(photo by Margo Weinstein)

will give the following series of lectures

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.

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.

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.