Abstract:

Date: 01/19/2000 (Wednesday)
Title: Modularity of the Mirror Map III
Speaker: Charles F Doran, Penn State
Location: TBA
Time: 5:30 PM

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Date: 02/02/2000 (Wednesday)
Title: Duality for lagrangian torus fibrations
Speaker: Jean-Luc A Brylinski, Penn State
Location: 103 McAllister Building
Time: 3:30 PM

Abstract: A conjecture of Strominger-Yau-Zaslow (SYZ) says that any Calabi-Yau manifold admits a lagrangian torus fibration with some singular fibers and that mirror symmetry consists in replacing each fiber by the dual torus. Many examples of such fibrations have now been constructed by Ruan and others. Hitchin proposed an extension of the SYZ conjecture involving so-called gerbes on tori. I will discuss the geometry of the torus fibration and its dual in the case of complex dimension 3 from the point of view of line bundles and gerbes over tori. For flat tori these geometric objects can be constructed explicitly. The correspondence between geometric objects on dual tori can be viewed as a Fourier transform in Grassmann variables and its kernel is the formal exponential of some line bundle.

Date: 02/09/2000 (Wednesday)
Title: Compactifications of configuration spaces
Speaker: Alexander P Ulyanov, Penn State
Location: 103 McAllister Building
Time: 3:30 PM

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Date: 02/16/2000 (Wednesday)
Title: The equivariant cohomology for toric varieties
Speaker: Bin Zhang, Penn State
Location: 103 McAllister Building
Time: 3:30 PM

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Date: 02/23/2000 (Wednesday)
Title: Loop Space, Vertex Algebras and the Chiral De Rham Complex
Speaker: Jeffrey A Raven, Penn State
Location: 103 McAllister Building
Time: 3:30 PM

Abstract:

Date: 03/01/2000 (Wednesday) (postponed to 03/29/2000)
Title: Hamiltonian Loops and Quantum Cohomology
Speaker: Augustin Banyaga, Penn State
Location: 103 McAllister Building
Time: 3:30 PM

Abstract:

Date: 03/013/2000 (Monday)
Title: Nilpotent orbits and a vanishing theorem for bundles on cotangent spaces
Speaker: Eric Sommers, Harvard University
Location: Willard 302
Time: 3:35 PM

Abstract:

Date: 03/22/2000 (Wednesday)
Title: Chiral De Rham Complex
Speaker: Jeffrey A Raven, Penn State
Location: 103 McAllister Building
Time: 3:30 PM

Abstract:

Date: 03/29/2000 (Wednesday)
Title: Hamiltonian Loops and Quantum Cohomology
Speaker: Augustin Banyaga, Penn State
Location: 103 McAllister Building
Time: 3:30 PM

Abstract:

Date: 04/05/2000 (Wednesday)
Title: Equivariant deformation quantization of cotangent bundles
Speaker: Ranee Brylinski, Penn State
Location: 103 McAllister Building
Time: 3:35 PM

Abstract:

Date: 04/10/2000 (Monday)
Title: Non-commutative Weil algebras
Speaker: Eckhard meinrenken, University of Toronto, Canada
Location: 116 McAllister Building
Time: 12:20 PM

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Date: 04/12/2000 (Wednesday)
Title: Projective differential geometry old and new: differential invariants and Sturm theory
Speaker: Valentin Ovsienko, CPT Marseille-Luminy
Location: 103 McAllister Building
Time: 3:35 PM

Abstract: An elementary inroduction to projective differential gemetry will be given as well as a survey of recent results (of Arnold, Ghys, Tabachnikov et al) relating the classical 4-vertex theorem to the Schwarzian derivative and Lorentzian geometry.

Date: 04/17/2000 (Monday)
Title: Ghosts in conformal field theory
Speaker: Dima Tamarkin, Harvard
Location: 116 McAllister Building
Time: 12:20 PM

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Date: 04/19/2000 (Wednesday)
Title: Construction of elliptic algebras
Speaker: Alexandrer Odesskii, Moscow and Chicago University
Location: 103 McAllister Building
Time: 3:35 PM

Abstract: This is a report on a joint work with B. Feigin. We study a class of finitely generated associative algebras which we call elliptic algebras. We introduce a construction of these algebras, called functional realization, where an algebra is defined by an explicit product on some functional space, such as the space of theta functions. Our algebras are deformations of algebras of polynomials. We compute symplectic leaves of the corresponding Poisson structures. We discuss a generalization of our construction, a deformation quantization of the moduli space of holomorphic P-bundles on an elliptic curve where P is a parabolic subgroup.

Date: 04/21/2000 (Friday)--Special Colloquium
Title: A few interrelated aspects of a century in fundamental physics and mathematics: From quantization and deformations to deformation quantization and its latest developments
Speaker: D. Sternheimer, Uni. Bourgogne, France
Location: 104 McAllister Building
Time: 4:30 PM

Abstract:Towards the end of 19$^{\mathrm{th}}$ century, physics seemed to have achieved our understanding of the world, with classical mechanics for the motion of rigid bodies, electromagnetism for waves and the Lorentz force to describe their interactions. It was only a plateau because then the `deformation daemon' started to hit, in 1887 when two American physicists discovered that the speed of light is a limit and when (in 1900) Planck came to his quanta hypothesis. In 1905, Einstein solved the first riddle when he showed (among others, and in our terminology) that the Galilei invariance group of Newtonian mechanics has to be {\it deformed} into the Poincar\'e group of relativistic mechanics and contributed to the solution of second by his theory of the photoelectric effect. The latter eventually led (in 1925) Louis de Broglie to his duality between waves and particles and what he called `m\'ecanique ondulatoire', which several German and Austrian physicists (Weyl, Heisenberg and Schr\"odinger) transformed into quantum mechanics, based on operators in Hilbert spaces and the `Copenhagen' (Bohr) probabilistic interpretation which Einstein and de Broglie hated. Around 1960 mathematicians (Kodaira-Spencer and Gerstenhaber) developed a theory of deformations and others introduced pseudodifferential operators. Around 1974, Mosh\'e Flato came to the conclusion that physics evolves in stages (when it hits a paradox), the passage from one level of scales (e.g. velocities and distances) to another being mathematically described by a deformation, in an appropriate category. This led us to the formulation of quantum mechanics (and quantum theories) on the same observables as classical mechanics (functions on a phase space) but with a deformed composition law, a star product, quantization being understood as a deformation -- what is now called deformation quantization and in effect reconciles Einstein and de Broglie with Bohr. It turned out that mathematicians had introduced such a deformed law with the composition of symbols of pseudodifferential operators in connection with index theorems, that quantum groups are in fact an avatar of star products (in the Hopf algebra category) and that physicists and mathematicians were around deformation quantization since a long time, but nobody dared (or could) look at the deformation aspect. Taking that aspect seriously on the symmetry group level led also to star representations of Lie groups. Deforming the Poincar\'e group to the anti De Sitter group by the introduction of a tiny negative curvature, we started with Fr\o nsdal and are developing a theory of `elementary' particles (massless in the beginning) as composed of two Dirac singletons (massless particles in a 2+1 Minkowski space), which can describe quantum electrodynamics and (in three flavors) could explain e.g. neutrino oscillations. In 1997 Maxim Kontsevich put what seemed at first a cherry on the cake that we had cooked by proving his formality conjecture and giving a complete solution to deformation quantization on general Poisson manifolds. It now appears that this has roots going very deep into modern mathematics, using in part methods (e.g. graphical) inspired by physics and extending, in particular via deformations of algebras over operads, far into seemingly unrelated notions like operads, Feynman path integrals, periods and Grothendieck's unfinished symphony of algebraic geometry. All this will develop well into the 21$^{\mathrm{st}}$ century.

Date: 04/26/2000 (Wednesday)
Title: Lefschetz formula for foliations
Speaker: Benameur Moulay, Institut Desargues, Lyon (France)
Location: 103 McAllister Building
Time: 3:35 PM

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