Conference in Cortona, Italy June 6-11, 2011

Titles and abstracts, slides (of Banagl, Friedman, Nest, Wockel) and notes of talks, as taken by Pierre Albin For the notes: many thanks to Pierre Albin who did a teriffic job with them.

- ALBIN

Title: The signature operator on stratified pseudomanifolds

Abstract: The signature operator of a Riemannian metric is an important tool for studying topological questions with analytic machinery. Though well-understood for smooth metrics on compact manifolds, there are many open questions when the metric is allowed to have singularities. We will report on joint work of Pierre Albin, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza on the signature operator on stratified pseudomanifolds and some of its topological applications. -
Banagl

Title: Singularities and Intersection Spaces: A Topological-Geometric Panopticon

Abstract: In many situations, it is homotopy theoretically possible to associate to a singular space in a natural way a generalized geometric Poincare complex, called its intersection space, whose cohomology turns out to be a new cohomology theory for singular spaces, not isomorphic in general to intersection cohomology or Cheeger-L2-cohomology. An alternative description of the new theory by a de Rham complex of global differential forms is available. The talk will discuss the properties of the new theory, stressing differences to intersection cohomology, as well as indicate numerous applications of these methods, even outside singularity theory. We will consider the K-theory of intersection spaces. We will see how, as a by-product, one obtains results on equivariant cohomology and flat bundles, closely related to results of X. Dai and Jörn Müller. The theory also addresses questions in type II string theory and mirror symmetry. While intersection cohomology is stable under small resolutions, the new theory is often stable under deformations of singularities. The latter result is partly joint work with L. Maxim. An analytic description remains to be found, but we shall indicate a partial result based on results of Melrose and Hausel-Hunsicker-Mazzeo. -
Bismut

Title: Bott-Chern cohomology and Riemann-Roch-Grothendieck

Abstract: The Bott-Chern cohomology of a complex manifold is a refinement of its ordinarycohomology. In the talk, I will explain the proof of a theorem of Riemann-Roch-Grothendieck for proper holomorphic submersions with values in the Bott-Chern cohomology, the considered manifolds not being assumed to be Kähler. The proof is based on an exotic hypoelliptic deformation of Hodge theory. - BUNKE

Title: The topological contents of eta the invariant (U. Bunke)

Abstract: The talk is about the universal bordism invariant obtainable from eta invariants. I will present its construction and main properties. The main result is an index theorem which identifies the analytic construction of the universal eta invariant with a homotopy theoretic one. I will also discuss intrinsic formulas for the universal eta invariant and how classical invariants can be derived as special cases. -
CARRON

Title: Subelliptic boundary conditions (with Ballmann and Brüing)

Abstract : I will introduce a general class of boundary condition for Dirac type operator that imply subelliptic estimate. Within this class, a relative/Splitting index theorem can be easily proved. -
Davis

Title: Torus bundles over lens spaces and topological K-theory (joint with Lück)

Abstract: For an odd prime p, there is an (p-1)-integral representation of Z/p which is free away from the origin, which gives an action of Z/p on the (p-1)-dimensional torus T with p fixed points. Let M be the quotient of the diagonal Z/p-action on S \times T where S is an odd-dimensional sphere with a free action. Let \Gamma be the fundamental group of M. We compute the equivariant K-(co)homology of the torus, and, using the Baum-Connes and Farrell-Jones Conjectures, the K-theory of the C^*-algebra of \Gamma and the structure set of the classifying space for \Gamma. As geometric consequences we prove the Gromov-Lawson-Rosenberg Conjecture for Spin manifolds with fundamental group \Gamma and classify manifolds having the homotopy type of M. This is the first computation of a structure set for a manifold whose fundamental group is not torsion-free, finite, or built from such using amalgamated products of HNN extensions. -
Deeley

Title: A relative construction in geometric K-homology and R/Z-valued index theory

Abstract: Given a unital *-homomorphism between (unital) C*-algebras, we construct a "relative" K-homology group via geometric cycles defined using spin^c manifolds with boundary and Hilbert C*-module bundles. This group fits into a natural six-term exact sequence and has a natural index mapping to the K-theory of the mapping cone of the *-homomorphism. A prototypical example occurs when the *-homomorphism is the inclusion of the complex numbers into a II_1-factor. In this case, the construction gives a geometric realization of K-homology with coefficients in R/Z. -
DEGERATU

Title: Singular Spin Structures

Abstract: Unlike a 3-dimensional manifold, a higher dimensional manifold need not be spin. On an oriented Riemannian manifold the obstruction to having a spin structure is given by the second Stiefel-Whitney class. I will show that even when this obstruction does not vanish, it is still possible to define a notion of singular spin structure and associated singular Dirac operator. I will conclude with insights this approach gives into the Positive Mass Theorem for higher dimensional non-spin-manifolds. This is joint work with Richard Melrose and Mark Stern. -
GOETTE

Title: Generalized Kreck-Stolz invariants and 2-connected homology 7-spheres

Abstract: We generalise the Kreck-Stolz invariants $s_2$ and $s_3$ by defining a new invariant, the $t$-invariant, for quaternionic line bundles over closed spin-manifolds $M$ of dimension $4k-1$ with $H^3(M; \Q) = H^4(M; \Q) = 0$. The $t$-invariant classifies closed smooth oriented $2$-connected rational homology $7$-spheres up to almost-diffeomorphism, that is, up to connected sum with an exotic sphere. It also detects exotic homeomorphisms between such manifolds. The $t$-invariant also gives information about quaternionic line bundles over a fixed manifold and we use it to give a new proof of a theorem of Feder and Gitler about the values of the second Chern classes of quaternionic line bundles over $\H P^k$. The $t$-invariant for $S^{4k-1}$ is closely related to the Adams $e$-invariant on the $(4k-5)$-stem. -
Friedman

Title: Cup and cap products and symmetric signatures in intersection (co)homology (Coauthors: James McClure)

Abstract: Intersection homology was introduced by Goresky and MacPherson in order to extend Poincare duality and related invariants to singular spaces, such as algebraic varieties. We'll provide a quick review of the basics and then discuss recent joint work with McClure extending other classical algebraic topology techniques such as cup and cap products to the world of intersection homology. In particular, this work allows us to construct a stratified-homotopy invariant symmetric signature for Witt spaces. -
Hambleton

Title: Gauge theory and smooth group actions on 4 manifolds

Abstract: we discuss which groups can act non-trivially on manifolds, in particular on 4-manifolds, of a given topology. For 4-manifolds, this involves analysis of the Yang-Mills moduli spaces. - HANKE

Title: Homotopy groups of the moduli space of metrics of positive scalar curvature.

Abstract: This is joint work with B. Botvinnik, T. Schick and M. Walsh. We show by explicit examples that in many degrees in a stable range the homotopy groups of the moduli spaces of Riemannian metrics of positive scalar curvature on closed smooth manifolds can be non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov-Lawson to certain nonlinear smooth sphere bundles constructed by Hatcher. - Kahle

Title: T-duality and differential K-theory

Abstract: I will give a precise formulation of T-duality for Ramond-Ramond fields. This gives a canonical isomorphism between the "geometrically invariant" subgroups of the twisted differential K-theory of certain principal torus bundles. Our result combines topological T-duality with the Buscher rules found in physics. -
Natsume

Title:The trace formulae for Toeplitz operators and noncommutative geometry.

abstract: For $C^\infty$-functions $f$ and $g$ on the circle $S^1$, the commutator $[T_f, T_g]$ of the Toeplitz operators is a trace class operator, and the trace formula says $$ Tr([T_f, T_g]) = \frac{1}{2\pi i}\int_{S^1}fdg.$$ This formula describes Helton-Howe's fundamental trace form $W$ in terms of a cyclic cocycle. We generalize this formula to higher dimensional cases. Suppose that $D$ is a self-adjoint elliptic differential differential operator of order one on a closed connected manifold $M$ of dimension n. As above, denote by $T_f$ the Toeplitz operator with the symbol $f\in C^\infty(M)$. Then $$W(T_{f_0}, T_{f_1}, \cdots, T_{f_n})= C_n\int_Mf_0 df_1\wedge\cdots\wedge df_n.$$ -
NEST

Title: Formal deformations of gerbes

Abstract: This talk is an overview of the results on the deformations of gerbes leading to the gerbe version of the algebraic index theorem This is a joint work of P. Bressler, A. Gorokhovsky, R. Nest and B. Tsygan -
MORIYOSHI

Title: Eta cocycles

Abstract: On a complete manifold with cylindrical end there exists a short exact sequence of C*-algebra associated to Roe's cyclic 1-cocycle, which is equivalent to the Wiener-Hopf extension for the group C*-algebra of the real numbers. On such a sequence we construct a relative cyclic cycle by exploiting the notion of b-trace due to Melrose and a relative index class for the Dirac operator. We also exhibit that the pairing of those elements yields the Atiyah-Patodi-Singer index theorem. Here the eta invariant is understood as the transgression term. With this a basic framework we shall generalize the APS theorem to a foliated manifold with boundary. We establish an index formula for longitudinal Dirac operators on a foliated bundle associated with the Godbillon-Vey cyclic cocycle due to Moriyoshi-Natsume, and introduce the Godbillon-Vey eta invariant. The talk is a joint work with P. Piazza. -
Reich

Title: The Farrell-Jones Conjecture for SL_n(Z) (joint work with Arthur Bartels, Wolfgang Lück and Henrik Rüping)

Abstract: The Farrell-Jones conjecture predicts that the algebraic K-theory of a group ring RG can be assembled from the algebraic K-theory of the rings RH, where H runs over all virtually cyclic subgroups of G. There is an analogous conjecture for L- theory. Among the most important applications of this conjecture is the Borel conjecture, which asserts that aspherical manifolds with fundamental group G are topologically rigid. A large part of the talk will give an introduction to the Farrell-Jones conjecture and its applications. We prove the conjecture for the groups SLn(Z). In fact we prove both the K- and L-theory conjecture in the more general version with coefficients in an additive category (with involutions). Since these generalized versions have good inheritance properties one can easily deduce the following statement. The K- and L-theoretic Farrell-Jones conjecture with coefficients in an additive category holds for groups that are subgroups or finite index overgroups of GLn(S), where S is any ring (not necessarily commutative) whose underlying abelian group is finitely generated. We would like to remark that the Borel conjecture for torsionfree discrete subgroups of GLn(R) is covered by work of Farrell and Jones and that the Baum- Connes conjecture is not known for SLn(Z) if n ≥ 4. - ROCHON

Title: Pseudodifferential operators on manifolds with foliated boundaries

Abstract: After describing the pseudodifferential calculus that Mazzeo and Melrose introduced on manifolds with fibred boundary, we will indicate how to generalize their construction to situations where the fibration on the boundary is replaced by a foliation. The operators obtained in this way have nice mapping properties. We will provide some simple criteria to determine when such operators are compact or Fredholm (when acting on suitable Sobolev spaces). Time permitting, we will conclude by exhibiting a formula for the index of certain Dirac-type operators arising in this context. - STEIMLE

title: K-theoretic obstructions to fibering a manifold

Abstract: Given a map f: M --> B between compact manifolds, is it homotopic to the projection map of a fiber bundle whose fibers are compact manifolds? Obstructions in higher algebraic K-theory to fibering the given map f will be defined. These refine the obstructions recently defined by Farrell-Lueck-Steimle; the vanishing of these obstructions has a concrete geometrical meaning: the obstructions are zero if and only if f fibers stably, i.e. after crossing M with a high-dimensional disk. The methods also provide a classification of the different ways of stably fibering f in terms of algebraic K-theory. -
VAN ERP

Title: Fredholm operators in K-homology: Beyond ellipticity

Abstract: The isomorphism between topological K-homology (Baum-Douglas) and analytic K-homology (Atiyah, Kasparov) provides a powerful context in which to study the index theory of elliptic operators. In recent work with Paul Baum we showed that the index theory of certain non-elliptic Fredholm operators is greatly clarified by placing it in the K-homology framework. We present the computation of a geometric K-cycle (and Chern character) for a class of hypoelliptic operators on contact manifolds. This K-cycle has interesting features that are not seen in the elliptic case. This is joint work with Paul Baum. -
WAHL

Title: Higher rho-invariants for the signature operator and the surgery exact sequence

Abstract: Using their higher Atiyah-Patodi-Singer index theorem Leichtnam and Piazza constructed a commuting diagram mapping part of Stolz's positive scalar curvature exact sequence to noncommutative de Rham homology. The vertical maps are given by the higher index on manifolds with boundary and the higher rho-invariant for the spin Dirac operator, respectively. Chang and Weinberger constructed a similar commutative diagram for the surgery exact sequence using L^2-signatures and L^2-rho-invariants. We show how to define the latter diagram in the higher case. It is still an open question whether the diagram commutes. - WANG

Title: Stringy product on orbifold K-theory

Abstract: Motivated by orbifold string theory models in physics, Chen and Ruan discovered a new product called the stringy product on the cohomology for the inertia orbifold of an almost complex orbifold. This cohomology with the stringy product is now called the Chen-Ruan cohomology, as a classical limit of orbifold quantum cohomology theory. Later, various versions of stringy product on orbifold K-theory of the inertia orbifold were proposed. In recent joint work with Hu, we define a stringy product on the orbifold K-theory for the orbifold itself and show that a modified de-localized Chern character to the Chen-Ruan cohomology is an isomorphism over the complex coefficient. As an application, we find a new product on the equivariant K-theory of a finite group with the conjugation product, which is different to the well-known Pontryajin (or fusion) product. -
Weiss

TITLE: Smooth maps to the plane and Pontryagin classes (joint work with Rui Reis)

The long-term goal is to prove that the Pontryagin classes for fiber bundles with fiber R^n, due to Novikov and Thom, satisfy the standard relations that we expect from Pontryagin classes of vector bundles of dimension n. Namely, square of Euler class is a Pontryagin class, and as a consequence of that, Pontryagin class in degree 4i of a bundle of fiber dimension n vanishes when 4i>2n. Using smoothing theory and some functor calculus, we translate this into a hypothesis in differential topology, concerning spaces of regular (nonsingular) smooth maps from D^n times D^2 to D^2 which extend the standard projection on the entire boundary of D^n times D^2. The talk is in part about the translation, and then about ideas from concordance theory, and again from functor calculus, which lead to a strategy for proving the hypothesis on regular maps to D^2. -
Wockel

Title: A smooth model for the string group

Abstract: There have been various constructions of the string group that are suited for differential geometric applications. It cannot be a finite-dimensional Lie group, so one of the most natural things to expect would be an infinite-dimensional Lie group. Surprisingly, such a model did not exist in the past. In this talk I will explain how to construct such an infinite-dimensional model, based on the construction of a topological model by Stolz. Moreover, it will be shown how this can be extended to a Lie 2-group model, making explicit comparisons between ordinary and categorical differential geometry possible.