Our seminars are on Mondays in the Bahen building, room BA6180, 5:10–6:00, unless otherwise indicated.
Giuseppe
Savaré, University of Pavia
Generation of the heat semigroup in metric-measure spaces with
lower curvature bounds
We will discuss some recent results on the metric-variational approach to gradient flows and its application to Wasserstein spaces of probability measures. We will mainly focus on the construction and the stability of gradient flows generated by geodesicallly convex functionals in a wide class of metric spaces, including Alexandrov spaces satisfying a lower curvature bound and the corresponding L2–Wasserstein spaces. (Partly in collaboration with L. Ambrosio.)
Alexander
Lytchak, University of Bonn
Affine structures in metric spaces
A map between geodesic metric spaces is called affine if it sends all linearly parameterized geodesics to linearly parameterized geodesics. In this talk I am going to discuss structural results about affine functions and affine maps and provide some applications.
Stefan
Wenger, UIC
Compactness for manifolds with bounded volume and diameter
Gromov’s compactness theorem for metric spaces asserts that every uniformly compact sequence of metric spaces has a subsequence which converges in the Gromov–Hausdorff sense to a compact metric space. I will show in this talk that if one replaces the Hausdorff distance appearing in Gromov's theorem by the flat distance then every sequence of oriented k-dimensional Riemannian manifolds with a uniform bound on diameter and volume has a subsequence which converges in this new distance to a countably k-rectifiable metric space. (In general, such a sequence need not have a subsequence which converges with respect to the Gromov–Hausdorff distance.) I will then sketch some applications of this theorem to the large scale geometry of non-positively curved spaces. The new distance mentioned above was first introduced and studied by Christina Sormani and myself. I will explain the basic properties of this distance and its relationship with other distances.
Paul Lee,
University of Toronto
Generalized Ricci curvature and measure contraction for 3D contact
sub-Riemannian manifolds
The measure contraction property (MCP), introduced by K. T. Sturm, is a generalization to more general metric measure spaces of Ricci curvature bound. However, whether a given space possesses the MCP is not computable in general. In this talk, I'll discuss computable sufficient conditions for a three-dimensional contact sub-Riemannian manifold to satisfy this property. This is joint work with Andrei Agrachev.
Jianguo Cao,
University of Notre Dame
A simplified proof to Perelman's
collapsing theorem for 3-manifolds
We discuss a simplified proof of a theorem of Perelman for locally volume-collapsed 3-manifolds with curvature bounded from below. Perelman’s collapsing theorem is as follows: Let {Mi3} be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by −1. Suppose that all unit metric balls in Mi3 have very small volume vi convergent to zero as i goes to infinity and suppose that Mi3 has possibly convex incompressible tori boundary. Then Mi3 must be a graph-manifold for sufficiently large i.
We will simplify the earlier proofs of this theorem given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's local convexity lemma to construct the desired local Seifert fibration structure on Mi3. Globally, we also use the splitting theorem and convex analysis to classify all singular surfaces with non-negative curvature. The earlier proof of classification of singular surfaces by Shioya-Yamaguchi used a “singular” version of the Gauss-Bonnet formula instead.
The verification of Perelman’s collapsing theorem is the last step in Perelman’s work solving Thurston's Geometrization Conjecture for 3-manifolds. This is a joint work with Jain Ge.
Anton
Petrunin, Penn State/University of Muenster
On
intrinsic isometries to Euclidean space
We consider compact length spaces which admit intrinsic isometries to Euclidean d-space. The class of these spaces is quite rich; it includes all d-dimensional Riemannian and sub-Riemannian manifolds, Euclidean polyhedra and more. The main result roughly states that the class of these spaces coincides with class of inverse limits of d-dimensional Euclidean polyhedra. This is work in progress, joint with D. Burago and S. Ivanov.
Bruce Kleiner,
NYU/Yale University
Local collapsing with a lower curvature
bound and the proof of Thurston's geometrization conjecture
A Riemannian manifold is collapsed with a lower curvature bound if the sectional curvature is at least −1 and the volume of every unit ball is small. Riemannian 3-manifolds satisfying a localized variant of this collapsing condition appear in Perelman’s proof of Thurston's geometrization conjecture; Shioya and Yamaguchi showed that such 3-manifolds have very restricted topology (they are graph manifolds), and this topological information is used in the last part of the proof of the conjecture.
In his preprint on Ricci flow with surgery, Perelman stated (without proof) a weaker local collapsing result which is sufficient to complete the proof of geometrization. Although it is weaker than the theorem of Shioya-Yamaguchi, it has the advantage of being easier to prove, as it uses only standard compactness theorems, well-known facts about the geometry of nonnegatively curved Riemannian manifolds, critical point theory, and elementary constructions.
The lecture will explain the relevant background, and discuss some highlights from the proof.
This is joint work with John Lott.
Florent
Balacheff, University of Lille
On some inequalities on
surfaces
We will survey universal inequalities (meaning without curvature bound) on Riemannian surfaces between length of some closed geodesic obtained by variational principles and the area or the diameter.
Benoit
Charbonneau, Duke University
Singular monopoles on the
product of a circle and a surface
In this talk, I will discuss work done with Jacques Hurtubise (McGill) to relate singular solutions to the Bogomolny equation on a circle times a surface to pairs [holomorphic bundle, meromorphic endomorphism] on the surface. The endomorphism is meromorphic and generically bijective, and corresponds to a return map. Its poles and zeros are related to the singularities of the corresponding solution to the Bogomolny equation. This talk is based on arXiv:0812.0221.
Dan
Mangoubi, University of Montreal
A Local Version of
Courant’s Nodal Domains Theorem
Let (M, g) be a compact Riemannian manifold. Given a smooth real function F on M, any connected component of the open set {F ≠ 0} is called a nodal domain of F. In 1923 Courant proved that the number of nodal domains which correspond to the kth eigenfunction of the Laplacian on M cannot exceed k. In 1990 H. Donnelly and C. Fefferman proved a nonsharp local version of Courant’s theorem. We will explain this, and show how one can obtain a sharp form of the local Courant theorem on real analytic manifolds.
Jonathan Tyler
Whitehouse, University of Minnesota
Discrete
Curvatures, Least Squares Errors, and “Quantitative
Geometry” of Measures
We introduce some discrete curvatures for simplices based on multiway generalizations of the ordinary sine function, and we show how their integrals can be used to estimate various least squares errors for measures on Euclidean spaces. This work relates the geometry of discrete structures within the support of the measure to its least squares geometry. One such result is a characterization of the uniform rectifiability of a d-regular measure in terms of such discrete curvatures.
Larry Guth, University of Toronto
Estimating Hopf invariants
Given a Riemannian 3-sphere (S3, g) and a map from it to the standard 2-sphere with Lipschitz constant L, how can we bound the Hopf invariant of this map? We discuss this problem and its cousins.
Chad Groft, University of Toronto
Isoperimetric spectra for various spaces
Let X be a finite CW complex or compact manifold with universal cover Y, let M be an orientable manifold with nonempty boundary, and consider the following question: If a function f from ∂M to Y is filled with a function g on M with minimum volume, is it possible to bound the volume of g in terms of the volume of f? If Y is highly connected, this is possible, and the bounding function essentially depends only on the fundamental group of X. We will discuss the dependence of this function on the manifold M: briefly, manifolds M and M' yield the same isoperimetric spectra assuming dim M = dim M' ≥ 4, or dim M = dim M' = 3 and ∂M = ∂M'. However, the same cannot be said in greater generality.
Igor
Belegradek, Georgia Tech
On uniqueness and
nonuniqueness of souls
Very little is known about the moduli space of complete nonnegatively curved metrics on an open manifold. In the talk I will focus on metrics with souls of codimension one and two. There the geometry is quite rigid, yet the moduli space may be disconnected. In studying these matters one is quickly led to nontrivial problems from higher-dimensional topology, and most of the talk will have strong topological flavor.
Regina Rotman, University of Toronto
Lengths of geodesic loops on complete Riemannian manifolds
I will present volume and diameter upper bounds for the length of a shortest geodesic loop and for lengths of geodesic loops at each point of a Riemannian manifold. In particular, I will show that on any complete noncompact Riemannian manifold of a finite volume there exists a geodesic loop of arbitrarily small length. I will also show that at each point of a closed Riemannian manifold there exist at least k geodesic loops of length at most c(n)k2d, where n is the dimension and d is the diameter of the manifold. (The latter result is joint with A. Nabutovsky).
Vladimir
Gol'dshtein, Ben Gurion University
Conformal de Rham
complex
One of the important questions in the theory of quasiconformal homeomorphisms is to design invariants of Riemannian manifolds which can be used to distinguish non-quasiconformally-equivalent manifolds. We describe a version of a de Rham complex adapted to quasiconformal geometry. It is a special case of so-called Lq,p-cohomology. As an application we prove that the 3-dimensional Lie group SOL is not quasiconformaly equivalent to the hyperbolic space H3.
Alex
Nabutovsky, University of Toronto
Local minima of the
length functional on loop spaces
Let M be a closed, simply-connected Riemannian manifold, and p an arbitrary point of M. We demonstrate that, if the length functional on the space of loops on M based at p has a very deep non-trivial local minimum, then it has many deep local minima. To prove this result, we introduce a notion of “effective universal coverings” which has other potential applications in Riemannian geometry.
Larry Guth, University of Toronto
The volume function and the homology of the space of cycles
To begin, we recall Marston Morse's theory of the space of based loops in a manifold. The space of based loops in Sn is an interesting topological space. It is equipped with a natural function—the energy—which is closely related to the length of the loops. The critical points of the energy are geodesics, and the critical points are in turn related to the homology groups of the loop space.
In the 1970’s, Gromov took a new look at this subject from a more metric point of view. For each homology class in the space of loops, he asked what is the minimum length L so that this homology class can be realized by a family of loops of length at most L. On a simply connected manifold, Gromov proved that this length is roughly the dimension of the homology class.
Next we introduce the space of k-cycles in a closed Riemannian manifold M. These cycles are closed k-dimensional surfaces with mild singularities that fit together into a topological space whose components are Hk(M). The topology of the space of cycles was studied by Almgren in his thesis in 1960’s, and is even more interesting than the topology of the space of loops. The space of k-cycles is also equipped with a natural function: the k-dimensional volume.
We study the analogue of Gromov’s question: for each homology class in the space of cycles, what is the smallest volume V so that we can realize this homology class by a family of cycles of volume at most V? We present some good estimates for the case of mod-2 cycles.
If you would like to give a talk, please contact one of the organizers: Chad Groft, Larry Guth, Vitali Kapovitch, Alex Nabutovsky, and Rina Rotman.
The Geometry & Topology seminar was also held during the 2006–2007 and 2002–2003 school years. In 2007–2008 we combined with the Fields Analysis Working Group.
University of Toronto | Arts & Science | Mathematics | Seminar series
Last updated 23 Sept 2008