Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, April 26, 2018.

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events for the
events containing

More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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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.