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for events the day of Tuesday, April 2, 2019.

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Tuesday, April 2, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, April 2, 2019

Weil-Petersson analogs for graphs

Tarik Aougab (Brown University)

Abstract: The Weil-Petersson metric is a Riemannian metric on the Teichmuller space which is natural in the sense that it comes from and reflects the geometry of the hyperbolic metrics on the underlying surface. Motivated by foundational work of McMullen, Pollicott-Sharp (and later Kao) proposed an analogous metric for the moduli space of metrics on a fixed graph. We study this metric and completely characterize its completion in the case of a rose. In this talk, we’ll introduce the Weil-Petersson metric and do our best to motivate the definitions so that no advanced prior knowledge of the subject will be necessary. This represents joint work with Matt Clay and Yo’av Rieck.

1:00 pm in 345 Altgeld Hall,Tuesday, April 2, 2019

Max-Min theorems for weak containment, square summable homoclinic points, and completely positive entropy

Ben Hayes (University of Virginia)

Abstract: I will present a max-min theorem for weak containment in the context of algebraic actions (i.e. actions of a discrete group by automorphisms of a compact group). Namely, given an algebraic action of $G$ on $X$, there is a maximal, closed $G$-invariant subgroup $Y$ of $X$ so that the action of $G$ on $Y$ is weakly contained in a Bernoulli shift. This subgroup is also the minimal subgroup so that any action weakly contained in a Bernoulli shift is "$G$$X/Y$-ergodic in the presence of $G$$X$" (this will be defined in the talk). Time permitting, I will discussion applications. These include showing that many algebraic actions are weakly contained in a Bernoulli shift, as well as applications to complete positive entropy of algebraic actions.

1:00 pm in 347 Altgeld Hall,Tuesday, April 2, 2019

Direct Scattering and Small Dispersion for the Benjamin-Ono Equation with Rational Initial Data

Alfredo Wetzel (Wisconsin-Madison)

Abstract: The Benjamin-Ono (BO) equation describes the weakly nonlinear evolution of one-dimensional interface waves in a dispersive medium. It is an integrable equation, with a known Lax pair and inverse scattering transform, that may be viewed as a prototypical problem for the study of multi-dimensional integrable equations and Riemann-Hilbert problems with a non-local jump condition. In this talk, we propose explicit formulas for the scattering data of the BO equation with a rational initial condition. For this class of initial conditions, the recovery of the scattering data can be done directly by exploiting the analyticity properties of the Lax pair solutions. Our procedure validates previous well-known formal results and provides new details concerning the leading order behavior of the scattering data in the small dispersion limit. In the small dispersion limit, we are able to derive formulas for the location and density of the eigenvalues, magnitude and phase of the reflection coefficient, and density of the phase constants.

2:00 pm in 243 Altgeld Hall,Tuesday, April 2, 2019

On two problems, related to additive combinatorics

Jozsef Balogh (Illinois Math)

Abstract: In the talk I will present two short results:

(a) Define $T=T(k)$ the minimal $t$ for which there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of the numbers $1,\dots, tn$ for all $n$, where equinumerous means that each color used the same number of times. Almost answering a question of Jungic, Licht (Fox), Mahdian, Nesetril and Radoicic, we almost determine the function $T$. It is a joint work with Linz.

(b) Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollobas in 1968. In this process, we start with initial "infected" set of edges $E(0)$, and we infect new edges according to a predetermined rule. Given a graph $H$ and a set of previously infected edges $E(t)$ subset of $E(K_n)$, we infected a non-infected edge $e$ if it completes a new copy of $H$ in $G=([n],E(t) + e)$. A question raised by Bollobas asks for the maximum time the process can run before it stabilizes. In 2015, Bollobas, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where $H$ is the $r$-vertex complete graph. They answered the question for $r > 3$ and gave a lower bound for every $r \ge 5$. In their paper, they also conjectured that the maximal running time is subquadratic for every integer $r$. In this paper we disprove their conjecture for every $r$ at least 6 and we give a better lower bound for the case that $r=5$. In the proof of the case $r=5$ we use the Behrend construction. Joint result with Kronenberg, Pokrovskiy and Szabo.

4:00 pm in 243 Altgeld Hall,Tuesday, April 2, 2019

Generalizing Koopman Theory to Allow for Inputs and Control

Kim, Hee Yeon (University of Illinois, Urbana-Champaign)

Abstract: The Koopman Operator (Bernard Osgood Koopman "Hamiltonian systems and transformation in Hilbert space", PNAS 17 (1931) 315-318) has emerged in Machine Learning as a tool to reformulate nonlinear dynamics in a linear framework. I will present the paper by Proctor, Brunton, and Kutz in SIAM J.App.Dyn.Sys. (with this title) vol. 17, No. 1, 909-930.

The authors introduce the Koopman Operator with inputs and control (KIC) which generalizes Koopman's spectral theory to allow for systems with nonlinear input-output characteristics. They show how this generalization is connected to dynamic mode decompositions with control (DMDc). They demonstrate KIC on several nonlinear dynamical systems, such as the standard epidemiological SIR-model for susceptible-infectious-recovered, hence resistant subjects (e.g. measles).

5:00 pm in Ballroom, Alice Campbell Alumni Center,Tuesday, April 2, 2019

Department Awards Ceremony