Time and Place: Thursday January 3rd, 14:30-15:30, Mathematics
Building, Lecture Hall 2
Speaker: Dror Bar Natan (Toronto)
Title: Meta-Groups, Meta-Bicrossed-Products, and the Alexander
The a priori expectation of first year elementary school students who
were just introduced to the natural numbers, if they would be ready
to verbalize it, must be that soon, perhaps by second grade, they'd
master the theory and know all there is to know about those numbers.
But they would be wrong, for number theory remains a thriving
subject, well-connected to practically anything there is out there in
I was a bit more sophisticated when I first heard of knot theory. My
first thought was that it was either trivial or intractable, and most
definitely, I wasn't going to learn it is interesting. But it is, and
I was wrong, for the reader of knot theory is often lead to the most
interesting and beautiful structures in topology, geometry, quantum
field theory, and algebra.
Today I will talk about just one minor example, mostly having to do
with the link to algebra: A straightforward proposal for a
group-theoretic invariant of knots fails if one really means groups,
but works once generalized to meta-groups (to be defined). We will
construct one complicated but elementary meta-group as a
meta-bicrossed-product (to be defined), and explain how the resulting
invariant is a not-yet-understood yet potentially significant
generalization of the Alexander polynomial, while at the same time
being a specialization of a somewhat-understood "universal finite
type invariant of w-knots" and of an elusive "universal finite type
invariant of v-knots".
Light refreshments will be served after the colloquium in the faculty
lounge at 15:30.
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