Abstracts:
Alejandro Adem, University of British Columbia, Canada
Title: Equivariant K-theory and Spaces of Commuting Elements in a Compact Lie Group
Abstract: Let G denote a compact connected Lie group with torsion-free fundamental group acting on a compact space X such that all the isotropy subgroups are connected and of maximal rank. We derive conditions on the action which imply that the equivariant K-theory of X is a free module over the representation ring of G. This can be applied to compute the equivariant K-theory of spaces of ordered commuting elements in certain compact Lie groups. This is joint work with J.M.Gomez.
C. S. Aravinda, TIFR Center for Applicable Mathematics, India
Title: Twisted doubles and nonpositive curvature
Abstract: In this talk we address the question of whether it is possible to always prescribe a nonpositively curved metric on a closed manifold which is homeomorphic to a closed nonpositively curved manifold. In particular, we discuss this question on certain examples which arise as twisted doubles of finite volume real hyperbolic manifolds.
Sergio Ardanza-Trevijano, University of Navarra, Spain
Title: Pontryagin-van Kampen duality theorem beyond locally compact Abelian groups
Abstract: Given a topological Abelian group G we denote by G∧ its group of characters endowed with the compact open topology, where by character we mean a continuous homomorphism from G into R/Z. The evaluation map αG : G → G∧∧ is defined by αG(g) = ⟨−, g⟩, where ⟨χ, g⟩ = χ(g) for each character χ∈ G∧ . A topological Abelian group G is reflexive αG is a topological isomorphism. Pontryagin-van Kampen duality theorem states that locally compact Abelian groups are reflexive. We will present in this lecture different ways to generalize this theorem to other classes of groups, finishing with a recent example of a precompact noncompact reflexive group. This is a joint work with M.J. Chasco, X. Domínguez and M. Tkachenko.
Grigori Avramidi, University of Chicago, U.S.A.
Title: Homology generated by flat tori in some locally symmetric spaces
Abstract: I'll show that many finite covers of SL(3,Z)SL(3,R)/SO(3) have non-trivial homology classes generated by totally geodesic flat 2-tori. This is joint work with Tam Nguyen Phan.
Igor Belegradek, Georgia Tech, U.S.A.
Title: Ends of negatively curved manifolds
Abstract: I will present some existence and nonexistence results for the topology of ends of negatively curved manifolds of finite volume, as well as of infinite volume.
Michelle Bucher, University of Geneva, Switzerland
Title: Volume of representations of hyperbolic lattices
Abstract: Let G be a lattice in SO(n,1) and let h:G --> SO(n,1) be any representation. For cocompact lattices, the volume of a representation is an invariant whose maximal and rigidity properties have been studied extensively. We use bounded cohomology to define the volume of a representation also in the noncocompact case (a different definition by Francaviglia and Klaff also exists) and establish a rigidity result for maximal representations, recovering Mostow rigidity for hyperbolic manifolds.
In the cocompact case, the set of values for the volume of a representation is discrete. In even dimension, this follows from the fact that the volume form is an Euler class. In odd dimension, this was proven by Besson, Courtois and Gallot. The situation changes in the noncocompact case and in particular the discreteness of the set of value is not valid anymore in dimension 2 and 3. We prove that in even dimension greater or equal to 4, the set of value of the volume of a representation is, up to a universal constant, an integer.
This is joint work with Marc Burger and Alessandra Iozzi
Chris Connell, University of Indiana, Bloomington, U.S.A.
Title: Minimal and Simplicial Volume of Generalized Graph Manifolds
Abstract: We examine certain smooth topological invariants and estimate, or in some cases compute, these for a general class of higher dimensional graph manifolds. The invariants we consider include minimal entropy, minimal volume, simplicial volume and the Yamabe
James Davis, Indiana University, Bloomington, U.S.A.
Title: Equivariant Rigidity
Abstract: A group G has a cocompact manifold model for EfinG if there is a G-manifold M with M/G compact, with MF contractible for all finite subgroups F of G, and with M having the G-homotopy type of a G-CW-complex. Two such cocompact manifold models are G-homotopy equivalent. G satisfies equivariant rigidity if any such G-homotopy equivalence is G-homotopic to a G-homeomorphism. For G torsion-free, a group satisfies equivariant rigidity if and only if M/G satisfies the Borel Conjecture.
Frank Connolly, Qayum Khan, and myself undertake a systematic attack on the problem of equivariant rigidity, using, among other ingredients, the Farrell-Jones Conjectures in K- and L-theory. A sample result is:
Theorem: All H1-negative involutions on a torus Tn are conjugate to a smooth action. If n = 0,1 (mod 4) or if n = 2,3 then all H1-negative involutions on Tn are topologically conjugate. Otherwise there are an infinite number of conjugacy classes of such actions. A involution on a space is H1-negative if it induces multiplication by -1 on the first homology.
Michael Davis, Ohio State University, Columbus, U.S.A.
Title: The cohomology of random graph products of groups
Abstract: Erdos-Renyi developed a theory of random graphs. Hence, there is a theory of random graph products of groups. Random right-angled Coxeter groups are rational duality groups. The formal dimension of the random Coxeter group is 1/2 the dimension of its nerve + 1.
Wojtek Dorabiala, Penn State, Altoona, U.S.A.
Title: Secondary transfer and higher torsion.
Abstract: Higher torsion invariants are invariants of bundles of smooth manifolds that can distinguish between bundles that are not diffeomorphic even if they are fiberwise homotopy equivalent. The talk will describe a construction of higher torsion by means of the secondary transfer of bundles. I will also explain how this approach to higher torsion provides an insight into some of its properties. This is a joint project with B.Badzioch.
Steve Ferry, Rutgers University, U.S.A.
Title: Quantitative topology: Lipschitz maps
Abstract: We consider when it is possible to bound the Lipschitz constant a priori in a homotopy between Lipschitz maps. If one wants uniform bounds, this is essentially a fi niteness condition on homotopy. This contrasts strongly with the question of whether one can homotop the maps through Lipschitz maps. We also give an application to cobordism and discuss analogous isotopy questions.
Roberto Frigerio, University of Pisa. Italy
Title: Rigidity of high dimensional graph manifolds
Abstract: We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a cusped hyperbolic manifold of dimension at least 3. The various pieces are attached together via affine maps of the boundary tori. Our main goal is to study this class of graph manifolds from the viewpoint of rigidity theory. We show that, in high dimensions, the Borel conjecture holds for our graph manifolds, and smooth rigidity holds within the class. We also prove that in every dimension larger than 3 there exist examples of graph manifolds which do not support any locally CAT(0) metric.
Ross Geoghegan, Binghamton University, U.S.A.
Title: Horospherical Limit Points
Abstract: Let the group G act on a proper CAT(0)space M, and let A be a finitely generated ZG-module. With this double occurrence of G, geometric via M and algebraic via A, comes a way of defining horospherical limit points of A over M. The subset of the boundary of M at infinity consisting of all of horspherical limit points has interesting meanings in several areas: (1) in the case when M is Gromov-hyperbolic it gives a geometric criterion for deciding when A is finitely generated over a given normal subgroup of G; (2) in the flat case, where M is a Euclidean space and G acts by translations, it is essentially the Groebner Fan in tropical geometry - this has long been known to be related to the Bieri-Neumann-Strebel invariant. (3) in the case of G = SLn(Z) acting on its symmetric space, it appears, at least conjecturally, as an interesting associated building (when the boundary at infinity is given the Tits metric). I will talk about some of these ideas. This is joint work with Robert Bieri.
Duan Haibao, Institute Of Mathematics, Chinese Academy of Sciences, China
Title: Schubert calculus and cohomology of Lie groups
Abstract: TBA s
Lizhen Ji, Michigan University, U.S.A.
Title: Geometry and topology of Teichmüller spaces and mapping class groups
Abstract: Motivated by results on symmetric spaces, arithmetic subgroups, and locally symmetric spaces, we will discuss some results on the geometry and topology of Teichmuller spaces, mapping class groups, and the moduli spaces of Riemann surfaces.
Daniel Juan-Pineda, UNAM, Morelia, Mexico
Title: The K-theoretic Fibered isomorphism conjecture for braid groups
Abstract: We outline the ingredients to prove the K-theoretic Farrell-Jones isomorphism conjecture for braid groups.
Nondas Kechagias, University of Ioannina, Greece
Title: An approximation of H*(Q0S0,Z/pZ) by H*(BΣ∞, Z/pZ)
Abstract: Let A be the mod p Steenrod algebra. A free A-algebra V(SD#) is defined using Dickson subalgebras of various length and an algebra isomorphism V(SD#) ≅ H*(Q0S0,Z/pZ) is provided. Certain subalgebras Ak filter H*(Q0S0,Z/pZ) and their quotient Ak/Ak-1 is isomorphic as A-algebras with the universal enveloping algebra on a certain Dickson subalgebra. A consequence of this approach is a new basis for H*(BΣ∞, Z/pZ).
Qayum Khan, University of Notre Dame, U.S.A.
Title: Applications of the Farrell--Jones Conjecture to the topology of 4-dimensional manifolds
Abstract: TBA
Peter Linnell, Virginia Tech, U.S.A.
Title: The Atiyah conjecture and approximating Betti numbers
Abstractt: I will review the Atiyah conjecture and its relation to L2-cohomology. Then I'll describe some recent progress, which will include looking at pro-p groups.
Brita Nucinkis, University of Southampton, Britain
Title: Bredon cohomological finiteness conditions for generalisations of Thompson's groups
Abstract: We define a family of groups that generalises Thompson’s groups T and G, and also those of Higman, Stein and Brin. For groups in this family we describe centralisers of finite subgroups and show, that for a given finite subgroup Q, there are finitely many conjugacy classes of finite subgroups isomorphic to Q. We use this to show that our groups have a slightly weaker property, quasi-F∞, to that of a group possessing a finite type model for the classifying space for proper actions EG if and only if they posses finite type models for the ordinary classifying space (joint work with C. Martínez-Pérez and Francesco Matucci).
Boris Okun, University of Wisconsin-Milwaukee, U.S.A.
Title: On the Atiyah Conjecture for weighted L2-cohomology
Abstract: The classical Strong Atiyah Conjecture predicts possible denominators for the L2-Betti numbers of a space with a proper cocompact group action in terms of the torsion of the group. For reflection type actions by Coxeter groups there is a deformation of the usual L2 theory, the so-called weighted L2-cohomology. I will explain what should to be the appropriate generalization of the Strong Atiyah Conjecture in the weighted setting, and present some partial results. This a joint work work with Richard Scott (Santa Clara University)
Ivonne Ortiz, Miami University, Ohio, U.S.A.
Title: Lower Algebraic K-theory of Three-dimensional Crystallographic groups
Abstract: In this joint work with Daniel Farley, we compute the lower algebraic K-groups of all split three-dimensional crystallographic groups G. These groups account for 73 isomorphism types of three-dimensional crystallographic groups, out of 219 types in all. Alves and Ontaneda in 2006, gave a simple formula for the Whitehead group of a 3-dimensional crystallographic group G in terms of the Whitehead groups of the virtually infinite cyclic subgroups of G. The main goal in this work in progress is to obtain explicit computations for K0(ZG) and K−1(ZG) for these groups.
Erik Kjær Pedersen, University of Copenhangen, Denmark
Title: Actions on a sphere cross euclidean space
Abstract:
Tam Nguyen Phan, University of Chicago
Title: Gluing locally symmetric manifolds: asphericity and rigidity
Abstract: I will discuss a construction of aspherical manifolds by gluing the Borel-Serre compactification of locally symmetric spaces using the reflection group trick. I will also discuss the rigidity aspect of these manifolds.
Frank Quinn, Virginia Tech, U.S.A.
Title: An approach to the Farrell-Jones conjecture for K-theory
Abstract: A general geometric model for finite-isotropy actions and a long argument reduces the conjecture to estimates of the stability threshold in controlled K-theory. A new model for K has better estimates than old versions, but not yet good enough. I have no feeling about whether estimates good enough to prove the theorem are possible.
S. K. Roushon, Tata Institute, Mumbai, India
Title: The Farrell-Jones Isomorphism conjecture for a certain class of groups
Abstract:
Ben Schmidt, Michigan State University, U.S.A.
Title: Three manifolds of constant vector curvature
Abstract: A Riemannian manifold M has constant vector curvature K if every tangent vector to M belongs to a tangent plane with sectional curvature K. I will discuss the problem of classifying finite volume three-manifolds with constant vector curvature K and with all sectional curvatures bounded by K.
Wolfgang Steimle, University of Bonn, Germany
Title: Approximate fibrations and Farrell-Jones Conjectures
Abstract: An approximate fibration is a map between closed manifolds which satisfies an 'approximate' homotopy lifting property. In this talk we discuss whether a given map from some closed manifold to a non-positively curved manifold B is homotopic to an approximate fibration. This question is related to the Farrell-Jones conjectures about the algebraic K- and L-theory of the group ring over the fundamental group of B, which are known to hold for that class of target manifolds. It follows that one can give explicit conditions under which the given map approximately fibers. This is joint work with Tom Farrell and Wolfgang Lück.
Alina Vdovina, Newcastle University, Britain
Title: Hyperbolic buildings: explicit constructions, cocompact actions and Gromov-Schoen superrigidity
Abstract: We will present constructions of buildings as universal coverings of CW-complexes with certain properties. This will allow us to see explicitly interesting group actions. At the end we will discuss the Gromov-Schoen superrigidity, established by G. Daskalopoulos, C. Mese and the speaker.