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for events the day of Friday, February 5, 2016.

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Friday, February 5, 2016

2:00 pm in 445 Altgeld Hall,Friday, February 5, 2016

An introduction to metric-measure spaces supporting a Poincare inequality

Derek Jung (UIUC Math)

Abstract: A special case of the Sobolev Embedding Theorem states that if $u\in W^{1,p}(\mathbb{R}^n)$ and, $p\in[1,n)$ the $\frac{np}{n-p}$-integral of $u$ is dominated by the $p$-integral of its gradient. In Euclidean spaces, one has control over the size of a Sobolev function by the size of its gradient. Now suppose we consider metric-measure spaces on which we have similar control locally. We say that a metric-measure space supports a Poincare inequality if the size of every measurable function is controlled by the size of its upper gradients, in the sense of averaged integrals over balls. Examples of such spaces include Euclidean spaces and compact Riemannian manifolds. In this talk, I will state properties that these metric-measure spaces share and show the typical techniques one employs when working in these spaces. Knowledge of Math 540 is assumed.

2:00 pm in 447 Altgeld Hall,Friday, February 5, 2016

The Batalin-Vilkovisky formalism of the spinning particle

Ezra Getzler   [email] (Northwestern University)

Abstract: We show that the axiom of Felder and Kazhdan on the vanishing of the cohomology groups in negative degree associated to solutions of the classical master equation in the Batalin-Vilkovisky formalism is violated by the spinning particle in a flat background coupled to D=1 supergravity. In this model, there are nontrivial cohomology groups in all negative degrees, regardless of the dimension of the spacetime in which the spinning particle is propagating.

4:00 pm in 241 Altgeld Hall,Friday, February 5, 2016

Caucusing for Curvature Bounds

Bill Karr (UIUC Math)

Abstract: Alexandrov geometry is the study of non-smooth analogs of Riemannian manifolds with curvature bounded from below or above. Alexandrov spaces often arise as "limits" of Riemannian manifolds with sectional curvature bounds or as orbifolds where the underlying space is a Riemannian manifold with a sectional curvature bound. We will define Alexandrov spaces, Gromov-Hausdorff limits, and then look at some results as well as some open questions about spaces with curvature bounds. The talk is meant to be a showcase of some interesting topics that might be studied in an introductory reading group on Alexandrov geometry.

4:00 pm in 345 Altgeld Hall,Friday, February 5, 2016

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

Anton Bernshteyn (UIUC)

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.