© | Dror Bar-Natan: Academic Profile:

Summary of Proposal, 2012

(submitted to the NSERC, July 2012)

In the space provided below, state the objectives of the proposed research program and summarize the scientific approach, highlighting the novelty and expected significance of the work to a field or fields in the natural sciences and engineering. [...] Your summary must not exceed forty-five lines on the printed copy. [...]

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), notebook (web/pensieve), grants (web/here), 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.