Ottawa-Carleton joint algebra seminar (Fall 2012)

Formal Hecke algebras and algebraic oriented cohomology theories

Formal Hecke algebras and algebraic oriented cohomology theories

Abstract: Motivated by geometric realizations of (degenerate) affine Hecke algebras via convolution products on the equivariant K-theory (or homology) of the Steinberg variety, we define a "formal (affine) Hecke algebra" associated to any formal group law. Formal group laws are associated to algebraic oriented cohomology theories. When specialized to the formal group laws corresponding to K-theory and (co)homology, our definition recovers the usual affine and degenerate affine Hecke algebras. However, other formal group laws (such as those corresponding to elliptic and cobordism cohomology theories) give rise to apparently new algebras with interesting properties. This is joint work with Alex Hoffnung, José Malagón-Lopez, and Kirill Zainoulline.
 

Heegaard Floer homology solid tori

Heegaard Floer homology solid tori

Abstract: There is a rigorous sense in which sufficiently complicated three-manifolds decompose into simpler pieces along two-dimensional tori. Classical invariants of three-manifolds behave nicely with respect to such a decomposition, for example, at the level of the fundamental group this decomposition naturally gives rise to a free product with amalgamation. Recently, Lipshitz, Ozsv\'ath and Thurston have established how decomposition along surfaces manifests in Heegaard Floer homology -- the resulting algebraic structures are extremely rich. This talk will focus on the algebraic objects that arise in this context, and describe some of the simplest of these. As the title suggests, there are infinite families of distinct three-manifolds with torus boundary that behave like solid tori from the point of view of these new invariants. No familiarity with Heegaard Floer homology will be assumed.
 

Algebraic analogue of Atiyah's theorem

Algebraic analogue of Atiyah's theorem

Abstract: In topology there is a theorem of Atiyah, concerning K-theory of classifying space of connected compact Lie group. We consider an algebraic analogue of this theorem. We prove that for a split reductive algebraic group G over a field there is an isomorphism between K-theory of étale classifying space of group G and a completion of the G-equivariant K-theory of the base field.
 

Derivations of algebras obtained by étale descent

Derivations of algebras obtained by étale descent

Abstract: We will describe derivations of Lie algebras obtained by étale descent and discuss applications to multiloop algebras and extended affine Lie algebras. The talk is based on joint work with Arturo Pianzola.
 

Relatively hyperbolic groups and their quasiconvex subgroups

Relatively hyperbolic groups and their quasiconvex subgroups

Abstract: After reviewing Bowditch's approach to relatively hyperbolic groups, I will explain a combination theorem of relatively hyperbolic groups along malnormal, quasiconvex subgroups. I will then describe a criterion for detecting quasiconvexity of a subgroup of a relatively hyperbolic group which splits as a graph of groups. This is joint work with D. Wise.
 

Junior Algebra Seminar: Homogeneous projective varieties of rank 2 groups

Junior Algebra Seminar: Homogeneous projective varieties of rank 2 groups

Abstract: We will start by using two different kind of graphs to represent the Weyl group of a given root system. After drawing the graphs, we will study the algebraic groups corresponding to this root system. Then, we will use different techniques to give a geometric description of the homogeneous spaces G/P where G is the algebraic group corresponding to the root system and P is one of its parabolic subgroups. Finally, we will make a link between the graphs and the multiplication of some basis elements in the Chow group CH(G/P).
 

Yangians and quantizations of slices in the affine Grassmannian

Yangians and quantizations of slices in the affine Grassmannian

Abstract: We study slices to Schubert varieties in the affine Grassmannian, which arise naturally in the context of geometric representation theory. These slices carry a natural Poisson structure, and our main result is a quantization of these slices using subquotients of quantum groups called Yangians. We discuss also conjectural applications of these results to categorical representation theory. This is based on joint work with Joel Kamnitzer, Ben Webster, and Alex Weekes.
 

A lemma on singularity of components of automorphic representations

A lemma on singularity of components of automorphic representations

Abstract: In his representation theoretic formulation of the theory of theta series, Roger Howe proved the following statement (among many other things): let G=Mp_2n be the metaplectic group, A=Π Q_v be the ring of adeles, and π=⊗π_v be an automorphic representation of G(A). Then π_v is singular (that is, small in a suitable sense) for some v if and only if π_v is singular for all v. In this talk I will show that using the harmonic analysis on unipotent algebraic groups the above statement can be generalized to arbitrary simple algebraic groups G. I will also discuss potential applications of this generalization. I will try to give a self-contained talk.
 

Infinitely transitive group action, unirationality, and a question of J.-L. Colliot-Théléne

Infinitely transitive group action, unirationality, and a question of J.-L. Colliot-Théléne

Abstract: Let X be a rationally connected variety (over C, say). There is a recent conjecture of F. Bogomolov saying that X is unirational iff X × P^k admits an infinitely transitive regular action of an (infinite-dimensional ind-)algebraic group G on an open subset U ⊂ X × P^k for some k (depending on X), i.e., G ⊂ Aut(U), with additional assumption that G is generated by the additive groups (C, +). In order to prove/disprove the conjecture it is natural to ask (a question of Colliot-Thelene) whether the previous U is in fact always a compactification of a finite-dimensional algebraic group. I will talk on how to obtain the "No" answer to the latter question.
 

From Poincaré's limit set to the roots of tropical geometry

From Poincaré's limit set to the roots of tropical geometry

Abstract: The "geometric invariant of a group G" is a subset Σ(G) of the boundary sphere at infinity of the Euclidean space Eⁿ=G_ab⊗R. It was introduced some 30 years ago because it contains important information about finiteness properties of the group G. For instance, for a metabelian group G, one can read off from Σ(G) whether G has a finite presentation, and whether G has a K(G,1)-complex with finite skeleta. Moreover, the methods developed to compute Σ(G) in the mid-eighties anticipated what 15 years later would crystallize as the tropicalization of an algebraic variety. This suggest there is more interesting mathematics here.

Starting point is the observation that we can interpret Σ(G) as a certain horospherical limit set of G acting, by translation, on the space of finite subsets of  Eⁿ. This viewpoint relates Σ(G) to the classical limit set of a discrete group of Moebius transformations on the Riemann sphere. The aim now is to extract and develop the aspects of Σ(G) which are available in the more general setting when G acts by isometries on a proper non-positively curved space M. Both the translation action on Eⁿ and the Moebius action on hyperbolic n-space occur as the extreme special cases. The encouraging intermediate example between these extreme cases is G = SL_n(Z) acting on the symmetric space M = SL_n(Z)/SO(n): Grigory Avramidi and Dave Witte Morris recently proved Rehn's Conjecture: The corresponding horospherical limit set here is the complement of the rational spherical building in ∂_∞M.