Mathematics 600, Essential dimension

Term 1, 2009-10, TuTh 14:00 - 15:20, Math Annex 1118

  • Instructor : Zinovy Reichstein
    Office: 1224 Math Annex
    Phone: 2-3929
    E-mail: reichst at math dot ubc dot ca

  • Course description:
    The essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. This notion, at the crossroads of algebra, algebraic geometry and Lie theory, was introduced in the late 1990s, and has been extensively researched since then. It has rich connections to various classical problems, and can be studied by a range of algebraic, cohomological and geometric techniques.
    The course is intended to survey these techniques and related areas of algebra, using essential dimension as a unifying theme. I plan to expand on the 5-lecture mini-course on this topic I gave at the workshop on Essential and Canonical Dimension in June, 2008. The preliminary version of the notes I wrote then can be found here. The style of these notes is a cross between a survey article and an advanced textbook. In particular, it includes a set of exercises after each section. In the course I intend to fill in some of the background material on quadratic forms, central simple algebras, algebraic geometry, algebraic groups, Galois cohomology, and invariant theory.

  • Prerequisites:
    The only hard prerequisite is a firm grasp of abstract algebra (Math 501/502 or equivalent) but prior exposure to algebraic geometry will be helpful.

  • Marking:
    This is an advanced topics course. There will be no exams. I will assign homework problems from time to time. Registered students are expected to attend the lectures on a regular basis.

    Problem set 1. Due Tuesday, October 6.

    Problem set 2. Due Tuesday, October 20

    Problem set 3. Due Tuesday, November 10

    Problem set 4. Due Tuesday, December 1 (Problem 2(c) reworded November 27)

    Bibliographical references for the rationality problem.

    [1] H.W. Lenstra, Jr. Rational functions invariant under a finite abelian group. Invent. Math. 25 (1974), 299--325. MR0347788 (50 #289)

    [2] Swan, Richard G.(1-CHI) Noether's problem in Galois theory. Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982), 21--40, Springer, New York-Berlin, 1983. MR0713790 (84k:12013)

    [3] David J. Saltman, Noether's problem over an algebraically closed field. Invent. Math. 77 (1984), no. 1, 71--84. MR0751131 (85m:13006)

    [4] Arnaud Beauville, Jean-Louis Colliot-The'le`ne, Jean-Jacques Sansuc, Peter Swinnerton-Dyer, Varie'te's stablement rationnelles non rationnelles. (French) [Nonrational stably rational varieties] Ann. of Math. (2) 121 (1985), no. 2, 283--318. MR0786350 (86m:14009)

    [5] Lieven Le Bruyn, Centers of generic division algebras, the rationality problem 1965--1990. (English summary) Israel J. Math. 76 (1991), no. 1-2, 97--111. MR1177334 (93f:16024)