Department of

# Mathematics

Seminar Calendar
for events the week of Wednesday, April 25, 2018.

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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, April 23, 2018

3:00 pm in Altgeld Hall 345,Monday, April 23, 2018

#### Equivariant Homotopy Theory of Finite Spaces and Sylow Theorems

###### Joseph Rennie (UIUC Math)

Abstract: Finite topological spaces serve as great pedagogical tools, and not just because they are a source of counter examples. In this talk, I will go from zero to Sylow in about forty minutes, covering the essentials of the theory of finite spaces along the way. On our way to this unsurprising result are some rather surprising ones. In the remaining ten minutes, I will discuss a conjecture in Sylow Theory by Quillen.

4:00 pm in 245 Altgeld Hall,Monday, April 23, 2018

#### Risk Engineering: Mathematical Principles with Uncertainty in Insurance Business

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

Abstract: Natural disasters and human-made hazards are inevitable but their consequences need not be. Engineers respond by designing autonomous vehicles that prevent accidents, making earthquake-proof buildings, and developing life saving medical equipment. We actuaries and financial analysts answer by creating and managing innovative financial and insurance products to reduce and mitigate the financial impact of car accidents, earthquakes, and make healthcare available to those in dire need. The focus of this talk is to provide an overview of various research topics pertaining to quantitative risk management and engineering of equity-linked insurance products and personal retirement planning. It aims to demonstrate the mathematical fun with risk management problems as well as to offer a glimpse of technical development and challenges arising from these fields.

5:00 pm in 241 Altgeld Hall,Monday, April 23, 2018

#### Organizational meeting

Tuesday, April 24, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, April 24, 2018

#### Genus bounds in right-angled Artin groups

###### Jing Tao (University of Oklahoma)

Abstract: In this talk, I will describe an elementary and topological argument that gives bounds for the stable commutator lengths in right-angled Artin groups.

2:00 pm in 347 Altgeld Hall,Tuesday, April 24, 2018

#### Estimates of Dirichlet heat kernels for subordinate Brownian motions

###### Panki Kim (Seoul National University)

Abstract: In this talk, we discuss estimates of transition densities of subordinate Brownian motions in open subsets of Euclidean space. When open subsets are $C^{1,1}$ domain, we establish sharp two-sided estimates for the transition densities of a large class of killed subordinate Brownian motions whose scaling order is not necessarily strictly below 2. Our estimates are explicit and written in terms of the dimension, the Euclidean distance between two points, the distance to the boundary and the Laplace exponent of the corresponding subordinator only. We also establish boundary Harnack principle in $C^{1,1}$ open set with explicit decay rate. This is a joint with with Ante Mimica.

3:00 pm in 241 Altgeld Hall,Tuesday, April 24, 2018

#### Colorings of signed graphs - a short survey

###### Andre Raspaud (LaBRI, Bordeaux University)

Abstract: The signed graphs and the balanced signed graphs were introduced by Harary in 1953. But all the notions can be found in the book of König (Theorie der endlichen und unendlichen graphen 1935). An important, fundamental and prolific work on signed graphs was done by Zaslavsky in 1982. In this talk we are interested in coloring of signed graphs. We will give a short survey of the different existing definitions and the recent results on the corresponding chromatic numbers. We will also present new results obtained by using the DP-coloring.

4:00 pm in 1 Illini Hall,Tuesday, April 24, 2018

#### Moduli of Twisted Curve

###### Hao Sun (UIUC)

Abstract: I'll give an introductory talk about the twisted curves. Twisted curves are related to the study of r-spin Witten classes and r-spin geometry of the moduli space of curves.

Wednesday, April 25, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, April 25, 2018

#### Randomness in 3-Dimensional Geometry and Topology

###### Malik Obeidin (Illinois Math)

Abstract: The probabilistic method, pioneered by Paul Erdos, has proven to be one of the most powerful and versatile tools in the field of combinatorics. Mathematicians working in diverse fields, from graph theory to number theory to linear algebra, have found the probabilistic toolset valuable. However, 3-manifold topology has only been recently approached from this angle, though the field itself is full of intricate combinatorics. In this talk, I'll describe some of the ways one might define a "random 3-manifold" and the subtleties that arise in the definition. I'll also talk about how one can use these ideas to experiment computationally with 3-manifolds, to help us get a handle on what is "common", and what is "rare".

Thursday, April 26, 2018

11:00 am in 241 Altgeld Hall,Thursday, April 26, 2018

#### The Unreasonable Effectiveness of Benford's Law in Mathematics

###### A J Hildebrand and Junxian Li (Illinois Math)

Abstract: We describe work with Zhaodong Cai, Matthew Faust, and Yuan Zhang that originated with some unexpected experimental discoveries made in an Illinois Geometry Lab undergraduate research project back in Fall 2015. Data compiled for this project suggested that Benford's Law (an empirical "law" that predicts the frequencies of leading digits in a numerical data set) is uncannily accurate when applied to many familiar mathematical sequences. For example, among the first billion Fibonacci numbers exactly 301029995 begin with digit 1, while the Benford prediction for this count is 301029995.66. The same holds for the first billion powers of 2, the first billion powers of 3, and the first billion powers of 5. Are these observations mere coincidences or part of some deeper phenomenon? In this talk, which is aimed at a broad audience, we describe our attempts at unraveling this mystery, a multi-year research adventure that turned out to be full of surprises, unexpected twists, and 180 degree turns, and that required unearthing nearly forgotten classical results as well as drawing on some of the deepest recent work in the area.

1:00 pm in 347 Altgeld Hall,Thursday, April 26, 2018

#### Constraints on eco-evolutionary dynamics in bacterial communities

###### Seppe Kuehn (Physics, Illinois)

Abstract: Can we predict evolutionary and ecological dynamics in microbial communities? I argue that understanding constraints on biological systems provides a path forward to build predictive models. I present two vignettes which illustrate the power of elucidating constraints. First, we ask how constraints on phenotypic variation can be exploited to predict evolution. We select Escherichia coli simultaneously for motility and growth and find that a trade-off between these phenotypes constrains adaptation. Using genetic engineering, high-throughput phenotyping and modeling we show that the genetic capacity of an organism to vary traits can qualitatively depend on its environment, which in turn alters its evolutionary trajectory [eLife, 2017]. Our results suggest that knowledge of phenotypic constraints and genetic architecture can provide a route to predicting evolutionary dynamics. Second, in nature microbial populations are subjected to nutrient fluctuations but we know little about how communities respond to these fluctuations. Using automated long-term single cell imaging and custom continuous-culture devices we subject bacterial populations to nutrient fluctuations on multiple timescales. We find populations recover faster from large, frequent fluctuations. Our observation is explained by a model that captures constraints on the rate at which populations transition from planktonic and aggregated lifestyles.

2:00 pm in 241 Altgeld Hall,Thursday, April 26, 2018

#### The p-torsion of Ree curves

###### Dane Skabelund (UIUC)

Abstract: This talk will describe some recent computations involving the structure of 3-torsion of the Ree curves, which are a family of supersingular curves in characteristic 3.

4:00 pm in 245 Altgeld Hall,Thursday, April 26, 2018

#### Random Matrices, Heat Flow, and Lie Groups

###### Todd Kemp (UCSD)

Abstract: Random matrix theory studies the behavior of the eigenvalues and eigenvectors of random matrices as the dimension grows. In the age of data science, it has become one of the hottest fields in probability theory and many parts of applied science, from material deposition to wireless communication. Initiated by Wigner in the 1950s (with some key results going back further to Wishart and other statisticians in the 1920s), there is now a rich and well-developed theory of the universal behavior of random spectral statistics in models that are natural generalizations of the Gaussian case. In this talk, I will discuss a generalization of these kinds of results in a new direction. A Gaussian random matrix can be thought of as an instance of Brownian motion on a Lie algebra; this opens the door to studying the eigenvalues of Brownian motion on Lie groups. I will present recent progress understanding the asymptotic spectral distribution of Brownian motion on unitary groups and general linear groups. The tools needed include probability theory, functional analysis, combinatorics, and representation theory. No technical background is required; only an interest in trying to understand some cool and mysterious pictures.

Friday, April 27, 2018

4:00 pm in 241 Altgeld Hall,Friday, April 27, 2018

#### Cannon-Thurston Maps

###### Elizabeth Field (UIUC)

Abstract: When a map between two objects extends to a continuous map between their boundaries, this boundary map is called the Cannon-Thurston map. When the objects at play are a hyperbolic group and a hyperbolic subgroup, there are two main questions which arise from this definition. First, when does the Cannon-Thurston map exist? Further, if such a map exists, can the fibers of this map be described in some nice algebraic or geometric way? In this talk, we will introduce the background needed to understand these questions. We will then briefly explore some of what is known about their answers, as well as some open questions which remain.