CMI-IMSc Mathematics Colloquium, 2012.

Talks

Arun Ram, Views from 20 years trekking on the LS path

Abstract: My travels on the LS (Lakshmibai-Seshadri) path have brought many personal realizations and its vistas have allowed me to look out over many beautiful structures. As best I can, I shall give a brief summary (1 hour instead of 20 years) including: Kashiwara crystals, combinatorial aspects of Schubert calculus, affine Hecke algebras and structure of Chevalley groups.

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Michel Brion, The cohomology algebra of an algebraic group.

Abstract: The topic of the talk is the cohomology algebra of an algebraic group G with coefficients in the structure sheaf O_G. This graded Hopf algebra has been described by Serre when G is an abelian variety: it is an exterior algebra on g = dim(G) primitive elements of degree 1. I shall present an analogous result for an arbitrary algebraic group G. The proof is based on known structure theorems for such groups, and on the Fourier-Mukai correspondence.

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Corrado DeConcini, Index, infinitesimal Index of transversally elliptic operators and splines.

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Claudio Procesi, Algebraic and geometric aspects of the non linear Schroedinger equation

Abstract: (Joint work with Michela Procesi). We discuss a class of normal forms of the completely resonant non-- linear Schr\"odinger equation on a torus. \begin{equation} iu_t-\Delta u=\kappa |u|^{2q}u +\partial_{\bar u}G(|u| ^2),\quad q\geq 1\in\N .\end{equation} Where $u:= u(t,\varphi)$, $\varphi\in \T^n$ and $G(a)$ is a real analytic function whose Taylor series starts from degree $q+2$. The case $q=1$ is of particular interest and is usually referred to as the {\em cubic NLS}. We stress the geometric and combinatorial constructions arising from this study. We aim at applications to quasi--periodic solutions. These require a careful study of the first 3 Melnikov non--degeneracy conditions in order to apply a KAM algorithm. Of particular relevance is the fact that the infinite-dimensional quadratic form appearing in the normal form is described by a finite number of combinatorially defined graphs which produce interesting polynomials and problems on their nature.

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C.S. Rajan, Spectrum and Arithmetic

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Patrick Polo, On torsion in integral intersection cohomology of Schubert varieties

Abstract: For purposes of representation theory, intersection cohomology with coefficients in Z of certain varieties has been investigated in recent works by Juteau, Juteau-Mautner-Williamson and Williamson-Braden. In particular, Braden discovered, several years ago, an example of 2-torsion in the intersection cohomology of a certain Schubert variety in the symmetric group S(8). Other examples of 2-torsion for S(8) were later obtained via a computer calculation by Williamson. We show, by studying the geometry of one of these examples, that it can be generalized to give an example of n-torsion for a certain Schubert variety in S(4n).

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Ravi Vakil, Stabilization of discriminants in the Grothendieck ring

Abstract: We consider the ``limiting behavior'' of discriminants, by which we mean informally the closure of the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the collection of unordered points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the "ring of motives", as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization can be described in terms of the motivic zeta values. The results extend parallel results in both arithmetic and topology. Even in the case of points on the line (polynomials in a single variable) there are consequences that are new to us. I will also present a new conjecture (on ``motivic stabilization of symmetric powers'') that was naturally suggested by our work, and try to make a case that it is compelling (although Daniel Litt has shown that it contradicts the well-known "cut-and-paste" conjecture). This is joint work with Melanie Wood.

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Jochen Heinloth, TBA

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M. S. Raghunathan, TBA

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V. Srinivas, Abelian Varieties and Theta Functions associated to certain compact Riemannian manifolds: constructions inspired by superstring theory

Abstract:This lecture will outline a consruction of abelian varieties and theta functions appearing in papers of Witten, and Moore and Witten, associated to the topological K-groups of certain Riemannian manifolds, and some related constructions. This is based on joint work with Chris Peters and Stefan M\"{u}ller-Stach.

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Jonathan Weitsman, Semiclassical analysis, loop group characters, and the modular group action (joint with Victor Guillemin and Shlomo Sternberg)

Abstract: Quantized systems can display symmetry not arising from invariances of the underlying symplectic manifolds. One example of this type of "enhanced symmetry" is the appearance of modular invariance for characters of loop group representations; this group action does not arise from any known symmetry of the coadjoint orbit (though it does of course appear in conformal field theory). We show that a modular group action appears geometrically in the corresponding semiclassical category. This indicates that semiclassical analysis may make it possible to find enhanced symmetry in other situations where constructing the quantization may be difficult or unattainable.

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Laurent Lafforgue, TBA.

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N. Mohan Kumar, Spaces of rational curves in general hypersurfaces.

Abstract: This is joint work with Roya Beheshti. I will discuss the irreducibility of the space of smooth rational curves in general hypersurfaces of degree d in n-space. In particular, they are irreducible if d<2n/3 and n>20

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Adrian Langer, On a positive equicharacteristic version of the Grothendieck-Katz conjecture

Abstract: I would like to talk about the p-curvature conjecture. The equicharacteristic zero version of this conjecture was obtained by Yves Andre. I will focus on possible generalizations to positive equicharacteristic. This is a report on a joint work with Helene Esnault.

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Constantin Teleman, Geometric Langlands correspon- dence after Beilinson-Drinfeld and deformed opers

Abstract: Let G be a complex reductive group and C a smooth compact Riemann surface. Two decades ago, Beilinson and Drinfeld offered a geometric construction of a part of the geometric Langlands correspondence, by assigning automorphic perverse sheaves ("Hecke eigensheaves") on the moduli of G-bundles over C to certain local systems on C, with structure group the Langlands dual group G^ ("G^-opers"). I will discuss joint work with E. Frenkel, in which we use (much older, but still unpublished!) cohomological calculations on the moduli of G-Higgs bundles on C in order to extend the Beilinson-Drinfeld construction to (formal) deformations of opers systems. The commutative limit of the construction seems closely related to holomorphic symplectic deformations of the moduli of Higgs bundles.

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Tadao Oda, (Semi)stability and (quasi)crystals.

Abstract: In our joint paper (Trans. AMS, 1979), Seshadri and I constructed compactifications of the generalized Jacobian variety of a nodal curve. The combinatorial structure of the nodal curve is described by the finite graph whose vertices (resp. edges) correspond to the irreducible components (resp. nodes).

Through GIT, each polarization of the nodal curve gives rise to a compactification of the generalized Jacobian variety whose combinatorial structure is described in terms of the toric geometry associated with a convex polyhedral tiling of the one-homology space of the graph.

In this talk, my recent work in progress is explained on the surprising relevance of what we did more than 30 years ago to crystals and quasicrystals:

A lattice in a Euclidean space gives rise to (facet-to-facet and space-filling) convex polyhedral tilings called the Voronoi tiling and its dual Delaunay tiling of the Euclidean space. Given a subspace of the Euclidean space, one can systematically construct convex polyhedral tilings of the subspace called the Namikawa tilings, which are generalization of the Delaunay tiling.

In the orthonormal setting where the lattice is spanned by an orthonormal basis of the ambient Euclidean space, the method of Namikawa tilings turns out to be the ``cut and project'' method relevant not only to periodic convex polyhedral tilings hidden in crystals but also to aperiodic convex polyhedral tilings hidden in quasicrystals. One can amplify the results in our 1979 paper that dealt with the orthonormal setting arising naturally out of finite graphs.

In discrete geometric analysis, M. Kotani and T. Sunada constructed in their paper (Trans. AMS, 2000) a crystal associated to a finite connected graph (possibly with multiple edges and loops) as the "standard realization" of an abelian covering of the graph. A crystal is a one-dimensional complex of points and line segments in a Euclidean space periodic with respect to the translation action of a lattice.

For the maximal abelian covering, the crystal is in the one-homology space and is shown to have hidden in it a convex polyhedral tiling obtained as a Namikawa tiling.

More generally, the crystal obtained as the standard realization of a (not necessarily maximal) abelian covering of the graph is hoped to have hidden in it a convex polyhedral tiling obtained as a generalization of the Namikawa tilings.

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Vikram Mehta, TBA

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