Algebra Seminar Fall 2011

Organizers: Yuli Billig (Carleton) and Kirill Zainoulline (UOttawa)

Speaker: Changlong Zhong (UOttawa)
Title: Comparison of dualizing complexes
Abstract: In this talk I will introduce four dualizing complexes defined by M. Spiess, T. Moser, S. Bloch (duality proved by T. Geisser) and K. Sato, and compare them in the derived category. We show that Bloch's complex is quasi-isomorphic with all the three in the situation when they are properly defined (and assuming some well-known conjectures).

Speaker: Andrei Rapinchuk (Virginia)
Title: On division algebras having the same maximal subfields
Abstract: The talk will be built around the following question: let $D_1$ and $D_2$ be two central quaternion division algebras over the same field $K$; when does the fact that $D_1$ and $D_2$ have the same maximal subfields imply that $D_1$ and $D_2$ are actually isomorphic over $K$? I will discuss the motivation for this question that comes from the joint work with G.~Prasad on length-commensurable locally symmetric spaces, and will then talk about some available results. One of the results states that if the answer to the above question is positive over a field $K$ (of characteristic not 2) then it is also positive over any finitely generated purely transcendental extension of $K$. I will also discuss some generalizations to algebras of degree $> 2$ and some recent finiteness results. This is a joint work with V.Chernousov and I.Rapinchuk.

Speaker: Baptiste Calmes (Universite d'Artois)
Title: Equivariant methods and oriented cohomologies of homogeneous spaces
Abstract: I'll mention recent developments in the construction of equivariant cohomology theories in algebraic geometry and explain how they can be used to interpret and generalize known computations of various (oriented) cohomology theories applied to projective homogeneous varieties.

Speaker: Hadi Salmasian (UOttawa)
Title: Categories of unitary representations of Lie supergroups and a GNS construction.
Abstract: I will talk about global realizations of unitary representations of infinite dimensional Lie supergroups. Example include super-loop groups and super-analogues of Virasoro groups. We prove a stability result required to obtain well-behaved categories. As an application of the stability result, we obtain a variation of the classical GNS construction in this context. This is joint work with Karl-Hermann Neeb and Stephane Merigon.

Speaker: Kirill Zainoulline (UOttawa)
Title: Equivariant pretheories and invariants of torsors
Abstract: We will introduce and study the notion of an equivariant pretheory. Basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes of the respective Tits algebras. This is a joint project with S. Gille.

Speaker: Rick Jardine (Western Ontario)
Title: Cosimplicial spaces and cocycles
Abstract: Cosimplicial spaces were introduced by Bousfield and Kan in the early 1970s as a technical device in their theory of homology completions. These objects have since become fundamental tools in much of homotopy theory, but the original theory remains rather mysterious. The point of the talk is that cosimplicial spaces are quite amenable to study with modern methods of sheaf theoretic homotopy theory and cocycle categories. Non-abelian cohomology theory has a particularly interesting and useful interpretation in this context.

Speaker: Inna Bumagin (Carleton)
Title: On a class of relatively hyperbolic groups that are CAT(0).
Abstract: It has been a long standing open question whether all (Gromov) hyperbolic groups are CAT(0). It extends naturally to the question whether groups hyperbolic relative to finitely generated free abelian subgroups are CAT(0). Whereas the answer is not known in general, recent work of E.Alibegovic and M.Bestvina and the theory of groups with quasi-convex hierarchy developed by D.Wise provide numerous examples of (relatively) hyperbolic CAT(0) groups. We generalize the theorem proved by E.Alibegovic and M.Bestvina and show that every finitely generated group acting freely on a Z^n-tree is CAT(0). These groups are hyperbolic relative to finitely generated free abelian subgroups, by results due to V.Guirardel and F.Dahmani. Necessary definitions will be given during the talk. The talk is based on joint work with Olga Kharlampovich.

Speaker: Jean Fasel (Munich)
Title: Splitting projective modules using Chern classes
Abstract: Let X be a smooth affine variety of dimension d over a field k and let E be a vector bundle of rank r. If E splits off a free bundle of rank 1, then the Chern class c_r(E) is trivial. If the base field k is algebraically closed, and r=d then M. P. Murthy (with N. Mohan Kumar when d=3) proved that the converse statement holds. In this talk, we will discuss more general situations, namely r=d over arbitrary fields and r=d-1 over algebraically closed fields.