09:00 to 10:00 A Molev ([Sydney])Higher Sugawara operators and the classical W-algebra for gl n We give an explicit description of the center of the affine vertex algebra for gl n at the critical level. The results are applied to construct generators of the classical W-algebra for gl n and calculate the eigenvalues of the central elements in the Wakimoto modules at the critical level. INI 1 10:00 to 11:00 Non-reductive lie algebras and their representations with good invariant-theoretic properties One of the main topics of Invariant Theory is to describe finite-dimensional representations of complex reductive groups with good properties of algebras of invariants. For instance, it is known that, for simple algebraic groups, most of irreducible representations with polynomial algebras of invariants occur in connection with periodic automorphisms of semisimple Lie algebras. (These are the so-called "$\Theta$-groups" of Vinberg.) In my talk, I will discuss several constructions of non-reductive algebras having polynomial algebras of invariants for the adjoint or coadjoint representations (e.g. contractions of semisimple algebras and iterated semi-direct products). Some of these coadjoint representations can also be understood as $\Theta$-group associated with periodic automorphisms of non-reductive Lie algebras. There are also possibilities for constructing more general representations with polynomial algebras of invariants using bi-periodic gradings and contractions. INI 1 11:00 to 11:30 Coffee and Posters 12:30 to 13:30 Lunch at Wolfson Court 15:00 to 15:30 Tea and Posters 15:30 to 16:30 V Serganova ([UC at Berkeley])Geometric induction for algebraic supergroups Let G be a classical algebraic supergroup, and H be its subgroup. The geometric induction functor is the derived functor from the category of H-modules to the category of G-modules. It is defined as the cohomology of vector bundles on G/H. We study this functor in detail in case when H is a parabolic subgroup and G=SL(m,n) or OSP(m,2n) and use this result to find the characters of all irreducible representations of G. INI 1 16:30 to 17:30 P Etingof ([MIT])Representation theory in complex rank The subject of representation theory in complex rank was started by P.Deligne, who defined complex rank analogs for the classical complex groups GL(N),O(N),and Sp(N), and (later) for the symnmetric group S_N. These are certain symmetric tensor categories of superexponential growth (i.e. non-Tannakian), in which the dimension N of the generating object is a generic complex number. Later F.Knop generalized the latter construction to a large class of finite groups (such as general linear groups over a finite field). For generic N, these categories are semisimple, so one may think of these results as "compact" representation theory in complex rank. I will speak about "noncompact" representation theory in complex rank. The discussion will include classical real groups (i.e., classical symmetric pairs), degenerate affine Hecke algebras, rational Cherednik algebras, affine Lie algebras, Yangians, and so on. By definition, a representation of a "noncompact algebra" of complex rank is a representation of its "maximal compact subalgebra" (i.e. an (ind)-object of the corresponding tensor category) together with some additional structure (morphisms satisfying some relations). I will discuss the explicit form of such morphisms and relations in several special cases (such as rational Cherednik algebras). This approach leads to a multitude of new interesting representation categories, which, in a sense, capture the phenomenon of "stabilization with respect to rank" in representation theory of classical groups and algebras. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)