Seminars Winter 2009

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Fall 2009

Tuesday, January 06, 2009

Xinwen Zhu, Berkeley

Every Flat Bundle on the Punctured Disk has an Oper Structure (+ pretalk)

Let G be a complex reductive group. A G-oper on a complex curve is a G-(de Rham) local system on it with a reduction of the underlying G-bundle to the Borel satisfying certain conditions. It is known that not every G-local system on a complete curve has an oper structure. I will show, on the contrary, that every G-local system on the punctured disc admits an oper structure. This result plays an important role in a version of local geometrical Langlands correspondence proposal by Frenkel and Gaitsgory. This is a joint work with Edward Frenkel.

Tuesday, January 13, 2009

Kobi Kremnizer, University of Chicago

Tate Spaces and Central Extensions (+pretalk)

In this talk I will explain the notion of a Tate vector space and how to associate to it a determinant theory. Using this we will get the universal central extension of the group of automorphisms of a Tate space. This universal central extension will also give central extensions of loop groups. I will then explain the higher categorical versions of this constructions: 2-Tate spaces and 2-central extensions. This is joint work with Sergey Arkhipov.

Tuesday, January 20, 2009

Erik Carlsson, Northwestern

Vertex Operators and Hilbert Schemes (+pretalk)

For a long time people have recoginized that there is a (perhaps mysterious) connection between the cohomology of the Hilbert scheme of points on a surface and 2-d conformal field theory. The direct sum $\bigoplus_n \mbox{Hilb}_n X$ is isomorphic to some Fock space in CFT, and the important operators on the CFT side make for valuable auxilliary gadgets on the Hilbert scheme side. I'll give a geometric construction for the "vertex operator" in CFT for any smooth surface, and show how this can be used for some simple Hilbert scheme calculations.

Tuesday, January 27, 2009

Abhishek Banerjee, Johns Hopkins University

Periodicity in Cyclic Cohomology and Monodromy at Archimedean Infinity

The cohomology of the "fibre at infinity" of an arithmetic variety can be computed by means of a complex first introduced by Consani. At archimedean infinity, this complex replaces Steenbrink’s complex for the cohomology of the universal fibre of a degeneration over a disc. The nearby cycles complex associated to this degeneration carries a monodromy operator $N$ and we can show that the graded pieces of the filtration on the cohomology of the nearby cycles complex by $\mbox{Ker}(N^j)$, $j\geq 0$, are isomorphic to the cyclic homology of a sheaf of differential operators (using some results of Wodzicki). Further, we can show that, under this isomorphism, the periodicity operator in cyclic homology coincides with the (logarithm of) the monodromy on the nearby cycles complex.

In this talk, we will do the same at archimedean infinity, where we have to work with "global sections" rather than with sheaves, and therefore show that there is a natural map from the cyclic cohomology of the ring of differential operators to the graded pieces of a filtration on the cohomology of the fibre at infinity, and that in this framework, the periodicity operator in cyclic cohomology is again the counterpart of the monodromy operator on Consani’s complex. The switch between cyclic homology and cohomology is a consequence of the fact that the monodromy operators on the nearby cycles complex and on Steenbrink’s complex are equal only upto homotopy. This is followed up by defining a complex with monodromy that plays the role of a nearby cycles complex for the fibre at infinity. Again, the monodromy operator on the latter is equivalent to the monodromy on Consani’s complex upto homotopy.

Finally, we consider the long exact sequence of Connes and Karoubi involving the algebraic, topological and relative $K$-theories of a Frechet Algebra. This long exact sequence lies above the periodicity sequence in cyclic homology. In this talk, we will construct the same sequence for the $K$ theories of the sheaf of differential operators, using the cohomologies of simplicial sheaves as defined by Brown and Gersten. This long exact sequence then lies above the periodicity sequence of cyclic (hyper)cohomologies.

Tuesday, February 17, 2009

Ralph Kaufmann, Purdue University

Correlators, Graphs and Cyclic Operads (pre-talk)

Starting from graphs with additional data, we will construct multi-linear functions called correlation functions. In the presence of a duality for the underlying vector space these functions can be turned into operations, These operations and their concatenations neatly assemble into a larger structure called a cyclic operad. Moreover in certain situations, these operations can be lifted to a chain or even a geometric level. A good tool for this are CW complexes. If one is able to construct a suitable differential in the graph setting one can use it to construct a space roughly by assigning a cell to each graph, as we will explain.

Stabilization and Semisimplicity

We discuss the relationship between our moduli space actions and the semi-simplicity of the underlying Frobenius algebra. One upshot is that a quantum string topology bracket vanishes.

Thursday, February 26, 2009

Ben Webster, MIT

Representation Theory and a Strange Duality for Symplectic Varieties (+pretalk)

In recent work with Braden, Licata and Proudfoot, we showed that certain algebras constructed from hyperplane arrangements have a number of nice properties which are surprisingly reminiscent of the BGG category O; in particular, they are Koszul, and Koszul duality corresponds to a well known combinatorial duality. I'll explain why we think properties are connected to a geometric origin for both these categories, and how this suggests an underlying duality between pairs of symplectic varieties.

Tuesday, March 3, 2009

Boris Tsygan, Northwestern

Oscillatory Modules

Tuesday, March 10, 2009

Bohan Fang, Northwestern University

T-duality and the Coherent-constructible Correspondence for Toric Varieties

Given any equivariant holomorphic line bundle on a toric variety, I will describe how to apply T-duality to produce a Lagrangian in its mirror space. This Lagrangian can be further equated with a constructible sheaf by Nadler-Zaslow's microlocalization theorem. This correspondence between equivariant line bundles and constructible sheaves turns out to be a coherent-constructible functor which is an equivalence. Passing to K-theory recovers the isomorphism given by Morelli.

Tuesday, March 17, 2009

Michael Movshev, Stony Brook University

An L-infinity Extension of Atiyah-Ward Correspondence (+pretalk)

In the talk I will explain why the self-dual four-dimensional Yang-Mills theory is perturbatively equivalent to the BF-theory on the twistor space. An extension of this construction to the full Yang-Mills theory will be also explained if time permits.

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