J. Bernstein
Tel Aviv University

A remark about the definition of pseudo-differential operators
Abstract: Usually the definition of pseudo-differential operators is given in terms of Fourier transform. This definition is based on a local approximation of a manifold by a linear space. In some geometric situations our manifold has additional geometric structures that do not have linear approximation - and then Fourier transform can not be used.

I will describe another definition of pseudo-differential operators that is formulated in more geometric terms and does
not use Fourier transform. I hope that some generalizations of this definition would allow to extend the theory of pseudo-differential operators to more complicated geometric situations.

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Y. Eliashberg
Stanford University

Fuchs quillenization and Madsen-Weiss theorem
Abstract: 35 years ago D.B. Fuchs wrote a paper "Quillenization and bordism" where he suggested reformulations of certain homotopy-theoretic problems as problems of desingularization of some geometric objects. In the talk I will reinterpret Madsen-Weiss's proof of the Mumford conjecture in this spirit. This is a joint work with S. Galatius and N. Mishachev.

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E. Frenkel
UC Berkeley

Quantization of soliton systems and Langlands duality
Abstract: TBA

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S. Gindikin
Rutgers University

Harmonic analysis on symmetric spaces: between algebra and analysis
Abstract: E.Cartan and H.Weyl developed two approaches to harmonic analysis on complex semisimple Lie groups --- algebraic and analytic (transcendental). In the focus of the analytic approach of H.Weyl was the idea to replace complex groups by maximal compact subgroups --- ``unitary trick." He avoided to build an analysis directly on complex groups. What would be changed if to choose this possibility? We discuss from this point of view some old and new problems.

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A. Givental
UC Berkeley

Cobordism-valued intersection theory on \bar{M}_{0,n}
Abstract: TBA

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M. Khovanov
Columbia University

Categorification of quantum groups
Abstract: I'll review the joint work with Aaron Lauda on realization of positive half of the quantized universal enveloping algebra of a simple Lie algebra as the Grothendieck biring of a certain tower of graded algebras, as well as graphical calculus for a categorification of the Beilinson-Lusztig-MacPherson form of quantum sl(n).

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A. Khovanskii
University of Toronto

Newton Convex Bodies
Abstract: Any semigroup of integral points in $n$-dimensional space could be approximated by the semigroup of all the points in a sublattice and lying in a convex cone. This result allows to generalize the notion of Newton polyhedron and define the Newton convex body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. Applications to intersection theory include a far-reaching generalization of Kushnirenko theorem (from Newton polyhedra theory), a new version of Hodge inequality, elementary proofs of Alexandria-Fenchel inequality and its corollaries in convex geometry and their analogues in algebraic geometry. We show that for a wide class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfies a Brunn-Minkowski type inequality. We also found a generalization of Fujita approximation theorem and its analogues for semigroups of integral points and graded algebras. My talk based on a join work with Kiumars Kaveh.

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A. Kirillov
University of Pennsylvania

Family algebras and generalized exponents for multivector representations of SO(2n+1)
Abstract: We compute explicitly the generalized exponents for multivector representations of SO(2n+1), using the structure of corresponding family algebras.

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S. Novikov
University of Maryland

New Discretization of Complex Analysis
Abstract: TBA

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V. Retakh
Rutgers University

From factorizations of noncommutative polynomials to combinatorial topology
Abstract: Unlike their commutative counterparts, polynomials over noncommutative rings admit many different factorizations which makes their theory much harder and more interesting.

In 1995 I. Gelfand and the speaker constructed $n!$ different factorizations of a "generic" noncommutative polynomial in one variable with $n$ distinct roots. Later with R. Wilson we studied "algebras of pseudo-roots" or "noncommutative splitting algebras" associated with such factorizations. Such algebras can be described in terms of special directed graphs called layered graphs.

To any cell complex one can also associate a layered graph and a "splitting algebra" defined by this graph. There are surprising connections between properties of cell complexes and the corresponding splitting algebras. In my talk I will construct a bridge between the splitting algebras associated to layered graphs and combinatorial topology.

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C. Roger
University of Lyon

Lie algebra cohomology, infinite dimensional symplectic structures and hamiltonian systems
Abstract: We shall describe some relations between parts of some kind of a trilogy:
1/ Infinite dimensional Lie algebras and their cohomology (algebra),
2/ Symplectic structures on the coadjoint orbits on their dual (geometry),
3/ Partial differential equations as hamiltonian systems (hopefully integrable!) on those symplectic structures (dynamics).

The basic and most well known example is :
Virasoro cocycle(Gel'fand-Fuchs cohomology) --> Quadratic densities on the circle and Schwarzian derivative --> Korteweg-De Vries equation and hierarchy.

We shall discuss various generalizations of the above scheme:
a/ change of metrics modifies the hamiltonian (but not the geometry) and the associated equations (results by Misiolek and Khesin)
b/ some new Lie algebras appear naturally as symmetries, and their cohomological properties imply unexpected hamiltonian action (results by J.Unterberger and the author)
c/adding loops in the scenario gives equations with one more variable (results by V. Ovsienko and the author)
d/ the Lie algebra of unimodular vector fields and its various extensions sheds light on Magnetohydrodynamics (MHD), and Chromohydrodynamics (CHD) (results by F. Gay-Balmaz).

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G. Segal
University of Oxford

The Lie algebra cohomology of some groups related to loop groups
Abstract: The usual algebraic techniques used in studying loop groups depend on the rotational symmetry, and so they do not work very well in cases like that of the smooth functions with compact support on the affine line. I shall describe a method which allows one to treat some of these cases: it goes back to ideas used by Gelfand and Fuchs in the 1960s to study Lie algebras of vector fields. The method was developed over ten years ago in the thesis of my student Yunhyong Kim, but it has never been published as far as I know.

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S. Tabachnikov
Pennsylvania State University
(Public Lecture)

Flavors of "bicycle mathematics"
Abstract: The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This mapping of a circle to a circle is always a Moebius transformation. Moebius transformations belong to one of the three types: elliptic, parabolic and hyperbolic. A 100 years old Menzin's conjecture (now, a theorem) states: if the front wheel track is an oval with area at least $\pi$ then the respective monodromy is hyperbolic. A proof of this result will be outlined.

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O. Viro
Stony Brook University

Boundary value Khovanov homology
Abstract: In the talk the Khovanov homology is generalized from links to tangles. Unlike the previous generalizations, which are due to Khovanov and Bar Natan, this one categorifies a well-known generalization of the Jones polynomial, the Reshetikhin-Turaev functor, in the same way as the Khovanov homology categorifies the Jones polynomial: the homology groups have graded Euler characteristics equal to matrix elements of the map corresponding to the tangle.

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