Linear algebra of quaternionic matrices with applications to quaternionic analysis and

geometry.

S. Alesker

In different contexts there are different notions of non-commutative determinants (super, quantum, Gelfand-Retakh...). For matrices with quaternionic entries the Dieudonne and Moore determinants are known. We will discuss old and new properties of them. Using these linear algebraic results we develop a basic theory of plurisubharmonic functions of quaternionic variables. Some relations to quaternionic-Kahler geometry will be mentioned.

An Index theorem for transversally elliptic operators on open manifolds.

M. Braverman

Let M be a an open manifold without boundary acted upon by a compact connected Lie group G. Given a G-invariant Dirac operator on M and an equivariant map v : M → Lie(G), such that the corresponding vector field on M does not vanish outside of a compact subset, one can construct an element of compactly supported transversal K-theory. Hence, a topological index of the pair (D, v) is defined.

We introduce a deformed Dirac operator on M, whose index is defined as a distribution on G. Our index theorem states that this analytical index equals the topological index. As a main step in the proof we show that the analytical index is invariant under non-compact cobordism of the type considered by Guillemin, Ginzburg and Karshon. We will also present applications to index theory of compact manifolds and to discrete series representations of reductive groups.

Riemann Surfaces and 3-Regular Graphs

R. Brooks

We develop a model for strdying the behavior of a typical Riemann surface of large genus. The model is based on a construction of Riemann surfaces from finite #-regular graphs. Using this model, we show that a typical Riemann surface has large first eigenvalue and Cheeger constant, and has a point p with large injectivity radius about p.

Joint work with Eran Makover.

Dirichlet series, Novikov theory and spectral geometry

D. Burghelea

We consider a system (M, g, ω) where (M, g) is a closed Riemannian manifold and ω a closed one form whose all zeros are nondegenerate. For t ∈ R large enough, the Witten complex (Ω^{∗}, d^{∗}(t) = d^{∗}+tω∧) decomposes canonically as a direct sum of two one parameter family of cochain complexes (Ω_{sm}(t), d^{∗}(t)) and (Ω_{la}(t), d^{∗}(t)).

We show that the inverse Laplace transform of (Ω_{sm}(t), d^{∗}(t)) is exactly the Novikov complex associated to (M, g, ω) and the inverse Laplace transform of the analytic torsion of (Ω_{la}(t), d^{∗}(t)) properly corrected, is exactly the counting function of closed trajectories of −grad_{g}(ω) interpreted as an element in the Novikov ring associated to the cohomology class [ω]. This is done by interpreting the elements of the relevant Novikov ring as Dirichlet series. There are minor hypotheses which are generically satisfied both for the formulation of the concepts and for the proof.

Joint work with S Haller.

Classification of certain torus bundles over lens spaces.

J. Davis

We classify up homeomorphism the total spaces of torus bundles over lens spaces, w here the lens spaces have fundamental group C_{p} where p is an odd prime and the action of C_{p} on the fundamental group of the torus has only zero as a fixed point. The proof uses the Farrell-Jones Isomorphism Conjecture and the p-chain spectral sequence of Davis-Lueck. This classification is of interest for two reasons, first, it is the first classification of any manifold whose fundamental group is not built up from finite groups and torsion free groups using amalgamated free products and HNN extensions. The second reason why this is of interest is that the answer is so nice, the homeomorphism type (within a simple homotopy type) depends only on the splitting invariants of p-fold covers and the ρ-invariants of the fundamental group obtain by mapping to C_{p}-quotients.

This is joint work with Wolfgang Lueck.

Equivariant cohomology of buildings

T. Januszkiewicz

We provide a computation of cohomology with coefficients in a unitary representation for a class of spaces including Tits-Kac-Moody buildings of very large thickness. A special case is the L^{2} cohomology of a building; an amplification of the method provides computation of L^{p} cohomology.

Based on joint work with Jan Dymara.

Quandle homology and knot invariants

S. Kamada

For a quandle, we can define three homology groups (rack homology, degeneration homology and quandle homology). A relationship among them is given. For a given quandle and its quandle cocycle, an invariant of a knotted surface in 4-space (or a knot in 3-space) is constructed. Some applications of this invariant are introduced.

Joint work with J. Scott Carter and Masahico Saito.

Mass inequalities associated to cup products

M. Katz

The stable norm in the homology groups of a closed manifold X with a Riemannian metric g gives rise to invariants of g called stable systoles. We show that to every nontrivial top dimensional cup product in X, is associated a geometric inequality involving these invariants and the total volume, satisfied by every metric g on X. Our key tool is the existence of quasiorthogonal bases in real homology with respect to the stable norm, and F. John’s theorem on approximating Banach norms by Euclidean ones.

Joint work with V. Bangert.

On pseudo-ribbon surface-links

A. Kawauchi

A pseudo-ribbon surface link is a surface-link with only the double line singularities in the 3-space projection. We first show that there are lots of pseudo-ribbon, non-ribbon surface-links with non-zero total genus, generalizing a known example on a surface-knot of genus one. Then by studying a canonical Seifert hypersurface of a pseudo-ribbon surface-link, we show that the torsion linking of every pseudo-ribbon surface-link is the zero form, which generalizes the fact for a ribbon surface-link. Further, we introduce and algebraically estimate the triple point cancelling number of a surface-link to show that every surface-link is concordant to a surface-link with triple point cancelling number equal to any previously given integer greater than that of the given surface-link.

Tate cohomology, Poincaré duality and string topology

John R. Klein

Tate cohomology was first introduced to exploit the similarities between the cohomology of a finite group and its homology. Farrell later extended Tate theory to groups having finite virtual cohomological dimension.

My talk has three parts.

Part 1: I will show to construct Tate cohomology for any topological group (hence any discrete group) and any generalized cohomology theory. I will show that this construction is characterized by four axioms.

Part 2: In the solution to Part 1, the so-called “dualizing spectrum” of the group plays a crucial role. I will show how the dualizing spectrum is intimately related to Poincare duality and its generalizations. I will exploit this connection to prove theorems about Poincare duality spaces.

Part 3: Recent work of Chas and Sullivan shows how intersection theory on the free loop space of a d-dimensional manifold enables one to describe a degree −d product on the homology of the free loop space. This, together with the circle action on the free loop space is shown to equip its homology with the structure of a Batalin-Vilkovisky algebra.

The goal of this part of my talk is to show that this picture can be deduced from Parts 1 and 2 of my talk. In particular, I claim that the Chas and Sullivan picture can be extended to the free loop space of a Poincare duality space.

Finite order invariants of immersions

Tahl Nowik

We will present foundations for the study of finite order invariants of immersions of a closed orientable surface into R^{3}, analogous to chord diagrams and the 1-term and 4-term relations of knot theory. We will classify all order one invariants, showing that the space of all such invariants is extremely large.

The Yamabe invariant for manifolds with finite fundamental group

J. Rosenberg

The Yamabe invariant of a manifold, defined to be the supremum of the Yamabe constants^{(∗)} of all conformal classes of Riemannian metrics on the manifold, is a numerical measure of the possible values of the scalar curvature. It is positive if and only if the manifold admits a metric of positive scalar curvature. In dimensions 2 and 4, it can take any sign and is fairly well understood (via Gauss-Bonnet and Seiberg-Witten, resp.). But in dimensions greater than 4, Petean recently proved the surprising fact that ALL simply connected manifolds have nonnegative Yamabe invariant. We have extended this to many manifolds with finite fundamental group, and have also proved some new results on positive scalar curvature, by means of a new construction of Riemannian metrics on Toda brackets.

(∗) The Yamabe constant of a conformal class is the smallest value of the scalar curvature in the class, among all metrics with constant scalar curvature and unit volume.

Joint work with Boris Botvinnik, Univ. of Oregon.

Diophantine geometry over groups and the elementary theory of a free group

Z. Sela

We study sets of solutions to systems of equations defined over a free group, projections of such sets, and the structure of elementary sets defined over a free group. The structure theory we develop answers some questions of A. Tarski, and enable us to answer those f.g. groups that are elementary equivalent to a free group.

Magnetic Schrödinger operators: discreteness of spectrum and strict positivity criteria

M. Shubin

For the magnetic Schrödinger operators with semi-bounded below scalar potential in R^{n} or on a manifold of bounded geometry we give a necessary and sufficient condition for the discreteness of spectrum in terms which include energy of the magnetic field in balls of fixed radius and a capacity term characterizing the scalar (electric) potential in these balls.

In R^{n} under condition that the scalar potential is non-negative, we also give a necessary and sufficient condition for the operator to be strictly positive i.e. condition that 0 is not in the spectrum of this operator.

This is a joint work with V. Kondratiev and V. Maz’ya.

Algebraically integrable systems and finite groups.

L. Smolinsky

In this talk the notion of integrable system and algebraically integrable system will be discussed. The flow of many of these systems is described in terms of a group action by the Weyl group of the associated Lie algebra.

Positive Hopf algebras and set-theoretical Yang-Baxter equation

Min Yan

Motivated by the problem of finding set-theoretical solutions of the Yang-Baxter equation, we introduce the concept of Positive Hopf algebras and positive quasi-triangular structures. We will discuss the classifications of these objects and the consequence for knot invariants.

L^{2} Betti numbers of closed manifolds

A. Zuk

We present a computation of the spectrum of the lamplighter group obtained using theory of finite automata. We explain how this computation yields a counterexample to a strong version of the Atiyah conjectures about the range of L^{2}-Betti numbers of closed manifolds.

Transversal Twistor Spaces of Foliations

Izu Vaisman

The transversal twistor space of a foliation F of an even codimension is the bundle Z(F) of the complex structures of the fibers of the transversal bundle of F. On Z(F) there exists a foliation F̂ by covering spaces of the leaves of F, and any Bott connection of F produces an ordered pair (I_{1}, I_{2}) of transversal almost complex structures of F̂. We discuss the projectability of I_{1}, I_{2} along the leaves of F̂ and the integrability of the projected complex structures. I_{2} never leads to an integrable structure. The essential projectability and integrability condition of I_{1} is the existence of a transversal, flat, projective structure of F.