NSF/CBMS Conference

Abstracts of Other Talks

Abstracts of Prof. Rosenberg's lectures can be found here.

Matthew Ando - Twisted generalized cohomology and twisted elliptic cohomology
Abstract: The Umkehr map in twisted K-theory plays an interesting role in the study of D-branes in string theory. I'll explain how twisted generalized cohomology and twisted Umkehr maps arise in stable homotopy theory. I'll also explain how twisted equivariant elliptic cohomology is related to the Verlinde algebra, and why H Sati and I speculate that the twisted Umkehr map in elliptic cohomology arises in M-theory.

Paul Baum - Equivariant K Homology
Abstract: In this talk a geometric model (along the lines of Baum-Douglas) for equivariant K homology will be given when the group G is a compact Lie group or a countable discrete group. The twisted version of this has been defined by Bai-Ling Wang and is tantamount to the D-branes of string theory. A recent development is the geometric realization of the Tate theory associated to equivariant K theory.
The above is joint work with N. Higson, J. Morava, H. Oyono-Oyono, and T. Schick.

Jonathan Block - Homological mirror symmetry and noncommutative geometry.
Abstract: We reformulate homological mirror symmetry (in some cases) as an equivalence of derived categories of modules associated to noncommutative objects.

Peter Bouwknegt - The geometry behind non-geometric fluxes
Abstract: By considering T-duality for strings moving in a geometric background, i.e. in the presence of curvature and H-fluxes, one arises at situations in which the string is coupled to, what is known in the literature as, non-geometric fluxes. In this talk we will consider T-duality in the context of generalized geometry and unravel the geometry behind these non-geometric fluxes.

Alan Carey - Twisted Geometric Cycles
Abstract: I will try to explain a recent paper of my collaborator Bai-Ling Wang in which he proves that there is a generalisation of the Baum-Douglas geometric cycles which realise ordinary K-homology classes to the case of twisted K-homology. We propose that these twisted geomtric cycles are D-branes.

Charles Doran - Algebraic cycles, regulator periods, and local mirror symmetry
Abstract: We construct classes in the motivic cohomology of certain 1-parameter families of Calabi-Yau hypersurfaces in toric Fano n-folds, with applications to local mirror symmetry (growth of genus 0 instanton numbers) and inhomogeneous Picard-Fuchs equations. In the case where the family is classically modular the classes are related to Belinson's Eisenstein symbol; the Abel-Jacobi map (or rational regulator) is computed for both kinds of cycles. For the "modular toric" families where the cycles essentially coincide, we obtain a motivic (and computationally effective) explanation of a phenomenon observed by Villegas, Stienstra, and Bertin. This is joint work with Matt Kerr.

Jacques Distler - Geometry and Topology of Orientifolds I

Dan Freed - Geometry and Topology of Orientifolds II

Nigel Higson - The Baum-Connes Conjecture and Parametrization of Representations
Abstract: The Baum-Connes conjecture posits a sort of duality between the (reduced) unitary dual of a group and a variant of its classifying space. The conjectured duality occurs at the level of K-theory. For example, for free abelian groups the conjecture amounts to a K-theoretic form of Fourier-Mukai duality.
This talk will be about the Baum-Connes conjecture for Lie groups (for these, like the free abelian groups, the conjecture is proved). Associated to any connected Lie group is its so-called "contraction" to a maximal compact subgroup. This is a smooth, one-parameter family of Lie groups, and a consequence of the conjecture is that the duals of the groups in this family are all the same, at the level of K-theory. A rather surprising development is that in key cases the duals are actually the same, at the level of sets. I shall examine this phenomenon, characterize it more precisely, and prove it in some cases. What is missing at the moment is a conceptual explanation for it. Very likely, this will require a refinement of the Baum-Connes machinery.

Sooran Kang - The Yang-Mills functional and Laplace's equation on quantum Heisenberg manifolds
Abstract: In this talk, we discuss the Yang-Mills functional and a certain family of its critical points on quantum Heisenberg manifolds using noncommutative geometrical methods developed by A. Connes and M. Rieffel. In the main result, we construct a certain family of connections on a projective module over a quantum Heisenberg manifold that gives rise to critical points of the Yang-Mills functional. Moreover, we show that this set of critical points can be described as a set of solutions to Laplace's equation on quantum Heisenberg manifolds.

Marc Rieffel - Vector bundles for "Matrix algebras converge to the sphere"
Abstract: In earlier papers I gave a precise meaning to statements in the literature of high-energy quantum physics to the effect that sequences of matrix algebras of increasing dimension converge to the 2-sphere, or to other spaces (coadjoint orbits). Associated statements in that literature then say that on the sphere there are monopole bundles, and that "here are the associated monopole bundles on the matrix algebras". This talk is a preliminary report on my attempt to give that statement a precise meaning. For a given equivariant vector bundle over the sphere (or over a coadjoint orbit), I will exhibit certain projective modules over the corresponding matrix algebras that were proposed earlier by Eli Hawkins. I will show that there is a natural generalization to this setting of the usual Berezin transform that goes from the matrix algebras to the algebra of functions on the sphere. This strengthens the correspondence. I will indicate how I hope to strengthen the correspondence further.

Hisham Sati - Fivebrane structures in string theory and M-theory
Abstract: We consider topological and geometric aspects of the `dual formulations' of string theory and M-theory. We observe that the magnetic dual version of the Green-Schwarz anomaly cancelation condition can be read as a higher analog of String structure, which we call Fivebrane structure. This involves lifts to higher connected covers of the structure groups. Such structures also appear when considering the dual of the C-field in M-theory. We explain the topological obstructions to the existence of Fivebrane structures and describe some aspects of their geometry. We also describe twists of such structures which show up in the description of certain NS-branes and M-branes. This is joint work with U Schreiber and J Stasheff and with M Ando.

Claude Schochet - An Update on the Unitary Group
Abstract: Let A be a unital C*-algebra. Its unitary group, UA, contains a wealth of topological information about A. However, the homotopy type of UA is out of reach even for A = M2(C). There are two simplifications which have been considered. The first, well-traveled road, is to pass to π(U(A ⊗ K )) which is isomorphic (with a degree shift) to K(A). This approach has led to spectacular success in many arenas, as is well-known.
A different approach is to consider π(UA)⊗Q, the rational homotopy of UA. We report on progress in the calculation of this functor for the cases A = C(X)⊗ Mn(C ) and A a unital continuous trace C*-algebra. The results in these cases suggest the possibility of a two-dimensional dimension function which tracks separately the spatial and matrix dimension of these algebras.
These results are joint work with J. Klein, G. Lupton, N.C. Phillips, S. Smith and work in progress with A. Toms.

Eric Sharpe - GLSMs, gerbes, and Kuznetsov's homological projective duality
Abstract: In this talk we shall briefly outline how noncommutative resolutions arise physically as target spaces of string compactifications. We shall outline examples of ``gauged linear sigma models'' which describe moduli spaces of string vacua in which noncommutative resolutions are smoothly connected to more standard string compactifications, giving a physical realization of Kuznetsov's ``homological projective duality.'' The notion of string compactifications on gerbes will play an essential role in the analysis.

Mathai Varghese -The index of projective families of elliptic operators
Abstract: I will talk about ongoing research with Melrose and Singer, where we recently established an index theorem for projective families of elliptic operators. In this case, the index takes values in a smooth version of the twisted K- theory of the parametrizing space.

Dana Williams - Proper Actions on C*-algebras
Abstract: In 1990, Rieffel formulated the notion of a proper C*-dynamical system (A,G, α). Under reasonable hypotheses, the corresponding reduced crossed product A×α,rG is Morita equivalent to a ``generalized fixed point algebra'' Aα in the multiplier algebra M(A). In this talk, I will discuss a number of results concerning proper actions and their generalized fixed point algebras. These results are most efficiently stated by showing that our constructions are functorial. This is joint work with Astrid an Huef, Steve Kaliszewski and Iain Raeburn.

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