Geometry/Topology Seminar

Spring 2010

Thursdays (and sometimes Tuesdays) 2-3pm, in Eckhart 308

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• 03/30/10
• 4/1/10
• 4/6/10
• 04/8/10
• 04/13/10
• 04/22/10
• 04/23/10
• 04/29/10
• 05/6/10
• 05/13/10
• 05/14/10
• 05/18/10
• 05/27/10
• 06/10/10

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• contacts

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• previous quarters
• W2010
• F2009
• S2009
• W2009
• F2008
• S2008
• W2008
• F2007
• S2007
• W2007
• F2006
• S2006
• W2006
• F2005
• S2005
• W2005

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• • Tuesday March 30 at 2pm in E308
  • Ana Rechtman, University of Illinois at Chicago and Conacyt
  • Existence of periodic orbits of geodesible vector fields on closed 3-manifolds
  • Abstract: A vector field on a closed manifold M is geodesible if there is a Riemannian metric making its orbits geodesics. After discussing some examples of geodesible vector fields and results on the existence of periodic orbits for vector fields on closed 3-manifolds, I will sketch the proof of the existence of a periodic orbit when M is either diffeomorphic to S³ or has nontrivial \pi₂ and X is either real analytic or C^∞ and preserves a volume. On 3-manifolds, the class of geodesible vector fields contains Reeb vector fields, defined by a contact form, and vector fields that admit a global cross section.
• • Thursday April 1 at 2pm in E308
  • Frank Calegari, Northwestern University
  • Mod-p homology growth in 3-manifolds
  • Abstract: There exist 3-manifolds X with infinite fundamental group whose homology is that of the 3-sphere. If X is a hyperbolic 3-manifold, however, then Lubotzky explained how to find covers Y of X such that the homology H₁(Y,Z/pZ) becomes arbitrarily large. Already this is enough to prove that lattices in SL₂(C) do not satisfy the congruence subgroup property. In this talk, I will study the growth of H₁(Y,Z/pZ) in some special towers of covers of X corresponding to p-adic analytic groups. The special case when G is the additive group of p-adic numbers will be related to the classical Alexander polynomial. The main theorem will be an exact asymptotic estimate for the Z/pZ-rank of cohomology in these towers. This is joint work with Matthew Emerton.
• • Tuesday April 6 at 2pm in E308
  • G. Ben Simon, E.T.H. Zurich
  • Causal Representations of Surface Groups
  • Abstract: I will give a new characterization of classical Teichmuller spaces for closed surfaces. This new description will be the basis of a generalizations (we call Causal Representations and is due to Hartnick and myself) of the classical Teichmuller space. Further, causal representations description carries a strong geometrical ingredient. If time will permit I will describe roughly the main steps in the proof of (one) the key statement(s). Joint work with : M.Burger, T.Hartnick, A.Iozzi, A.Wienhard.
• • Thursday April 8 at 2pm in E308
  • Kevin Wortman, University of Utah
  • Non-nonpositive curvature of some non-cocompact arithmetic groups
  • Abstract: Leuzinger-Pittet conjectured that any irreducible, non-cocompact, arithmetic subgroup of a real semisimple Lie group G has an exponential lower bound to its isoperimetric inequality in the dimension that is 1 less than the R-rank of G. Thus, arithmetic groups in higher rank Lie groups would not admit a nonpositively curved geometry in any reasonable sense. I'll show a proof of the conjecture for the case of arithmetic groups whose relative Q-type is A_n, B_n, C_n, D_n, E₆, or E₇.
• • Tuesday April 13 at 2pm in E308
  • David Constantine, University of Chicago
  • Compact forms of homogeneous spaces and group actions
  • Abstract: Given a homogeneous space J\H, is there a discrete subgroup \Gamma in H such that J\H/\Gamma is a compact manifold? This is the compact forms question and it is expected that, apart from a few simple cases, compact forms are rather rare. I will speak on an approach to the problem using dynamics -- specifically the action of the centralizer of J. If this group is higher-rank and semisimple, Zimmer noticed that cocycle superrigidity could be used to prove no compact form exists. In this talk I build on this work and use tools from hyperbolic dynamics, Ratner's theorems on unipotent flows to eliminate some fairly strong algebraic conditions that Zimmer and his coauthors must assume.
• • Double Header - Part I
  • Thursday April 22 at 2pm in E308
  • John Mackay, University of Illinois at Urbana-Champaign
  • Boundaries of hyperbolic groups and conformal dimension
  • Abstract: The boundary at infinity of a Gromov hyperbolic group carries a canonical topological structure; in fact, it has a canonical metric defined up to quasisymmetric maps. I will survey some of the links between the algebraic structure of the group and a metric invariant of the boundary: Pansu's conformal dimension. We will focus on cases where the boundary has topological dimension zero or one.
• • Double Header - Part II
  • Thursday April 22 at 3:15pm in E308
  • Julien Paupert, University of Utah
  • New non-arithmetic lattices in SU(2,1)
  • Abstract: The first examples of non-arithmetic complex hyperbolic lattices were constructed by Mostow in 1980. These examples are generalized triangle groups generated by complex reflections of orders 3, 4 or 5. We will see how to parametrize such triangle groups and how to identify which of them are lattices. We will then survey an ongoing project with Deraux and Parker whose goal is to use this idea to construct (many) new non-arithmetic lattices in SU(2,1).
• • Special seminar
  • Friday April 23 at 2pm in E207
  • Albert Fathi, ENS Lyon
  • A weak KAM approach to a theorem of Sullivan
  • Abstract: In his 1976 Paper in Inv. Math. Sullivan has obtained a huge amount of results. We will discuss one of them: If on a compact manifold M we give a continuous family of closed convex cones in the tangent space, give a necessary and sufficient conditions to find a closed form which is positive a each point in the cone. Our goal after explaining the problem from scratch is to describe an application of weak KAM methods already used in joint work with A. Siconolfi to give a proof of that theorem.
• • Thursday April 29 at 2pm in E308
  • Ben McReynolds, University of Chicago
  • Geometric spectra
  • Abstract: I will discuss some joint work with Alan Reid on recent generalizations of some classical results in spectral geometry. For context, I will give an elementary review of classical spectral geometry. This will cover the geodesic length spectrum and eigenvalue spectrum for a closed Riemannian manifold. I will discuss rigidity type results and methods for producing counterexamples to full spectral rigidity in this classical setting. I will then discuss generalized versions of these spectra that arise from basic geometric problems. For these geometric spectra, I will discuss rigidity type results and methods for producing counterexamples full spectral rigidity. An emphasis will be placed on examples and making the material accessible to a general geometrically minded audience.
• • Thursday May 6 at 2pm in E308
  • Simon Brendle, Stanford University
  • Blow-up phenomena for the Yamabe equation
  • Abstract: The Yamabe problem asserts that any Riemannian metric on a compact manifold can be conformally deformed to one of constant scalar curvature. However, this metric is not, in general, unique, and there are examples of manifolds that admit many metrics of constant scalar curvature in a given conformal class. It was conjectured by R. Schoen in the 1980s (and later by Aubin) that the set of all metrics of constant scalar curvature 1 in a given conformal class is compact, except if the underlying manifold is conformally equivalent to the sphere Sⁿ equipped with its standard metric. I will discuss counterexamples to this conjecture in dimension 52 and higher. I will also describe joint work with F. Marques, which extends these counterexamples to dimension 25 and higher. The condition n \geq 25 turns out to be optimal.
• • Thursday May 13 at 2pm in E308
  • Peter Storm, Jane Street Capital
  • Infinitesimal rigidity of hyperbolic manifolds with totally geodesic boundary
  • Abstract: Using the Bochner technique, Steve Kerckhoff and I recently proved the following theorem. Let M be a compact hyperbolic manifold with totally geodesic boundary. If M has dimension at least four, then the holonomy representation of M is infinitesimally rigid. This is an infinite volume analog of the Calabi-Weil rigidity theorem. I will explain some of the background and ideas used in the proof.
• • Friday May 14 at 2pm in E207
  • Iosif Polterovich, Universite de Montreal
  • Spectral function on a Riemannian manifold: dynamical features and average growth
  • Abstract: The talk focuses on asymptotic properties of the spectral function of the Laplacian on a Riemannian manifold. In particular, I will show that, roughly speaking, the dimension of the manifold determines the average growth of the spectral function. At the same time, its pointwise behavior is closely related to dynamics of the geodesic flow. Some open problems concerning almost periodicity of the spectral function will be discussed. The talk is based on a joint work with H. Lapointe and Y. Safarov.
• • Tuesday May 18 at 2pm in E308
  • Roland Roeder, Indiana University Purdue University Indianapolis
  • Expanding Blaschke Products for the Lee-Yang zeros on the Diamond Hierarchical Lattice
  • Abstract: In a classical work, Lee and Yang proved that zeros of certain polynomials (partition functions of Ising models) always lie on the unit circle. Distribution of these zeros control phase transitions in the model. We study this distribution for a special “Migdal-Kadanoff hierarchical lattice”. In this case, it can be described in terms of the dynamics of an explicit rational function in two variables. More specifically, we prove that the renormalization operator is partially hyperbolic and has a unique central foliation. The limiting distribution of Lee-Yang zeros is described by a holonomy invariant measure on this foliation. These results follow from a general principal of expressing the Lee-Yang zeros for a hierarchical lattice in terms of expanding Blaschke products allowing for generalization to many other hierarchical lattices. This is joint work with Pavel Bleher and Mikhail Lyubich.
• • Thursday May 27 at 2pm in E308
  • David Fisher, University of Indiana
  • A Hodge-de Rham theorem for group cohomology and applications
  • Abstract: I will develop a Hodge - de Rham type theorem for the cohomology of a (locally compact group) with coefficients in a representation by bounded operators on Hilbert spaces. A main goal will be to generalize various results of Mok, Korevaar-Schoen and Shalom relating vanishing of cohomology in families of representations to vanishing of reduced cohomology. I will give applications to the cohomology of semisimple Lie groups and their lattices and also to local rigidity of group actions.
• • Thursday June 10 at 2pm in E308
  • Dave Morris, University of Lethbridge
  • Survey of invariant orders on lattice subgroups
  • Abstract: At present, there are more questions than answers about the existence of an invariant order on a lattice in a semisimple Lie group. We will discuss four different versions of the problem: the order may be required only to be invariant under multiplication on one side, or on both sides, and the order may or may not be required to be total, rather than only partial.

For questions, contact

• Benson Farb, farb at math.uchicago.edu
• Alex Eskin, eskin at math.uchicago.edu
• Shmuel Weinberger, shmuel at math.uchicago.edu
• Ben McReynolds, dmcreyn at math.uchicago.edu
• Pierre Py, pierre.py at math.uchicago.edu