# Algebraic Geometry, Group Cohomology, Representation Theory (AG-GC-RT)

### Overview

Algebraic geometry is a mathematical discipline which uses the techniques and tools of algebra (e.g. rings, ideals and fields) to attack geometric problems. The fundamental objects which algebraic geometers study are algebraic varieties, the common zeros of a collection of polynomials. In the last four decades, beginning with the ground breaking work of Alexandre Grothendieck, the discipline has undergone phenomenal growth and has had a profound influence on the development of modern mathematics. Many of its celebrated works have led to Fields Medals: the proofs of the Weil Conjectures by Deligne, Mumford's work on geometric invariant theory, Hironaka's work on the resolution of singularities, Mori's work on the classification of algebraic varieties in dimension three and Wiles' proof of Fermat's Last Theorem which used arithmetic algebraic geometry. Furthermore, the work of Kazhdan, Lusztig, Kashiwara and others has made algebraic geometry an indispensable tool for representation theory. In the last fifteen years, exciting new connections between algebraic geometry and physics emerged, which led to unexpected new mathematical theories such as mirror symmetry and quantum cohomology and to many important developments in the field of mathematical string theory.

For the most part, these advances have been brought about by the fact that algebraic geometry poses intrinsically interesting and relevant problems, and has the property of developing the mathematical tools to solve them. It has therefore attracted many talented mathematicians, many of whom are not formally trained in the area, but have realized its value. This has further stimulated new connections between algebraic geometry and other disciplines: e.g. combinatorics, cryptography, statistics, and quantum computing.

Algebraic geometry has also given us new insight into the nature of algebraic groups and Galois cohomology. During the last two decades many exciting fundamental theorems have been established due to the introduction of new powerful techniques from algebraic topology and algebraic geometry. For instance, Voevodsky's use of homotopy and cobordism theory have resulted first in the solution of Milnor conjecture and, more recently, the Bloch-Kato conjecture. Further development of these ideas is crucial.

The PIMS CRG has many people working in the cutting edge in several of the above areas. Among the specialties represented by our varied group are algebraic stacks, geometric invariant theory, algebraic group actions, toric varieties and torus actions, algebraic cycles, Gromov-Witten theory, arithmetic algebraic geometry, classification theory, algebraic representation theory, Lie theory and Schubert varieties, group cohomology.

### PIMS Distinguished Chair

- Michael Thaddeus (Columbia, New York), September 2005 - August 2006

### Faculty

CRG Leaders:

- Arturo Pianzola (Alberta)
- Jim Bryan (UBC)

SFU:

U. Alberta:

- Xi Chen
- Gerald Cliff
- Vladimir Chernousov
- Terry Gannon
- Jim Lewis
- Arturo Pianzola

UBC:

- Alejandro Adem
- Kai Behrend
- Jim Bryan
- Jim Carrell
- Bill Casselman
- Kalle Karu
- Dale Peterson
- Zinovy Reichstein

U. Calgary:

U. Washington:

- Eric Babson
- Sara Billey
- Chuck Doran
- Amer Iqbal
- Sandor Kovacs
- Paul Smith
- Rekha Thomas
- James Zhang

**Postdoctoral Fellows**

Postdoctoral fellows associated with the algebraic geometry group include Jacob Shapiro (PIMS-UBC), Anca Mustata (UBC), Andrei Mustata (UBC).

Postdoctoral fellows associated with the cohomology/representation theory group include Jochen Kuttler (PIMS-UBC) and Kevin Purbhoo (UBC).

Two CRG sponsored PIMS postdoctoral fellows joined the group in the fall of 2005: Hsian-Hua Tseng (UBC) and Iulia Pop (U. Alberta). Shuang Cai (U. Alberta) has joined the CRG in the fall of 2006.

**Visitors**

- J. Carlson (U. Georgia)
- P. Gille (CNRS, Universite Paris-Sud)
- D. Harari (ENS Paris)
- Canon Leung (University of Science and Technology, Hong Kong)
- D. Maulik (Princeton)
- Jan Minac (U.Western Ontario)
- M. Roth (Queens)
- S. Smith (U. Illinois at Chicago)
- G. Soifer (Bar - Ilan University, Ramat Gan, Israel), March 2005: The Auslender Conjecture: history, results and open problems.
- O. Mathieu (University of Lyon I, France), May 2005: On the homotopy of geometric quotients.
- M. S. Raghunathan (Tata Institute of Fundamental Research, Mumbai, India), May 2005: Imbedding quasi-split groups of equal rank in isotropic groups.
- I. Panin (Steklov Institute, S. Pitersburg, Russsia), September 2005: A purity theorem for linear algebraic groups.
- Yongbin Ruan (Wisconsin-Madison), August 2-5 2005, UBC
- Steven Mitchell, Feb 22, 2006
- Ching-Li Chai, March 7-10, 2006
- K. Zainoulline (Bielefeld University, Germany), March 2006: Motivic decompositions of projective homogeneous varieties.
- A. Vishik (Independent University, Moskow, Russia), September 2006
- Jesper Grodal, Nov. 27 to Dec. 16, 2006