[Location] [Participants] [Organizers] [Schedule] [Talks]

Special Session on Geometric Topology AMS 2008 Spring Eastern Meeting Courant Institute of Mathematical Sciences New York University, March 15 to 16, 2008

This event is also part of the festivities organized in New York City to celebrate the sixtieth birthday of Sylvain Cappell, which include a conference on Singularities in Geometry and Topology also at the Courant Institute immediately following the AMS Meeting, from March 17 to 20.

• Invited Addresses at the AMS Meeting (including the Special Erdős Memorial Lecture by Timothy Gowers).
• Information about accommodations, registration, etc… (see also the “Singularities in Geometry and Topology” Conference's page).
• Online calendar with the Special Session's schedule.
• Official page of the Special Session on the AMS web-site.

Location

All talks in the Special Session will be in room 208, Silver Center. Invited Addresses will be in room 109, Warren Weaver Hall. The registration desk will be in the lobby, Warren Weaver Hall, and will be open 7:30 am-4 pm on Saturday, March 15, and 8 am-noon on Sunday, March 16.

Participants

The list of confirmed participants includes: Sylvain Cappell, James F. Davis, Pisheng Ding, Wojciech Dorabiala, Christopher Dwyer, Brad Forrest, Ian Hambleton, Mark W. Johnson, Keith Jones, Qayum Khan, Seonhee Lim, Wolfgang Lück, Gerald Marchesi, Francesco Matucci, Stephen F. Sawin, Rob Scheiderman, Dirk Schütz, Shmuel Weinberger, and Anna Wienhard.

Organizers

David Rosenthal (St. John's University) and Marco Varisco (Binghamton University, SUNY) [e-mail].

Schedule

Talks

• Sun 3-3:45 Sylvain Cappell (Courant Institute, New York, NY).
  L²-Betti-less manifolds.
  Joint work with Shmuel Weinberger.

  Abstract: I will explain joint work with Shmuel Weinberger on invariants of closed manifolds whose universal covers have no L² harmonic forms.

• Sat 9-9:45 James F. Davis (Indiana University, Bloomington, IN).
  Some remarks on Nil groups in algebraic K-theory.

  Abstract: We explain consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group.

• Sat 5-5:20 Pisheng Ding (St. John's University, Queens, NY).
  Mod-2 Equivalence of the Symbols of the de Rham and Signature Operators. Preliminary report.
  Joint work with James F. Davis.

  Abstract: We show that the symbol classes of the de Rham operator and the signature operator on a closed manifold of dimension 4n are equivalent mod 2 as KO-theory elements.

• Sun 2:30-2:50 Wojciech Dorabiala (Penn State Altoona, Altoona, PA).
  Secondary Transfer and Higher Reidemeister Torsion.
  Joint work with Bernard Badzioch, John Klein, and Bruce Williams.

  Abstract: Let p be a smooth bundle of compact manifolds. We will show that if the fundamental group of the base acts trivially on the rational homology of the fibers, then the bundle p gives rise to a secondary transfer map defined on the level of Whitehead spaces. This secondary transfer map can be used to give an alternative construction of the smooth torsion invariant of the bundle p defined by Dwyer, Weiss and Williams. As an application of this secondary transfer, we obtain a composition formula for the smooth torsion invariant.

• Sat 4:30-4:50 Christopher Dwyer (Binghamton University, SUNY, Binghamton, NY).
  Notes on an additive decomposition for twisted equivariant K-theory. Preliminary report.
  Joint work with Nansen Petrosyan.

  Abstract: I will discuss a decomposition of twisted K-theory in the case of a discrete group acting properly. This is work in progress in proving a conjecture by Alejandro Adem and Yongbin Ruan for twisted orbifold K-theory.

• Sat 10:30-10:50 Brad Forrest (Cornell University, Ithaca, NY).
  Degree Complexes in Auter Space.
  Joint work with Heather Armstrong and Karen Vogtmann.

  Abstract: Auter space, a cousin of Marc Culler and Karen Vogtmann's Outer space, was introduced by Allen Hatcher and Vogtmann in 1998. In this introduction, Hatcher and Vogtmann defined subspaces of Auter space called the degree complexes, which act like skeleta for Auter space. In this talk, I will present applications of the degree complexes including a presentation for Aut(F_n) and discuss minimality properties of these complexes.

• Sat 3-3:45 Ian Hambleton (McMaster University, Hamilton, ON, Canada).
  Aspherical 2-complexes and 4-manifolds. Preliminary report.
  Joint work with Matthias Kreck and Peter Teichner.

  Abstract: The talk will discuss the classification up to s-cobordism of closed, oriented, topological 4-manifolds whose fundamental groups have geometric dimension 2.

• Sun 10-10:20 Mark W. Johnson (Penn State Altoona, Altoona, PA).
  On Obstructions to Realizing Diagrams of Pi-algebras.
  Joint work with David Blanc and James M. Turner.

  Abstract: The category of Pi-algebras is the category of universal algebras containing, i.e. the natural categorical home for, the graded homotopy groups of a pointed topological space. Given a homotopy commutative diagram of pointed spaces, applying the graded homotopy groups functor yields a diagram of Pi-algebras of the same shape. One can then ask if there is a strictly commutative diagram of pointed spaces which yields the same diagram of Pi-algebras, called a realization of the diagram of Pi-algebras. In work which recently appeared, the authors established an obstruction theory for realizing diagrams of Pi-algebras, with the obstructions landing in Andre-Quillen cohomology groups. In the work to be discussed, the authors have since constructed various “local-to-global” spectral sequences for computing this Andre-Quillen cohomology in terms of the previously studied case of Andre-Quillen cohomology of a single Pi-algebra. Higher homotopy operations, such as Toda brackets, can also be described as obstructions to realizing certain diagrams of Pi-algebras. Thus, the broader collaborative project seeks to use the cohomology of diagrams as a tool to organize higher homotopy operations.

• Sat 8:30-8:50 Qayum Khan (Vanderbilt University, Nashville, TN).
  Algebraic K-theory over the infinite dihedral group. Preliminary report.
  Joint work with James F. Davis and Andrew A. Ranicki.

  Abstract: Bass defined an exotic Nil-summand of the algebraic K-theory of a polynomial extension. Later, Waldhausen extended the definition to tensor algebras and defined an exotic Nil-summand of the algebraic K-theory of an injective amalgam of groups. Our Nil-Nil theorem states, under a certain finiteness condition, that there is a natural isomorphism from the amalgam Nil to a tensor Nil.

• Sat 4-4:20 Seonhee Lim (Cornell University, Ithaca, NY).
  Volume entropy rigidity for buildings. Preliminary report.
  Joint work with François Ledrappier.

  Abstract: Volume entropy of a Riemannian manifold is the exponential growth rate of the volumes of balls. Entropy rigidity for rank-1 Riemannian manifolds is known: a theorem of Besson-Courtois-Gallot says that the locally symmetric metrics attain minimal volume entropy among all Riemannian metrics. In this talk, we are interested in entropy rigidity for buildings, especially hyperbolic ones. We will give several characterizations of the volume entropy, analogous to the ones for trees, that will help us to find some lower bound on volume entropy.

• Sun 9-9:45 Wolfgang Lück (Westfälische Wilhelms-Universität, Münster, Germany).
  The Burnside Ring, Equivariant Stable Cohomotopy and the Segal Conjecture for Infinite Groups.

  Abstract: In a paper from 2005 we propose several definitions for the Burnside ring of an infinite discrete group. We introduce stable equivariant cohomotopy for proper cocompact G-CW-complexes. We formulate a version of the Segal Conjecture for infinite groups in this setting. We have now finished the proof of it in the setting of infinite discrete groups. This is the topic of the talk.

• Sat 10:30-10:50 Francesco Matucci (Cornell University, Ithaca, NY).
  Centralizers in R. Thompson's groups. Preliminary report.
  Joint work with C. Bleak, A. Gordon, G. Graham, J. Hughes, M. Kassabov, H. Newfield-Plunkett, and J. Sapir.

  Abstract: We give a classification of centralizers of elements in R. Thompson's groups F, T and V . This is achieved by using solutions of the ordinary conjugacy problem both from a dynamical perspective, seeing elements as piecewise-linear homeomorphisms, and a combinatorial one, using tree pairs and strand diagrams. We give combinatorial and topological applications of this description.

• Sun 8:30-8:50 Stephen F. Sawin (Fairfield University, Fairfield, CT).
  Supersymmmetric Quantum Mechanics, Infinite-Dimensional Mathai-Quillen Formalism, and Gauss-Bonnet-Chern. Preliminary report.
  Joint work with Dana Fine.

  Abstract: We restrict the action for N=1 Supersymmetric Quantum Mechanics (SUSY QM) to the subset of all paths which are the concatenation of n short geodesic segments. On this subspace the path integral representing the time evolution operator is a perfectly rigorous finite-dimensional supersymmetric integral. Its kernel is represented by an n-fold kernel product of a simple kernel called the approximate kernel. The limit as n goes to infinity exists and can be naturally interpreted as the infinite-dimensional path integral that defines the time evolution operator and partition function of the theory. When applied to loops instead of paths, the action can also be interpreted as the infinite-dimensional Mathai-Quillen form one uses to get a “proof” of the Gauss-Bonnet-Chern theorem. We prove that this large n limit converges to the heat kernel for the Laplace-Beltrami operator on forms. We use this convergence to give a rigorous version of the nonrigorous proof using the Mathai-Quillen formalism (or equivalently using SUSY QM) of Gauss-Bonnet Chern. We suggest how the same techniques may be used to make rigorous the SUSY QM “proof” of the Local Index Theorem and some other nonrigorous math integral arguments.

• Sat 2:30-2:50 Rob Schneiderman (Lehman College, Bronx, NY).
  Stable concordance of knots in 3-manifolds.

  Abstract: Concordance invariants for knots in 3-manifolds can be defined by applying intersection invariants to a singular (immersed) concordance. Besides fitting into a general theory of Whitney towers, the resulting knot invariants generalize the Arf invariant and classify knots up to stable concordance, which allows taking connected sums with copies of S²xS².

• Sun 10:30-10:50 Dirk Schütz (Durham University, Durham, England).
  Cohomology of planar polygon spaces.
  Joint work with Michael Farber and Jean-Claude Hausmann.

  Abstract: We study the topology of the moduli space of polygonal planar curves with given side-length vector. By a conjecture of Walker the side-lengths are determined by the cohomology ring of the moduli space. We show that this conjecture is true for a large class of length vectors, and that an analogous conjecture holds if one considers polygonal curves in 3-space.

• Sun 4:30-5:15 Shmuel Weinberger (University of Chicago, Chicago, IL).
  Aspects of compact topological group actions.

  Abstract: Away from the prime 2, there is a quite good understanding of equivariant surgery for finite group actions, assuming certain gap hypotheses, or of isovariant surgery, without the usual gap hypothesis, that is quite parallel to the classical surgery theory of manifolds. This talk will explain some of the issues involved in going beyond these cases: i.e. understanding the prime 2, equivariant surgery without the gap hypothesis, and infinite compact groups. I plan to discuss concrete examples where the phenomena can be described without any reference to surgery theory.

• Sun 4-4:20 Anna Wienhard (Princeton University, Princeton, NJ).
  Rotation numbers, a generalized Maslov index and representation varieties.
  Joint work with Marc Burger and Alessandra Iozzi.

  Abstract: I will explain a construction of rotation numbers which generalizes both the classical rotation number of a homeomorphism of the circle as well as the symplectic rotation number. For Lie groups of Hermitian type these rotation numbers are intimately related to a generalization of the Maslov index and play an important role when studying the varieties of representations of such groups.

Marco Varisco - February 4, 2008.