, and even ideas from homology theory. I will only assume familiarity with the words graph and measure.

Tuesday, September 26, 2017

11:00 am in 241 Altgeld Hall,Tuesday, September 26, 2017

#### Introduction to Shimura curves

###### Yifan Yang (National Chiao Tung University)

Abstract: Shimura curves are generalizations of modular curves. The arithmetic aspect of Shimura curves bears a great similarity to that of modular curves. However, because of the lack of cusps on Shimura curves, it is difficult to do explicit computation about them. This makes Shimura curves both interesting and challenging to study. In this talk, we will give a quick introduction to Shimura curves.

12:00 pm in 243 Altgeld Hall,Tuesday, September 26, 2017

#### Limits of cubic differentials and projective structures

###### David Dumas (University of Illinois at Chicago)

Abstract: A construction due independently to Labourie and Loftin identifies the moduli space of convex RP^2 structures on a compact surface S with the bundle of holomorphic cubic differentials over the Teichmueller space of S. We study pointed geometric limits of sequences that go to infinity in this moduli space while remaining over a compact set in Teichmueller space. For such a sequence, we construct a local limit polynomial (in one complex variable) which describes the rate and direction of accumulation of zeros of the cubic differentials about the sequence of base points. We then show that this polynomial determines the convex polygon in RP^2 that is the geometric limit of the images of the developing maps of the projective structures. This is joint work with Michael Wolf.

1:00 pm in Altgeld Hall,Tuesday, September 26, 2017

#### New developments on subelliptic estimates

###### Martino Fassina (Illinois Math)

Abstract: We first recall some history of subelliptic estimates for d-bar. We then discuss a recent paper of Zaitsev concerning effectiveness of the Kohn algorithm and generalizations by the speaker. The talk should be accessible to graduate students.

1:00 pm in 345 Altgeld Hall,Tuesday, September 26, 2017

#### Asymptotics of parameterized exponential integrals given by Brownian motion on globally subanalytic sets

###### Tobias Kaiser (University of Passau)

Abstract: (Joint work with Julia Ruppert.) Understanding integration in the o-minimal setting is an important and difficult task. By the work of Comte, Lion and Rolin, succeeded by the work of Cluckers and Miller, parameterized integrals of globally subanalytic functions are very well analyzed. But very little is known when the exponential function comes into the game. We consider certain parameterized exponential integrals which come from considering the Brownian motion on globally subanalytic sets. We are able to show nice asymptotic expansions of these integrals.

3:00 pm in 241 Altgeld Hall,Tuesday, September 26, 2017

#### An improved lower bound for Folkman's theorem

###### József Balogh (Illinois Math)

Abstract: Folkman's Theorem asserts that for each $k \in \mathbb N$, there exists a natural number $n=F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a subset $A$ of $[n]$ of size $k$ with the property that all the sums of the form $\sum_{x\in B} x$, where $B$ is a nonempty subset of $A$, are contained in $[n]$ and have the same colour. In 1989, Erdős and Spencer showed that $F(k) \ge 2^{ck^2/\log k}$, where $c>0$ is an absolute constant; here, we improve this bound significantly by showing that $F(k) \ge 2^{2^{k-1}/k}$ for all $k \in \mathbb N$. Joint with Sean Eberhard, Bhargav Narayanan, Andrew Treglown, Adam Zsolt Wagner.

3:00 pm in 243 Altgeld Hall,Tuesday, September 26, 2017

#### To Be Announced

###### Mark Penney (University of Oxford / Max Planck Institute (Bonn))

Wednesday, September 27, 2017

3:00 pm in 243 Altgeld Hall,Wednesday, September 27, 2017

#### Applications of Mining Public Genome Data to Recover Statistical Trends using Geometric Combinatorics

###### Ruth Davidson (UIUC)

Abstract: Websites such as TreeBASE.org and datadryad.org provide public access to a wealth of genomic data released with peer-reviewed biological publications. Phylogenomics-the recovery of the common evolutionary history of a group of taxa from short gene samples recovered from long genomes-is a basic area of research that gives rise to many quantitative methods for mining data for evolutionary signals. In turn, myriad fields such as ecology, medicine, and linguistics consume these methods; thus improved methods have very broad scientific impact. We present a publication (joint work with MaLyn Lawhorn, Joseph Rusinko, and Noah Weber) that provides a baseline framework, built on geometric combinatorics, for studying statistical trends in genomic data. Further, we will outline future research directions that will (1) build on this framework to inform the development of new theory and methods for model-testing, and (2) improve the understanding of trends in phylogenomic data in the systematic biology, computer science, statistics, and mathematics communities.

4:00 pm in 245 Altgeld Hall,Wednesday, September 27, 2017

#### Sandia Labs Math/Stats Overview

###### TBA (TBA)

Thursday, September 28, 2017

11:00 am in 241 Altgeld Hall,Thursday, September 28, 2017

#### Equations of hyperelliptic Shimura curves

###### Yifan Yang (National Chiao Tung University)

Abstract: Because of the lack of cusps on Shimura curves, there are few methods to construct modular forms on them. As a result, it is very difficult to determine equations of Shimura curves. In a recent work, we devised a systematic method to construct Borcherds form. Together with Schofer's formula for values of Borcherds forms at CM-points, this enabled us to determine equations of all hyperelliptic Shimura curves. This is a joint work with Jia-Wei Guo.

12:00 pm in 243 Altgeld Hall,Thursday, September 28, 2017

#### Effective Twisted Conjugacy Separability of Nilpotent Groups

###### Mark Pengitore (Purdue Math)

Abstract: There has been a recent interest in providing effective proofs of separability properties such as residual finiteness and conjugacy separability. Unlike residual finiteness, conjugacy separability does not respect finite extensions. Thus, we introduce twisted conjugacy separability, originally defined by Fel'shtyn, in order to study effective conjugacy separability of finite extensions of conjugacy separable groups. In joint work with Jonas Dere, we provide an effective proof of twisted conjugacy separability of finitely generated nilpotent groups. That, in turn, provides an effective bound for conjugacy separability of all finite extensions of a fixed nilpotent group in terms of the asymptotic behavior of conjugacy separability of the base nilpotent group.

3:00 pm in 345 Altgeld Hall,Thursday, September 28, 2017

#### W-Operators and Combinatorial Hurwitz Numbers

###### Hao Sun (University of Illinois)

Abstract: We will introduce the W-operator and show some interesting combinatorial properties about the W-operator. Then, we will show how the W-operator relates to the study of combinatorial Hurwitz numbers.

4:00 pm in 245 Altgeld Hall,Thursday, September 28, 2017

#### Fall Department Faculty Meeting

Abstract: The Fall Department Faculty Meeting will be held at 4 p.m. in 245 Altgeld Hall, followed by a reception in 239 Altgeld Hall.

Friday, September 29, 2017

12:00 pm in 243 Altgeld Hall,Friday, September 29, 2017

#### To Be Announced

###### Fernando Roman Garcia (UIUC)

3:00 pm in 341 Altgeld Hall,Friday, September 29, 2017

#### Bott-Samelson varieties and combinatorics

###### Laura Escobar (UIUC)

Abstract: Schubert varieties parametrize families of linear spaces intersecting certain hyperplanes in C^n in a predetermined way. In the 1970’s Hansen and Demazure independently constructed resolutions of singularities for Schubert varieties: the Bott-Samelson varieties. In this talk I will describe their relation with associahedra. I will also discuss joint work with Pechenick-Tenner-Yong linking Magyar’s construction of these varieties as configuration spaces with Elnitsky’s rhombic tilings. Finally, based on joint work with Wyser-Yong, I will give a parallel for the Barbasch-Evens desingularizations of certain families of linear spaces which are constructed using symmetric subgroups of the general linear group.

4:00 pm in 345 Altgeld Hall,Friday, September 29, 2017

#### "Borel circle squaring" by Marks and Unger: Part 3

###### Anush Tserunyan (Illinois Math)

Abstract: In the previous talk of this series, we built a real valued Borel flow by taking the average of the "relative" matchings over each connected component. In the current talk, I'll describe how to obtain an integer valued Borel flow out of a real valued one, which is the main difficulty of the whole proof. Getting a Borel matching from an integer valued Borel flow is much easier and we will do so if time permits.

4:00 pm in 241 Altgeld Hall,Friday, September 29, 2017

#### Framed cobordisms in algebraic topology

###### Pedro Mendes De Araujo (UIUC)

Abstract: The Thom-Pontryagin construction was the first machinery developed to compute homotopy groups of spheres, in terms of framed cobordism classes of manifolds embedded in Euclidean space. Although much less successful at that than the algebraic machinery developed later, it has the advantage of being highly geometric and intuitive. In this talk, which will hopefully be fun, full of pictures, and requiring little more than an acquaintance with manifolds (and a high tolerance to geometric hand waving), we'll look at how it can be used to compute pi_{n+1} S^n. On the way, we'll give a very explicit proof of the Freudenthal suspension theorem.