Special Colloquium
Karen Vogtmann
Cornell University
Symmetries, ping-pong and Outer space
Abstract: The symmetries of any geometric object form a system which obeys a
small set of natural ``rules." Mathematicians have observed that
systems obeying the same set of rules occur throughout mathematics,
though with wildly different descriptions. Since such systems, called
groups, lie at the heart of many calculations in mathematics and
the sciences, we are very interested in trying to understand them
better. The branch of mathematics in which I work, called
geometric group theory, studies abstractly defined groups by trying
to find concrete geometric objects whose symmetry groups are
basically the same as the group under investigation. A close look at
such an object often reveals characteristics of the group which were
not obvious from the original description. A nice aspect of geometric
group theory is that the search for such objects and attempts to
visualize them often lead to very beautiful mathematical imagery.
One of the simplest of all types of groups are the so-called free
groups; here the word free can be interpreted as saying that
the group is free of any complicated internal structure.
It is often very useful to be able to recognize when a group which
looks like it might be very complicated is in fact secretly just a
free group. One very appealing way to solve this problem is to find
an appropriate geometric object, which we then use as a table on which
to play mathematical ``ping-pong"; the reason for this name becomes
clear after seeing an example. Much of my work involves trying to
understand symmetries of free groups. I will describe an
appropriate
ping-pong table, known as Outer space, which can sometimes be
used to settle this question.
Reception at 3 pm in SEO 300. This talk is sponsored by WISEST.
Monday October 13, 2008 at 4:00 PM in Lecture Center D4