Department of

# Mathematics

Seminar Calendar
for events the day of Friday, September 28, 2018.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, September 28, 2018

1:00 pm in 2 Illini Hall,Friday, September 28, 2018

#### Dynamics and Spectral Theory 1: Weyl’s Law and Geodesic Flow

Abstract: In this talk I’ll introduce some results relating the dynamics of geodesic flow and the eigenvalues of the Laplace operator. We’ll begin by investigating how the asymptotics of eigenvalue growth can change in the presence of dynamical hypotheses, and use these results to motivate this interplay of analysis and geometry. This talk will be focused on examples and should have minimal analytic prerequisites.

3:00 pm in 145 Altgeld Hall,Friday, September 28, 2018

#### Conservative Methods for Liberal ODE's

###### Nikolas Wojtalewicz (Illinois Math)

Abstract: A conservative method for a dynamical system is a numerical method of solving a dynamical system which preserves conserved quantities associated to that dynamical system. While many methods, such as symplectic or Runge-Kutta methods, have properties that allow them to preserve some types of conserved quantities for specific dynamical systems, few methods can preserve any type of conserved quantity for any given system. In this talk, we introduce the Multiplier method, a conservative method for solving a dynamical system which preserves any type of conserved quantity. The talk will be divided into three parts: first, discussing the basic theory and terms behind the Multiplier method; second, going over the proof on how to apply the Multiplier method; finally, if time permits, we will show some example applications of the Multiplier method, as well as compare the Multiplier method with another numerical method.

4:00 pm in 345 Altgeld Hall,Friday, September 28, 2018

#### On the model theory of group actions on probability measure algebras

###### Ward Henson (UIUC (Emeritus))

Abstract: We treat such group actions using continuous model theory. For a finite or countable set S, let L_S be the continuous signature for probability measure algebras expanded by unary function symbols, one for each element of S. In this language, let T_S be the set of axioms for probability algebras (which we denote Pr) together with conditions expressing that each of the unary function symbols is interpreted by an automorphism of the algebra. If G is a group generated by S, we consider the extension of T_S obtained by adding a condition for each word w on S that represents the identity in G, asserting that the composition of unary functions corresponding to w is the identity; denote this theory by T_S(G). The main result to be discussed in this talk is that each T_S has a model companion T*_S, for which we give explicit axioms; this model companion is complete and has quantifier elimination. Its models consist of very particular actions on atomless probability algebras by the free group generated by S. Expressing and justifying our axioms for T*_S requires some information about the model theory of atomless probability algebras, which will be discussed in the first part of the talk. It is also true that when G is an amenable group, then T_S(G) has a model companion, which is very well behaved, but it will not be discussed much in this talk. (This is work in progress with Alex Berenstein; especially, we are aiming to understand the models of T*_S better, when |S|>1.)

4:00 pm in 241 Altgeld Hall,Friday, September 28, 2018

#### Integrability of the Toda lattice

###### Matej Penciak (UIUC)

Abstract: I will introduce topics such as the Toda lattice, the Lax matrices, and integrability.