Technion, Department of Mathematics

GEOMETRY AND TOPOLOGY SEMINAR

5769 (2008/2009) Academic Year

• Thursday, 2 July 2009, Amadu 509
  
  15:30-16:30 - Chloe Perin (Hebrew U. of Jerusalem)
  Definable sets and cyclic subgroups of the free group
  Abstract:
  Definable sets on a group are a generalisation of varieties: while a variety is the set of elements of G satisfying some given finite system of equations, a definable set is given by more general formulas, which can include quantifiers. Thus for example, the set of elements g of a group G admitting a square root is definable, as it corresponds to the formula: "there exists an x such that g=x^2", but it is not a variety in general. The collection of definable sets on a group is called its definable structure.
  Given a cyclic subgroup C of the free group F, we want to study the definable structure induced by F on C, that is, the intersections of definable sets of F with C. To do this, we use techniques developped by Sela to find formal solutions to system of equations and inequations with parameters over free groups. These are based on analyzing actions on simplicial and real trees.
  
  16:35-17:35 - Moon Duchin (U. of Michigan)
  Flats and divergence
  Abstract:
  I'll start by describing joint work with Kasra Rafi where we studied the rate of divergence of geodesics in Teichmuller space and discovered that it was always at worst quadratic with respect to time. We found that the result carried over to the mapping class group almost directly. I will situate this in geometric context and explain in what other group settings one can expect to find similar "intermediate divergence" phenomena-- between the linear rates of flat spaces and the exponential rates of hyperbolic spaces.
  
  
  
• Thursday, 25 June 2009, Amadu 509, 15:30
  Igor Zelenko (Texas A&M University)
  Generalized Tanaka prolongations of filtered structures on manifolds with applications
  Abstract:
  The Tanaka prolongation procedure is a far reaching generalization of the prolongation of usual $G$-structures to filtered structure on manifolds such as distributions (subbundles of tangent bundles) and distributions with additional structures (for example, sub-Riemannian or more generally, sub-Finslerian, pseudo-product, and Cauchy-Riemann structures). First we will give a short informal review of the Tanaka prolongation procedure. Second we will give its generalization in several natural directions. Finally, combining these generalizations with the ideas from Optimal Control theory, we will present quite general results on finiteness of symmetry group and on construction of the canonical frames for distribution of any rank. The talk will not require from the audience the knowledge of the prolongation procedure of usual $G$-structures.
  
  
  
• Thursday, 4 June 2009, 15:30, Amadu 509
  Talia Fernos (UCLA)
  Reduced 1-cohomology and relative property (T)
  Abstract:
  The celebrated theorems of Delorme (1977) and Guichardet (1972) establish the equivalence between property (T) and the vanishing of 1-cohomology, where the coefficients are taken in a unitary representation. In 2000 Shalom proved that the (a priori) weaker condition of the vanishing of reduced 1-cohomology is in fact equivalent to property (T) for the class of compactly generated groups. In 2005-2006 de Cornulier, Jollisaint, and Fernos independently showed that the vanishing of the restriction map on 1-cohomology is equivalent to relative property (T). One may ask if the relative version of Shalom's theorem is true. In a joint work with Valette we exhibit a large class of non-compact amenable group-pairs where the restriction map on reduced 1-cohomology always vanishes. Since amenable groups can not have relative property (T) with respect to non-compact subgroups, our result gives a strong negative answer to the above question.
  
  
  
• Thursday, 2 April 2009, 15:30, Amadu 509
  Yael Algom-Kfir (University of Utah)
  Negative curvature phenomena in Outer Space
  Abstract:
  In a negatively curved space, closest-point projections to geodesics are "strongly contracting" in the following sense. Every ball disjoint from the geodesic projects to a set whose diameter is uniformly bounded (independently of the radius of the ball). While Teichmueller space is known not to be negatively curved, Minsky showed that geodesics in the thick part of Teichmueller space have uniformly strongly contracting projections. I will discuss the analogy between Teichmueller space and "Outer Space" which models the group Out(Fn), the outer automorphism group of the free group. I will then sketch a proof that an axis of a fully irreducible element in Out(Fn) has a strongly contracting projection.
  
  
  
• Thursday, 19 March 2009, 15:30-17:30 (a 2-hour talk!), Amadu 509
  David Blanc (University of Haifa)
  A crash course in stable homotopy theory (prerequisites for "Topological Modular Forms")
  Abstract:
  We will survey some of the basic definitions and facts about stable homotopy theory, with a goal to opening next week's workshop in Caesarea on Topological modular forms to a general geometric audience, as far as possible.
  Part I. Starting with the notion of the suspension of a space and Freudenthal's classical theorem on stabilization, we define stable homotopy classes of maps, the various forms of spectra and their basic properties, and the concept of a generalized (co)homology theory, with examples and an overview of their "classification".
  Part II. Spectral sequences are the basic computational tool of algebraic topology. In this part we will describe their motivation and construction, give several examples, and explain how they can be used even in the absence of complete data.
  In both parts we will try to present a conceptual point of view, without too many technicalities.
  
  
  
• Thursday, 12 March 2009, 15:30, Amadu 509
  Florian Deloup (University of Toulouse)
  Configurations of skew lines in 3-space and the stable equivalence problem
  Abstract:
  A configuration of skew lines in 3-space is a collection of disjoint lines in RP^3. Two such configurations are rigidly isotopic if there is an isotopy of the ambient space transforming the former into the latter such that any intermediate configuration remains a configuration of skew lines. We will survey the classification up to rigid isotopy of such configurations for small numbers of lines and review a related, simpler, question, that of stable equivalence. It will be accessible to students and non-specialists.
  
  
  
• Thursday, 5 February 2009, 14:30, Amadu 719
  Peter Storm (U. of Pennsylvania)
  Mapping class groups, part IV: Thurston's compactification of Teichmueller space
  
  
  
• Monday, 2 February 2009, 15:30, Amadu 919
  Peter Storm (U. of Pennsylvania)
  Mapping class groups, part III: Thurston's compactification of Teichmueller space
  
  
  
• Thursday, 29 January 2009, 15:30, Amadu 509
  Peter Storm (U. of Pennsylvania)
  Mapping class groups, part II: Teichmueller space
  
  
  
• Thursday, 15 January 2009, 15:30, Amadu 719
  Cornelia Drutu (Oxford U.)
  Kazhdan and Haagerup properties from the viewpoint of median spaces, applications to mapping class groups
  Abstract:
  Median spaces can be seen as non-discrete versions of CAT(0) cubical complexes. Both Kazhdan and Haagerup properties can be characterized in terms of actions of groups on median spaces. This allows to discuss homomorphisms of groups with property (T) into mapping class groups of surfaces. The latter application is due to the fact that every asymptotic cone of a mapping class group has a natural equivariant structure of median space. The talk is on joint work with I. Chatterji and F. Haglund (first part), and J. Behrstock and M. Sapir (second part).
  
  
  
• Thursday, 25 December 2008, 14:30, Amadu 719
  Sergey Matveyev (Chelyabinsk State U., Russia)
  Any 3-manifold admits an (1/2)-efficient spine
  Abstract:
  Let M be an irreducible boundary irreducible 3-manifold. We prove that one can construct a simple spine P of M such that all normal annuli in (M,P) are of a very specific type. The existence of 0-efficient triangulations and spines had been known long ago. The same problem for 1-efficient triangulations stated by W. Jaco and H. Rubinstein remains open.
  
  
  
• Thursday, 18 December 2008
  
  Lecture 1 - 14:30, Amadu 719
  Oren Ben-Bassat (Hebrew University)
  Gerbes and the Holomorphic Brauer Group of Complex Tori
  Abstract:
  Holomorphic gerbes are certain geometric objects whose isomorphism classes form the second cohomology group of the sheaf of nowhere vanishing holomorphic functions. Locally a gerbe on a small open set should be thought of as something isomorphic to the collection of all line bundles. Actually the line bundles act on gerbes similarly to the way to functions act on sections of line bundles. In this talk, we will present some aspects of the study of gerbes on complex tori. This study is analogous to the classical study of line bundles on complex tori. Concepts such as the Appell-Humbert theorem, and the Poincare bundle and more will be presented in this new setting.
  
  Lecture 2 - 15:45, Amadu 815
  Tal Poznansky (Scuola Normale Superiore di Pisa)
  Characterization of linear groups with simple reduced C*-algebra
  Abstract:
  One would like to understand what it means for a countable group to have a simple reduced $C^{*}$-algebra. We will discuss some elements of the proof of the following theorem, which characterizes such groups in the linear case. Let $\Gamma$ be a countable linear group. Then the following are equivalent:
  1. The reduced $C^{*}$-algebra of $\Gamma$ is simple.
  2. $\Gamma$ has no nontrivial normal amenable subgroups.
  
  
  
• Thursday, 11 December 2008, 14:30, Amadu 719
  Pierre-Emmanuel Caprace (University of Louvain, Belgium)
  Lattices of non-positively curved spaces
  Abstract:
  Non-positively curved spaces include Riemannian manifolds of non-positive curvature, Bruhat--Tits buildings as well as many other singular spaces which are typically cell complexes. I will discuss some recent developments which tend to show that most non-positively curved spaces admitting a lattice action are on the above list or split as direct products of spaces on the list.