Math 285F: Khovanov homology
Instructor: Sucharit Sarkar
MWF 11-11:50am, MS 6118.
By appointment, MS 6909.
, MyUCLA gradebook
We will study Khovanov homology, which is a very
modern invariant of knots and a categorification of the famous Jones
polynomial. Jones polynomial was one of the first invariants of knots
which was not geometrically defined and its precise geometric meaning
is still a mystery. Khovanov homology lifts the Jones polynomial one
level higher, and discovers surprising connections between
representation theory and knot theory. We will also study some
applications of these invariants, such as Kauffman's proof of Tait's
conjecture about alternating knots or Rasmussen's alternate proof of
Milnor's conjecture, first proved by Kronheimer-Mrowka, about torus
You should be familiar with the basics of topology (and
ideally a little bit about knot theory) and algebra (such as notions
of chain complexes). Anything else that we need, we will cover them in
class; so this will be a good learning opportunity for many modern
mathematical techniques. Since this a very new subject, there are no
good books written on the topic, so the lectures will follow papers
instead. Here is a tentative list of papers (a subset of) which we
plan to cover.
Exams and grading:
There will be a final take-home exam for the
undergraduate students who require a grade at the end of the course,
and the grade will be recorded in the MyUCLA gradebook. If you believe
that you have been graded incorrectly, or that your score was not
correctly recorded in the MyUCLA gradebook, you must bring this to the
attention of the instructor before the end of the quarter
(3/24). Grading complaints not initiated within this period of time
will not be considered. Please verify in a timely manner that your
scores are correctly recorded on MyUCLA.