Abstracts for Seminar on Algebra, Geometry and Physics

We define Lie algebroids over infinite jet spaces for vector bundles over smooth manifolds. We also establish their equivalent representation through homological evolutionary vector fields on certain infinite jet super-bundles. Our new definition is a nontrivial generalization of the classical one for usual manifolds, which is no longer applicable, and more fully grasps the geometry of strings in space-time. (The talk is based on the joint work in progress with J.van de Leur:math.DG/1006.4227v2.)

We consider the abstract pair 'group J with a distinguished subset C' where J is the K-points of a Jacobian of a smooth curve C, K algebraically closed. We prove that this pair determines the field, the curve and the Jacobian uniquely up to an isomorphism of fields and a bijective morphism of the varieties, which is a bijective isogeny on Jacobians. In characteristic 0 the bijective morphism is an isomorphism. Over finite fields the theorem proves a conjecture from the recent paper by Bogomolov, Korotaev and Tschinkel.

We consider certain $q$-series depending on parameters $(A,B,C)$, where $A$ is a positive definite $r \times r$ matrix, $B$ is an $r$-vector and $C$ is a scalar, and ask when these $q$-series are modular forms. Werner Nahm has formulated a partial answer to this question: he conjectured a criterion for which $A$'s can occur, in terms of torsion in the Bloch group. The conjecture was proved by Don Zagier for $r=1$. Recently me an Sander Zwegers found several counterexamples for $r\ge 2$, so the correct formulation of Nahm's conjecture became an interesting open question.

Various categories of integrable modules over the classical infinite- dimensional Lie algebras $sl(\infty), o(\infty), sp(\infty)$ have been studied in recent years. In this talk we construct a category based on the mixed tensor algebra (the tensor algebra of the natural and conatural modules). The indecomposable injectives in the category turn out to be simply indecomposable direct summands in the tensor algebra. In addition, we show that this category is antiequivalent to the category of locally unitary finite-dimensional modules over a Koszul algebra.