Department of

# Mathematics

Seminar Calendar
for events the day of Friday, September 4, 2015.

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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 4, 2015

2:00 pm in 143 Altgeld Hall,Friday, September 4, 2015

#### A category of open systems and networks without graphs

###### Eugene Lerman (UIUC Math)

Abstract: Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalize these kinds of structures as algebras over presentable colored operads. DeVille and Lerman, building on the groupoid formalism for coupled cell networks of Golubitsky, Stewart and collaborators, proposed an approach to the study of dynamics of complex systems which is based on graph fibrations. My goal is to describe an algebraic framework that encompasses both the operadic approache of Spivak et al. and fibrations of networks of manifolds of DeVille and Lerman.

4:00 pm in 345 Altgeld Hall,Friday, September 4, 2015

#### On "Baire measurable paradoxical decompositions via matchings" by A. Marks and S. Unger

###### Anton Bernshteyn (UIUC Math)

Abstract: The Banach--Tarski paradox states that the unit ball in $\mathbb{R}^3$ is equidecomposable with two copies of itself. Of course, there can be no such equidecomposition where each piece is measurable. Thus a natural question (first asked by Marczewski) is whether there exists such an equidecomposition where each piece has the Baire property. The answer is positive, as demonstrated by an intricate construction due to Dougherty and Foreman. This paper provides an alternative (short) proof of this result. In fact, a more general result is established, namely if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. The key ingredient of the proof is a Borel analogue of Hall's celebrated marriage theorem from graph theory. In this series of talks we will go over the proofs and enjoy the elegant transition back and forth between the original problem and its combinatorial counterpart.

4:00 pm in 243 Altgeld Hall,Friday, September 4, 2015

#### Classical Mechanics and Symplectic Geometry

###### Matej Penciak (UIUC Math)

Abstract: In this talk I will introduce the basics of the modern Lagrangian and Hamiltonian formulations of classical mechanics, and show the connection to symplectic and Poisson geometry. I will end with a discussion of how symmetries can be used to solve for the dynamics of classical systems and some formulations of Noether’s theorem on conserved quantities.