Define web := "http://www.math.toronto.edu/drorbn/Profile".
Over the next five years I plan to pursue following four projects:
* Knot Theoretic Algebra. There is a machine
that I like to call "projectivization", which takes certain problems in
knot theory and related topics, mostly related to finite type
invariants and their interaction with various topological operations, and
turns them into problems in graded algebra, often related to Drinfel'd
associators and other quantum-group-related constructs. A surprising amount
of algebra is in the image of the projectivization machine: associators (web/nat, web/ktgs), the
Grothendieck-Teichmuller group (web/gt), the
Kashiwara-Vergne conjecture (web/wko), and
more. One of my primary goals will be to show that the same is true for the
work of Etingof and Kazhdan on the quantization of Lie bialgebras. In some
sense this will mean that the theory of quantum groups not only has
applications to topology, but in fact, _it is_ topology.
* Algebraic Knot Theory. One output
of the "projectivization machine" is an invariant of knotted
objects which is well behaved ("homomorphic") with respect to
certain topological operations. This gives a handle towards the
study of topological properties that are expressible in terms of
those operations; these include the genus and unknotting number
of a knot, and whether or not it is ribbon. I plan to study
these said properties using such homomorphic invariants (web/akt).
* Computations in Knot and Tangle
Theory. I plan to continue developing the Knot Atlas (web/kat)
and the computer package KnotTheory` (web/kt). The
Rolfsen table of knots and subsequent enumerations by Hoste and
Thistlethwaite had enormous impact on knot theory; yet knots
are the "finished products" and the basic ingredients they are
made of, tangles, where never properly enumerated. I believe (web/infra)
an enumeration of tangles along with a "Tangle Atlas" of invariants
would have a comparable value, and I hope to create or participate in
the creation of that tool.
* An Alexander Knot Homology. Recently (web/regina) I
found a simple extension of the Alexander polynomial of knots to virtual
knots and virtual tangles, which has excellent behaviour with respect to
tangle compositions and which at all times remains of polynomial size
(similar previous extensions have target spaces of exponential size). There
are reasons to hope that my extension can be categorified in the spirit of
my extension of Khovanov homology to tangles (web/cob). Success
may reduce the known Alexander homologies from analytic-combinatorial
mysteries to natural algebra, and as in the Khovanov case (web/fast) may
speed up computations by an exponential amount.
In addition, I am committed to total communication and absolute
openness. Every handout I have distributed is on the web (web/portfolio),
videos of dozens of talks I gave are on the web (web/talks),
and my blackboard (web/bbs),
and even budget (web/budget)
are open. Proper maintenance of this requires time and money.
In science the predicted is often less interesting than
the unpredictable. With luck, at least some of my work in the next five
years will be on topics I haven't yet heard of.
This summary is at web/sum12.