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for events the day of Friday, September 11, 2015.

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Friday, September 11, 2015

2:00 pm in 447 Altgeld Hall,Friday, September 11, 2015

Grushin-type Surfaces

Matthew Romney (UIUC Math)

Abstract: The classical Grushin plane is a standard example of a sub-Riemannian manifold. It is known to be quasisymmetrically homeomorphic to the standard Euclidean plane, and to admit a bi-Lipschitz embedding in Euclidean 3-space. In this talk we will introduce a more general class of surfaces modeled on the Grushin plane. We prove that, under suitable assumptions, they can be embedded in some Euclidean space under a bi-Lipschitz map. Finally, we will discuss possibilities for further research.

2:00 pm in 143 Altgeld Hall,Friday, September 11, 2015

Groupoids, Fibrations and Rigidity

Matias del Hoyo   [email] (IMPA-Rio de Janeiro)

Abstract: Fibred categories were introduced in descent theory by A. Grothendieck. In this talk I will discuss fibred Lie groupoids, an incarnation of that formalism in differential geometry, studied by K. Mackenzie, among others. In a joint work with R. Fernandes we construct suitable metrics on fibred Lie groupoids, and present as an application a rigidity result, obtained independently by M. Crainic et al. I will discuss the rudiments of our theory, present the main theorems, and derive as corollaries several classic results in the geometry of actions, fibrations and foliations, by R. Palais, H. Rosenberg, and others.

4:00 pm in 241 Altgeld Hall,Friday, September 11, 2015

(Real) Schubert Calculus from Marked Points on $\mathbb{P}^{1}$

Jake Levinson   [email] (University of Michigan)

Abstract: I will describe a family S of Schubert problems on the Grassmannian, defined using flags tangent to $\mathbb{P}^{1}$ (or more generally, a stable curve) at a chosen set of points in the Veronese embedding. This family is nicely-behaved in general (for example, it is flat over the moduli space of stable curves), and has been of particular interest when the chosen points are all real. For zero-dimensional Schubert problems, work of Mukhin-Tarasov-Varchenko (2007) and Speyer (2014) showed that the real locus of S is a smooth cover of the moduli space of real stable curves; moreover, the monodromy of the cover has an elegant description in terms of Young tableaux and Schützenberger's jeu de taquin. I will give analogous results on real one-dimensional Schubert problems. In this case, S is a family of curves, whose real points turn out to be smooth, and whose real geometry is described by orbits of tableau promotion and a related operation involving evacuation.

4:00 pm in 345 Altgeld Hall,Friday, September 11, 2015

On "Baire measurable paradoxical decompositions via matchings" by A. Marks and S. Unger (continued)

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 11, 2015

Curvature bounds and bi-Lipschitz embeddings

Matthew Romney (UIUC Math)

Abstract: The first part of this talk will give an overview of uniformization and embedding problems in metric space geometry. In the second part, we consider these problems in the context of spaces of bounded curvature in the sense of Alexandrov. As the culmination, we present a new result giving sufficient conditions for a metric space to admit a bi-Lipschitz embedding in a finite-dimensional Euclidean space. The main requirement is that the space satisfy certain curvature bounds on the balls of a Whitney decomposition relative to a prescribed singular subset.