UIUC Dept. of Mathematics Seminar Calendar

    December 2006           January 2007          February 2007    
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Thursday, March 1, 2007

Differential Geometry Seminar
3:00 pm in 241 Altgeld Hall, Thursday, March 1, 2007

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Submitted by nevins.

Sean Keel (University of Texas at Austin)
Oort's conjecture for A_g
Abstract: I'll explain my proof, joint with Sadun, that there is no compact codimension g subvariety of the moduli space of Abelian varieties.

Tuesday, March 6, 2007

Friday, April 6, 2007

Thursday, April 12, 2007

Differential Geometry Seminar
3:00 pm in Altgeld Hall 241, Thursday, April 12, 2007

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Submitted by clein.

Christina Sormani (CUNY)
Almost Isotropy and Exponential Length Spaces
Abstract: In the Friedmann Model of the universe, cosmologists assume that spacelike slices of the universe are space forms (Riemannian manifolds of constant sectional curvature) because the universe is locally isotropic. Here we define a Riemannian manifold as almost locally isotropic by controlling the behavior of the exponential map in a sup norm sense. We then prove that such a manifold is Gromov Hausdorff close to a length space Y which is a collection of space forms joined at discrete points and that the exponential map is close to the exponential map on such a space. The proof involves an application of Grove-Petersen's Arzela Ascoli Theorem to control the limits of exponential maps on the limits of Riemannian manifolds. These limit spaces are called ``exponential length space" and we prove locally isotropic exponential length spaces are space forms.

Thursday, April 19, 2007

Differential Geometry Seminar
3:00 pm in 241 Altgeld Hall, Thursday, April 19, 2007

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Submitted by clein.

Roman Sauer (U Chicago Math)
Simplicial volume for locally symmetric spaces of finite volume and the proportionality principle for non-compact manifolds.
Abstract: We introduce the simplicial volume of manifolds and explain its applications and relations to other manifold invariants like minimal volume and L^2-Betti numbers. Then we discuss to what extent Gromov's proportionality principle, which compares the simplicial volume of closed Riemannian manifolds with isometric universal cover, holds for open Riemannian manifolds of finite volume. Finally, applications to locally symmetric spaces of finite volume (like e.g. positivity of the minimal volume) are presented.

Wednesday, May 2, 2007

Graduate Student Seminar in Analysis
3:00 pm in Altgeld Hall 143, Wednesday, May 2, 2007

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Submitted by jsnyder2.

Christopher Lee (UIUC Math)
Fredholm Operators and Differential Geometry
Abstract: I will introduce (or recall) the definition and basic properties of Fredholm operators on Banach spaces. Featured most particularly will be the stability properties such operators enjoy with respect to perturbations by compact operators. We will then go on to explore the usefulness of Fredholm operators in manifold theory, stating the infinite dimensional versions of the Implicit Function Theorem and Sard's Theorem. Time permitting, an application to Morse homology will be presented. I have every intention to make this talk accessible, but audience appreciation will be greatly enhanced by a run in (or two, or more!) with smooth manifold theory and algebraic topology.

Thursday, August 23, 2007

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, August 23, 2007

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Submitted by seminar.

John Sullivan (Department of Mathematics, University of Illinois and Technische Universit�t Berlin)
Two connections between combinatorial and differential geometry
Abstract: There is a rich interplay between combinatorial and differential geometry. We will give first a geometric proof of a combinatorial result, and then a combinatorial analysis of a geometric moduli space. The first is joint work with Ivan Izmestiev, Rob Kusner, Guenter Rote, and Boris Springborn; the second with Karsten Grosse-Brauckmann, Nick Korevaar and Rob Kusner. In any triangulation of the torus, the average vertex valence is 6. Can there be a triangulation where all vertices are regular (of valence 6) except for one of valence 5 and one of valence 7? The answer is no. We prove this geometrically, giving the torus the metric where each triangle is equilateral and then explicitly analyzing the holonomy. Indeed, techniques from Riemann surfaces can characterize exactly which euclidean cone metrics have full holonomy group no bigger than their restricted holonomy group (at least when the latter is finite). Next we consider the moduli space M_k of Alexandrov-embedded surfaces of constant mean curvature which have k ends and genus 0 and are contained in a slab. We showed earlier that M_k is homeomorphic to an open manifold D_k of dimension 2k-3, defined as the moduli space of spherical metrics on an open disk with exactly k completion points. In fact, D_k is the ball B^{2k-3}; to show this we use the Voronoi diagram or Delaunay triangulation of the k completion points to get a tree, labeled by logarithms of cross-ratios. The combinatorics of the tree are tracked by the associahedron, and the labels give us a complexification of the cone over its dual. We note similarities to the spaces of labeled trees used in phylogenetic analysis.

Thursday, September 13, 2007

CR Geometry Seminar
12:00 pm in 345 Altgeld Hall, Thursday, September 13, 2007

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Submitted by jlebl.

Jeremy Tyson (UIUC Math)
Sub-Riemannian differential geometry of hypersurfaces
Abstract: We survey the rapidly expanding literature on the intrinsic sub-Riemannian geometry of hypersurfaces in the Heisenberg group Hⁿ and more general Carnot-Caratheodory spaces. Analogs of classical differential geometric notions may be defined directly by projecting to the horizontal tangent bundle, or recovered in the limit from degenerating sequences of adapted Riemannian metrics. The development of such a theory has been motivated by questions in the sub-Riemannian calculus of variations, especially the Bernstein problem and Pansu's celebrated isoperimetric conjecture.

Tuesday, September 25, 2007

Women in Math
5:00 pm in 243 Altgeld Hall, Tuesday, September 25, 2007

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Submitted by chasman.

Andrea Barreiro (UIUC)
Mathematical analysis of a model of the oculomotor integrator
Abstract: We analyze a mathematical model for the oculomotor neural integrator, a brain-stem structure that ensures proper control of the eyes in all vertebrates. In order to ensure that the integrator produces effective eye position commands under changing circumstances, it is regulated by the cerebellum. We use tools from classical differential geometry to characterize the response of the network to cerebellar control. We find that our model is capable not only of normal function, but can also simulate a common eye movement disorder known as congenital nystagmus.

Tuesday, October 2, 2007

Differential Geometry Seminar
1:00 pm in 243 Altgeld Hall, Tuesday, October 2, 2007

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Submitted by ekerman.

Silvia Sabatini (MIT)
On the T−equivariant cohomology of a compact symplectic manifold
Abstract: Let a torus T act on a compact symplectic manifold M in a Hamiltonian fashion with isolated fixed points, and let g be a T−invariant Palais-Smale metric. In a recent work by R. Goldin and S. Tolman they introduced some canonical T−equivariant classes which generate the T−equivariant cohomology of M as a module over the T−equivariant cohomology of a point, and they gave explicit formulas to compute the restriction of these classes to the fixed points of the T−action. In this talk I will discuss some recent joint work with S. Tolman about a generalization of these formulas, and some consequences of this generalization. In particular we conjecture that this will allow us to prove Sara Billey�s formulas for any semisimple Lie Group.

Tuesday, October 9, 2007

Tuesday, October 16, 2007

Differential Geometry Seminar
1:00 pm in 243 Altgeld Hall, Tuesday, October 16, 2007

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Submitted by ekerman.

Stefan Wenger (Courant)
Gromov hyperbolic spaces and the sharp isoperimetric constant
Abstract: The classical isoperimetric inequality asserts that the area A enclosed by a rectifiable loop of length L in the plane satisfies 4\pi A \leq L^2 with equality exactly for circles. In this talk we will prove the following theorem: Let X be a complete geodesic metric space. If there exists an epsilon > 0 such that every sufficiently long closed curve in X (of length L, say) bounds a 2-chain whose area A satisfies 4\pi A \leq (1-epsilon)L^2 then X is Gromov hyperbolic. This result is sharp, new even for Riemannian manifolds, and strengthens theorems of Gromov, Papasoglu, Drutu, and Bowditch. Furthermore, a similarly optimal result can be obtained for the filling radius inequality.