Department of

Mathematics


Seminar Calendar
for events the day of Friday, February 12, 2016.

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Friday, February 12, 2016

2:00 pm in 445 Altgeld Hall,Friday, February 12, 2016

Dynamics of KdV-type Equations

Erin Compaan (UIUC Math)

Abstract: The talk will begin by discussing the wellposedness theory of the KdV equation. This theory took a significant step forward in the 90s when Bourgain proved wellposedness on low-regularity Sobolev spaces. These results relied on Fourier restriction norm spaces, which continue to be very important in the study of dispersive PDEs. After this history, I'll discuss some recent work on smoothing and dynamical properties of a coupled KdV-type system.

4:00 pm in 345 Altgeld Hall,Friday, February 12, 2016

Borel equivalence relations and cardinal algebras (Part III) by Alexander S. Kechris and Henry L. Macdonald

Anton Bernshteyn (UUC)

Abstract: For a Borel equivalence relation $E$ on a Borel space $X$, let $nE$ denote the direct sum of $n$ copies of $E$, i.e., the equivalence relation $F$ on $X \times \{1, \ldots, n\}$ such that $(x, i)F(y,j)$ if and only if $i = j$ and $xEy$. Suppose that $E$ and $F$ are countable Borel equivalence relations such that $nE$ is Borel reducible to $nF$. Does this imply that $E$ is Borel reducible to $F$? It turns out that the answers to this and several other questions about the Borel reducibility preorder can be obtained purely algebraically using the theory of cardinal algebras. This theory was developed by Tarski in the late 1940s in order to study the properties of cardinal addition in the absence of the full Axiom of Choice and has been largely forgotten since then. In this series of talks we will survey Tarski's results on cardinal algebras and see how they transfer to the Borel reducibility setting.

4:00 pm in 241 Altgeld Hall,Friday, February 12, 2016

De Rham homology, and some foliations

Daan Michiels (UIUC Math)

Abstract: While de Rham cohomology is standard material in differential geometry, de Rham homology isn't. We will define de Rham homology using the theory of currents, and give some elementary properties. To illustrate how de Rham homology can be useful, we discuss foliation currents and the way they relate to tautness of a codimension-1 foliation. The only prerequisite is some familiarity with differential forms and de Rham cohomology.