-->

Algebra Seminar Fall 2010

Organizers: Yuly Billig (Carleton) and Erhard Neher (Ottawa)

Talks in the Junior Algebra Seminar are aimed at graduate students in algebra.

Speaker: Alexander Merkurjev (University of California, Los Angeles)
Title: Essential dimension in algebra
Abstract: We introduce the notion of essential dimension and discuss its various applications in group theory, representation theory and algebraic geometry.

Speaker: Stefan Gille (München)
Title: Motives and cycle modules
Abstract: In the first part of the talk I will explain the construction of Voevodsky's category of effective motives which basic objects are sheaves with transfers. Then I will overview the relation of this category with Rost's category of cycle modules and show how this can be used for concrete computations of motivic cohomology groups.

Speaker: Jose Malagon-Lopez (Ottawa)
Title: Equivariant Algebraic Cobordism
Abstract: We define equivariant algebraic cobordism for a connected linear algebraic group over a field of characteristic zero. This construction is inspired by Totaro's construction for Chow groups of classifying spaces. We will see that our theory satisfies the analogues of the properties of an oriented cohomology theory (à la Levine-Morel), prove some of the expected properties from an equivariant cohomology theory, and make a few computations. This is joint work with J. Heller.

Speaker: Oxana Diaconescu (Montreal)
Title: Lie algebras and invariant integrals for polynomial differential systems
Abstract: The talk is devoted to application of Lie algebras of operators and of the theory of algebraic invariants to ordinary polynomial differential systems of first order. Lie's theorem on integrating factors is generalized for multi-dimensional polynomial differential systems. Lie algebras of operators were constructed for two-dimensional systems of Darboux type of degree m. With the help of these algebras particular invariant GL(2,R)-integrals were obtained. Recurrent formulas for some invariant integrals peculiar to a Darboux type system were constructed.

Speaker: Kirill Zaynullin (Ottawa)
Title: The Grothendieck gamma-filtration and the Rost invariant for linear algebraic groups
Abstract: Let X be the variety of Borel subgroups of a simple and strongly inner linear algebraic group G over a field k. We prove that the torsion part of the second quotient of Grothendieck's gamma-filtration on X is a cyclic group of order dividing the Dynkin index of G. As a byproduct of the proof we obtain an explicit cycle which generates this cyclic group; we provide an upper bound for the torsion of the Chow group of codimension-3 cycles on X; and we relate the generating cycle with the Rost invariant and the torsion of the respective generalized Rost motives. This is a joint project with Skip Garibaldi.

Speaker: Anne Queguiner (Paris 12)
Title: Invariant d'Arason pour les algèbres de degré 8 à involution orthogonale.
Abstract: On peut voir la théorie des algèbres à involution orthogonale comme un prolongement de la théorie des formes quadratiques. Il est donc naturel d'essayer d'étendre à ce contexte les invariants des formes quadratiques. Ainsi, Jacobson et Tits ont-ils défini le discriminant et l'algèbre de Clifford d'une involution orthogonale. Dans cet exposé, on étudie la question de l'extension de l'invariant d'Arason. En général, cet invariant n'est pas défini pour les algèbres à involution. Mais on peut cependant définir un invariant d'Arason relatif, qui détecte l'hyperbolicité des involutions orthogonales. Sa définition repose sur la trialité, qui permet de le décrire très concrètement comme l'invariant d'Arason d'une certaine forme quadratique.

Speaker: Sanghoon Baek (Ottawa)
Title: Introduction to cohomological invariants and essential dimension
Abstract: We introduce the notion of essential dimension and cohomological invariants. In particular, we discuss a useful connection between cohomological invariants and the essential dimension. The talk is aimed at graduate students.

Speaker: Enric Ventura (Univ. Politécnica, Barcelona)
Title: The conjugacy problem for some extensions of F_n, Z^m, B_n, and F.
Abstract: We will review the main idea in the solution of the conjugacy problem (CP) for free-by-cyclic groups given by Bogopolski-Martino-Maslavova-Ventura in 2006. A close analysis of the argument, having the classical Miller's groups in mind (which are free-by-free and have unsolvable conjugacy problem), gave rise to a subsequent and stronger result by the same authors giving an explicit characterization of the solvability of the conjugacy problem within the family of free-by-free groups. It turns out that the freeness of the base group is irrelevant in the whole proof, and the only crucial property is the solvability of the so-called twisted conjugacy problem (TCP). We shall give characterizations of the solvability of the conjugacy problem for certain families of extensions of groups with solvable TCP. We will discuss the particular cases of extensions of free groups F_n, free abelian groups Z^m, Braid groups B_n, and Thomson's group F.

Speaker: Alistair Savage (Ottawa)
Title: Hecke algebras and a categorification of the Heisenberg algebra
Abstract: In this talk, we will present a graphical category in terms of certain planar braid-like diagrams. The definition of this category is inspired by the representation theory of Hecke algebras of type A (which are certain deformations of the group algebra of the symmetric group). The Heisenberg algebra (in infinitely many generators), which plays an important role in the description of certain quantum mechanical systems, injects into the Grothendieck group of our category, yielding a "categorification" of this algebra. We will also see that our graphical category acts on the category of modules of Hecke algebras and of general linear groups over finite fields. Additionally, other algebraic structures, such as the affine Hecke algebra, appear naturally.
We will assume no prior knowledge of Hecke algebras or the Heisenberg algebra. The talk should be accessible to graduate students. This is joint work with Anthony Licata and inspired by work of Mikhail Khovanov.

Speaker: Kirill Zaynullin (Ottawa)
Title: Cohomology of flag varieties
Abstract: This is an introduction to flag varieties and their basic geometric properties. We discuss their cohomology and provide examples of computations.

Speaker: Olivia Dumitrescu (University of California, Davis)
Title: Degenerations Techniques and Interpolation Problems
Abstract: We introduce the interpolation problem and related conjectures. We approach it using algebraic geometry techniques, by exploiting toric degenerations of the projective plane. Using non-toric degenerations of the plane we present progress in Nagata's conjecture for ten points. We also explain how to generalize the method to any number of points.