## ABC Algebra Workshop

- Start Date: 04/08/2006
- End Date: 04/10/2006

Skip Garibaldi

Vladimir Chernousov

Alfred Weiss

Jan Minac

Alejandro Adem

University of British Columbia

**Speaker**: Skip Garibaldi

**Title**: Cohomological invariants

**Abstract**: The most familiar example of a cohomological invariant is the discriminant of a separable field extension or of a quadratic form. The subject of this talk is the fact that for many interesting objects--e.g., quadratic forms, etale algebras, and F_4-torsors--one can explicitly describe all the cohomological invariants. This talk is based on Serre's portion of the book "Cohomological invariants in Galois cohomology".

**Speaker**: Vladimir Chernousov

**Title**: Connectedness of classes of fields and zero cycles on projective homogeneous varieties.

**Abstract**: In the talk we present a new characteristic free uniform method of computing the Chow group of zero cycles on projective homogeneous varieties based on an idea of parametrization of splitting fields. These results were obtained jointly with A. Merkurjev.

**Speaker**: Patrick Brosnan

**Title**: Motives of Feynman diagrams

**Abstract**: For a graph G, Kirchhoff associated a homogeneous polynomial P_G (x) in as many variable as there are edges of the G and with degree equal to the first betti number of G. Motivated by quantum physics and the Weil conjectures, Kontsevich proposed the conjecture that there is a one-variable polynomial f_G such that the number of solutions of P_G over the field with q elements is given by f_G(q). Prakash Belkale and I disproved this conjecture by showing that the motives of the varieties defined by the P_G are, in a precise sense, generic. I will discuss our theorem, its proof and its aftermath.

**Speaker**: Alfred Weiss

**Title**: Galois invariants and p-adic L-functions

**Abstract**: A motivating heuristic principle on the structure of algebraic number fields is that their zeta functions know eveything about them. Perhaps the most convincing example of this phenomenon is the Main Conjecture of Iwasawa theory, which was proved by Mazur and Wiles. We present a refinement of this Main Conjecture, which can be formulated even in the non-abelian case, and discuss its relation to pseudomeasures. This is joint work with J. Ritter.

**Speaker**: Jan Minac

**Title**: Galois groups of maximal p-extensions and Galois modules

**Abstract**: The structure of absolute Galois groups of general fields is still a mystery. It is known epsilon more about the structure of their maximal pro-p-quotients but key questions are not yet solved. On the other hand recent advances in Galois cohomology should help. In this talk I will concentrate on the structure of Galois cohomology viewed as Galois modules over cyclic Galois p-groups. This work is a part of a program being pursued together with Adem, Karagueuzian, Labute, Lemire, Schultz, and Swallow.

**Speaker**: Alejandro Adem

**Title**: Commuting Elements and Spaces of Homomorphisms

**Abstract**: In this lecture we discuss basic topological properties of spaces of homomorphisms Hom(Q,G) where Q is a finitely generated discrete group and G is a Lie group. In particular, when Q is a free abelian group of rank n, this space can be thought of as the ordered commuting n-tuples in G. We describe the cohomology and stable homotopy type of the ordered commuting pairs and triples in SU(2).