Algebra Seminar Fall 2009

Organizers: Benjamin Steinberg (Carleton) and Kirill Zainoulline (UOttawa)

Speaker: Kirill Zainoulline, University of Ottawa
Title: Essential dimension of an Hermitian space
Abstract: Given an hermitian space we compute its essential dimension, Chow motive and prove its incompressibility in certain dimensions. (PDF )

Speaker: Rajender Adibhatla, Carleton University.
Title: Companion forms modulo p^n.
Abstract: Two modular forms (specifically normalized eigenforms) are said to be "companions" if they satisfy a certain congruence property. Analogously the Galois representations attached to them are locally split ( i.e diagonal w.r.t some basis.) Companion forms modulo p play a role in the weight optimization part of (the recently established) Serre's Modularity Conjecture . Companion forms modulo p^n can be used to reformulate a question of Greenberg about when a normalized eigenform has CM (complex multilplication.) This talk will introduce companion forms and outline in simple terms how they appear in the problems mentioned above.

Speaker: Alexander Nenashev, York Unversity
Title: From skew-symmetric forms on vector bundles to symplectically oriented cohomology theories.
Abstract: It is about symmetric and skew-symmetric bilinear forms on vector spaces and vector bundles, the Witt theory which classifies such forms, and some modern trends in Witt-like cohomology theories for algebraic varieties. I'll speak of quaternionic projective spaces, computation of their cohomology, Pontrjagin classes and other related matters.

Speaker: Luis Ribes, Carleton University
Title: Subgroup theorems and wreath products
Abstract: (Joint work with Benjamin Steinberg) I will describe the use of wreath products of groups (which I will remind you of) to provide simple proofs of some classical subgroup theorems (Nielsen-Schreier and Kurosh), as well as some more recent results, both for abstract and profinite groups.

Speaker: Bruce Allison, University of Victoria
Title: Simple Kantor Pairs
Abstract: A 5-graded Lie algebra is a Z-graded Lie algebra of the form L = L_{-2} + L_{-1}+ L_0+ L_1+ L_2. Given such a graded algebra, the pair P = (L_{-1}, L_1) has the structure of a Kantor pair with the property that the graded Lie algebra L can, up to central extension, be recovered from the Kantor pair. Examples of Kantor pairs are the well understood Jordan pairs which correspond to 3-graded Lie algebras. In this talk I will describe the structure of simple Kantor pairs. Other than Jordan pairs, there are 3 main classes. The pairs in one of these classes arise from Jordan pairs by a new process called reflection. This work is joint with Oleg Smirnov from the College of Charleston.

Speaker: Daniel Daigle, University of Ottawa
Title: Actions of G_a on affine surfaces and on the affine 3-space.
Abstract: I will discuss the following two open problems: (1) the classification of all actions of the algebraic group G_a = (k,+) on the affine 3-space, and (2) the classification of affine surfaces which admit many actions of G_a. Then I will explain the relation between the two problems, and how progress in (1) was made possible by progress in (2).

Speaker: Mark MacDonald, University of British Columbia
Title: Projective homogeneous varieties birational to quadrics
Abstract: An algebraic variety is homogeneous if it has a transitive action of a linear algebraic group. The projective ones can be classified in terms of subsets of Dynkin diagrams. A natural question is, when are two projective homogeneous varieties birational (i.e. isomorphic on open dense subvarieties)? In particular, quadrics, which are defined by degree 2 polynomials, are homogeneous for the orthogonal group. In this talk I will use reduced Jordan algebras to give new examples of explicit birational maps between quadrics and varieties which are homogeneous for non-orthogonal groups. An understanding of these maps, in terms of blow-ups and blow-downs, allows us to exhibit new motivic decompositions for some projective homogeneous varieties.

Speaker: Alistair Savage, University of Ottawa
Title: Quiver grassmannians, quiver varieties and the preprojective algebra
Abstract: Quivers play an important role in the representation theory of algebras with a key ingredient of the theory being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In this talk, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassmannians which are homeomorphic to the Demazure quiver varieties, and others which are homeomorphic to the graded/cyclic quiver varieties defined by Nakajima. The Demazure quiver grassmannians allow us to construct injective objects in the category of locally nilpotent modules of the preprojective algebra. We conclude by relating our construction to a similar one of Lusztig using projectives in place of injectives. This is joint work with Peter Tingley.

Speaker: Benjamin Steinberg, Carleton University
Title: Symbolic dynamics, profinite groups and the profinite completion of the free monoid.
Abstract: A symbolic dynamical system, or subshift, is a closed shift-invariant subspace of $A^{\mathbb Z}$ (with $A$ a finite set). Usually, one classifies subshifts up to conjugacy. Subshifts are precisely boundaries of factorial prolongable subsets of the free monoid. Recently, Almeida introduced a profinite group invariant of an irreducible subshift by considering the boundary of factorial prolongable subsets inside of the profinite completion of the free monoid. We survey some of the known results about these profinite group invariants. In particular, we have proved with A. Costa that the group associated to an irreducible sofic shift is free profinite.

Speaker: Manfred Kolster, McMaster University
Title: Special values of L-functions
Abstract: Classically, the special value of the Dedekind zeta-function of a number field at 1 (or 0) is related to the order of the class group and logarithms of units. The values at negative integers have a similar relation to algebraic K-groups and motivic cohomology groups, at least conjecturally. In the talk I will try to explain this relationship, the tools involved in proving some of the results, and discuss related questions.