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Topology Seminar, Spring 2003

The topology seminar will meet on Wednesdays 2-3:30 pm, in Room 226 Goldsmith Hall. The plan is to have a learning seminar on Geometric Group Theory. A very brief introduction and some references are given at the bottom of the page.

Schedule Spring 2003
Date Speaker/Title
April 30 Daniel Ruberman
Survey of Thurston's geometrization conjecture
Abstract: This will be a survey, for non-specialists, of the program put forth by Thurston in the late 70's for understanding all 3-manifolds. I will try to give a complete statement of Thurston's conjecture, and some idea of its implications, including the Poincare conjecture. As many people know, Grisha Perel'man has been lecturing on a proof of this conjecture using analysis and geometry (Ricci flow). I will not discuss Perel'man's work, except perhaps in passing.
Background: Basic knowledge of manifolds.
April 16,23
Spring Break
April 9
No Seminar (Perel'man talk at MIT)
April 2 Stefan Friedl
Introduction to CAT(0) spaces
March 26 Dmitry Kleinbock
Groups of Polynomial Growth (after Gromov)
March 19 Seminar cancelled.
March 12 Dmitry Kleinbock
Asymptotic invariants of infinite groups (after Gromov)
Abstract: The purpose of this talk is to go through a list of properties of finitely generated groups which are or are not invariant under quasi-isometries. Everyone is welcome and attendance of past meetings of the seminar is not required to follow the talk.
March 5 Spring Break; no seminar.
February 26 Daniel Ruberman
Groups acting on Graphs (after Bass-Serre).
February 12 Vladimir Chernov (Dartmouth)
Vassiliev invariants of Legendrian, Pseudo-Legendrian and framed knots in contact 3-manifolds.
February 5 Ophir Feldman
Definitions of hyperbolic groups.
January 29 Ophir Feldman
Introduction.

Introduction: A group with a set of generators can be regarded as a very symmetric metric space, using the `word metric'. This metric measures the difference betweem two elements by counting the number of multiplications by generators it takes to get from one to the other. Although groups are basically discrete objects, the geometry of this word metric is very rich. One can define notions of curvature and geodesics, and it turns out that these geometric notions reflect many deep algebraic properties of the group. A classical example is Dehn's use of the hyperbolic metric on a surface to solve the word problem (when a word in the generators is equal to the trivial element). The subject has exploded in the last 20 years, prompted by work of Gromov, Thurston, Cannon, and many others. We will try to learn some basic ideas and example, and generally get a feel for the subject.

Another introduction: Read the Mathematical Review of de la Harpe's book.

A few references:

  • Cannon, J. W. Geometric group theory. in Handbook of geometric topology, 261--305, North-Holland, Amsterdam, 2002.
  • Combinatorial and geometric group theory. Papers from the AMS Special Sessions on Combinatorial Group Theory and on Computational Group Theory held in New York, November 4--5, 2000 and in Hoboken, NJ, April 28--29, 2001. Edited by Sean Cleary, Robert Gilman, Alexei G. Myasnikov and Vladimir Shpilrain. Contemporary Mathematics, 296.
  • de la Harpe, Pierre Topics in geometric group theory. University of Chicago Press, Chicago, IL, 2000
  • Bridson, Martin R.; Haefliger, Andre Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999
  • Gromov, M. Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2 (Sussex, 1991), 1--295, London Math. Soc. Lecture Note Ser., 182,
  • Gromov, M. Hyperbolic groups. In Essays in group theory. Edited by S. M. Gersten. Mathematical Sciences Research Institute Publications, 8. Springer-Verlag, New York, 1987
  • Gersten, S. M. Introduction to hyperbolic and automatic groups. In Summer School in Group Theory in Banff, 1996, 45--70, CRM Proc. Lecture Notes, 17, Amer. Math. Soc., Providence, RI, 1999.

This page was last modified on September 02, 2005