Dartmouth 2006 Kemeny Lecture Series MATH

 Alan Weinstein

U. C. Berkeley

Photo by Margo Weinstein
(photo by Margo Weinstein)

will give the following series of lectures

Groupoids as generalized equivalence relations

Thursday, April 27, 2006
L01 Carson Hall, 4 pm
Tea 3:30 pm, Math Lounge

Abstract: The idea of what constitutes a "space" in geometry and analysis has evolved beyond the traditional set with a structure. In particular, the quotient X/G of an ordinary space X by the action of a group G is best represented not by the set of orbits, but by a groupoid which encodes the action.

When the group action is proper, its main "defect" is the presence of isotropy subgroups at special points of X. The quotient object is known as a stack or, when X is a smooth manifold and the isotropy groups are finite, an orbifold. In differential geometry and measure theory, it is frequently the case that the group action has dense orbits, and the quotient object is sometimes called a {\em quantum space}. The latter name arises because the algebra of functions on X/G is not taken to be the algebra of G-invariant functions on X, but rather a noncommutative (i.e. "quantum") algebra called the crossed product of G with X, built from the representing groupoid.

If X is a group and G is a dense normal subgroup, acting by translations, the quotient is a group, and the group multiplication should be encoded in the algebra A of G-invariant functions on X as a coproduct homomorphism A\mapsto A\otimes A, making this algebra into a Hopf algebra. But when we take A to be the crossed product algebra, it turns out that the appropriate encoding of the quotient group structure is a new structure, that of a "hopfish algebra". The quotient of the circle group U(1) by a dense cyclic subgroup provides a notable example.

Note: This talk will be accessible to graduate students.

Symmetry groupoids of crystals and networks

Thursday, April 27, 2006
Filene Auditorium, 7 pm

Abstract: Symmetry in nature is traditionally described mathematically in terms of the action of groups, such as the symmetry group of a crystal. But some materials, built of highly symmetric layers which may be stacked in an asymmetric way, admit only partial symmetry, for which the appropriate mathematical description is a more general object known as a groupoid. Groupoid symmetry has also been used to analyze coupled networks, such as those which control animal locomotion. In this talk, we will introduce the basic definition and elementary theory of groupoids, motivated by these examples from crystals and networks. No knowledge of standard group theory is assumed.

Note: This talk will be accessible to undergraduates.

NB: A PDF version of this announcement (suitable for posting) is also available.

The structure of proper groupoids

Friday, April 28, 2006
L01 Carson Hall, 4 pm
Tea 3:30 pm, Math Lounge

Abstract: The study of groupoids is limited by the absence of major structure theorems. In this talk, I will describe some progress (by Zung and myself) toward the understanding of groupoids in differential geometry, in which a linearization theorem shows that proper groupoids are built by gluing together proper actions of groups.

This talk will be accessible to graduate students.