Geometric Analysis and Topology Seminar

Fall 2014

The seminar's usual time is Friday at 11:00am in 517 Warren Weaver Hall (Directions). Special times and dates are marked in red. Click on the title of a talk for the abstract (if available).

Upcoming seminars:

Organizers: Sylvain Cappell, Jeff Cheeger, Robert Haslhofer, Bruce Kleiner, and Robert Young.

Previous semesters:


On the regularity of varifold mean curvature flow, Yoshi Tonegawa.  In his well-known book published in 1978, Brakke initiated the theory of mean curvature flow (MCF) using the notion of varifold and studied the existence and regularity properties. While there have been many advances in the understanding on his version of MCF, the full proof of his regularity theorem remained out of reach even for specialists of MCF. Recently we gave a new and complete proof of Brakke's regularity theorem which also comes with a natural generalization. The generalization fits well with its stationary counterpart, Allard's regularity theorem, and is useful to prove the partial regularity of Brakke's MCF in general Riemannian manifolds. Starting from the definition, I will explain the outline of the proof of the regularity theorem.
Structure of measures in Lipschitz differentiability spaces, David Bate.  This talk will present results showing the equivalence of two very different ways of generalising Rademacher's theorem to metric measure spaces. The first was introduced by Cheeger and is based upon differentiation with respect to another, fixed, chart function. The second approach is new for this generality and originates in some ideas of Alberti. It is based upon forming partial derivatives along a very rich structure of Lipschitz curves, analogous to the differentiability theory of Euclidean spaces. By examining this structure further, we naturally arrive to several descriptions of Lipschitz differentiability spaces.
Convergence of harmonic maps, Zahra Sinaei.  In this talk I will present a compactness theorem for a sequence of harmonic maps which are defined on a converging sequence of Riemannian manifolds. The sequence of manifolds will be considered in the space of compact n-dimensional Riemannian manifolds with bounded sectional curvature and bounded diameter, equipped with measured Gromov-Hausdorff topology.
Vector fields on metric measure spaces, and 1-rectifiable structure, Andrea Schioppa.  In this talk we describe a correspondence between two different structures associated to a metric measure space (X,mu): Weaver derivations and Alberti representations. The module of Weaver derivations is an algebraic structure which describes, roughly speaking, the measurable vector fields on (X,mu). An Alberti representation of the measure mu is a generalized Lebesgue decomposition of mu in terms of 1-rectifiable measures. As an application of this correspondence we obtain a characterization of the differentiability spaces in the sense of Cheeger which is, roughly speaking, a quantitative version of a recent characterization due to Bate.
Regularity of isometries of subRiemannian manifolds, Enrico Le Donne.  We consider Lie groups equipped with distances for which every pair of points can be join with an arc with length equal to the distance of the two points. These distances are generalizations of Riemannian distances. They are completely described as subFinsler structures, by the work of Gleason, Montgomery, Zippin, and Berestowski. We are interested in studying the isometries of such metric spaces. As for the Riemannian case, we show that a (global) isometry is uniquely determined by the blown-up map at a point. The blown-up map is an isometry between the tangent metric spaces, which in this case are particular groups called Carnot groups. Generalizing a result of U. Hamenst├Ądt, we also show that an isometry between open sets of Carnot groups are affine maps. A key point in the argument is in showing smoothness of such isometries. The work is in collaboration with L. Capogna and A. Ottazzi..
Gluing constructions for constant mean curvature (hyper)surfaces, Christine Breiner.  Constant mean curvature (CMC) surfaces are critical points to the area functional subject to an enclosed volume constraint. Classic examples include the round sphere, the cylinder, and a family of rotationally symmetric solutions discovered by Delaunay. More than 150 years later, Kapouleas determined a generalized gluing construction that produced infinitely many new examples of CMC surfaces. Building on and refining this work, we produce infinitely many new embedded CMC surfaces and hypersurfaces. In this talk I will outline the main steps of the gluing construction and explain some of the difficulties involved in solving such a problem. This work is joint with N. Kapouleas.
Ricci flow through singularities, Bruce Kleiner.  It has been a long-standing problem in geometric analysis to find a good definition of generalized solutions to the Ricci flow equation that would formalize the heuristic idea of flowing through singularities. I will discuss a notion in the 3-d case that has good analytical properties, enabling one to prove existence and compactness of solutions, as well as a number of structural results. It may also be used to partly address a question of Perelman concerning the convergence of Ricci flow with surgery to a canonical flow through singularities. This is joint work with John Lott.
Embedded minimal tori in S^3, Simon Brendle.  In 1970, Blaine Lawson constructed an infinite family of embedded minimal surfaces in S^3 which have higher genus. He also proved that there are many immersed minimal surfaces in S^3 of genus 1. We show that there is only one embedded minimal surface of genus 1 up to ambient isometries. The proof involves an application of the maximum principle to a function that depends on pairs of points.
The hypo elliptic Laplacian in real and complex geometry, Jean-Michel Bismut.  The hypoelliptic Laplacian is supposed to be a deformation of a standard elliptic Laplacian, that acts on the total space of the tangent bundle of a Riemann manifold, and interpolates between the elliptic Laplacian and the generator of the geodesic flow. Its construction involves a deformation of the underlying geometric and analytic structures. The hypoelliptic Laplacian is neither elliptic nor self-adjoint in the classical sense, but it is self-adjoint with respect to a Hermitian form of signature $(\infty,\infty)$. On locally symmetric spaces, the hypoelliptic deformation preserves the spectrum of the elliptic Laplacian. I will explain its construction in the case of the circle, its applications to Selberg's trace formula, and also to the proof of a Riemann-Roch-Grothendieck theorem in Bott-Chern cohomology.
Homotopical effects of k-dilation, Larry Guth.  The k-dilation of a map measures how much it stretches k-dimensional areas. If Dil_k f < L, then it means that for any k-dimensional submanifold S in the domain, Vol_k (f(S)) is at most L Vol_k(S). We discuss how the k-dilation restricts the homotopy type of a map. Our main theorem concerns maps between unit spheres, from S^{m} to S^{m-1}. If k > (m+1)/2, then there are homotopically non-trivial maps S^m to S^{m-1} with arbitrarily small k-dilation. If k is at most (m+1)/2, there every homotopically non-trivial map from S^m to S^{m-1} has k-dilation at least c(m) > 0. In this talk, I want to focus on the case k at most (m+1)/2. The non-trivial homotopy type of a map S^m to S^{m-1} is detected by a certain Steenrod square. The main issue is how to connect Steenrod squares with quantitative estimates about k-dimensional volumes. This involves a mix of topology and geometry -- on the geometrical side the tools are related to isoperimetric inequalities/geometric measure theory. (We don't assume familiarity with Steenrod squares.).
Cube complexes and the proof of Thurston's conjectures, Dani Wise.  In the 1970's, Thurston conjectured that every closed aspherical 3-manifold has a finite cover that has positive first Betti number, and that every closed hyperbolic 3-manifold has a finite cover that fibers over the circle. Surprisingly, the key to understanding these issues turned out to be the nonpositively curved cube complexes popularized by Gromov. Over the past 20 years, a program was developed to understand groups using cube complexes, incorporating a construction of Sageev and a connection of Haglund-Wise to Coxeter groups. The recent resolution of the surface subgroup problem by Kahn-Markovic supplied the important ingredient to make this program applicable to closed hyperbolic 3-manifolds, and after recent progress of the speaker, the exciting final ingredient was recently completed by Agol. I'll survey these developments without assuming extensive background from the audience.
Harmonic maps in rigidity problems, Chikako Mese.  The holomorphic rigidity conjecture of Teichmuller space states that the mapping class group uniquely determines the Teichmuller space of a compact surface as a complex manifold. In this talk, we discuss a harmonic maps approach to this conjecture.

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