|
Abstract: Group representations are
fundamental objects
of study in mathematics. They allow mathematicians (and
chemists and physicists, for that matter) to think of an
abstract group in terms of linear algebra—rotations,
reflections, and so forth. Given two representations,
one might ask whether it is possible to draw a "continuous
path" from one to the other. Using some
algebraic topology,
we can assign to each representation an object called a
"Toledo invariant." We prove that if two representations have
different
Toledo invariants, then it is impossible to draw a "continuous
path" from one to the other. Surprisingly, one
can (in some cases) use differential geometry and algebraic geometry
to compute
all the Toledo invariants that occur.
This talk will make use of the new color-coded "Prerequisite Alert
Level" system.
|