last updated: Oct. 19, 2002
The phrase ``research plan'' is almost an oxymoron, for if it can be
planned, it ain't research. Or at least, if it can be planned it's already
half done, and hence it isn't the best of research.
Thus my primary plan of mathematical research for the next few years is to
follow my nose. The scents that usually attract me most are
My backup plans are (in case I become anosmic, or if today's strong smells
remain dominant in the future):
- I feel there is a great need for a foundational book on finite
type invariants that will summarize the relationship between those
and Lie algebras, associators, Feynman diagrams and the Chern-Simons
theory, etc. I've been meaning to write such a book, but I still find
that even the foundations of the theory are not yet fully settled,
and hence a pre-condition for writing it is to do some foundational
research. Just as an example -- along with D. P. Thurston I'm trying to
rewrite Drinfel'd's theory of associators and the classical theory of
6j symbols in terms of knotted trivalent graphs, tetrahedra and
Pachner moves. Much is already done but much remains to be done. See
e.g. . Much foundational work also remains
regarding Feynman diagrams and configuration space integrals. And then
there's writing the book.
- I plan to continue following Khovanov's novel ``categorification''
- On the Vassiliev knot invariants,
Topology 34 423-472 (1995).
- Non-associative tangles,
in Geometric topology (proceedings of the Georgia international
topology conference), (W. H. Kazez, ed.), 139-183, Amer. Math.
Soc. and International Press, Providence, 1997.
- Polynomial invariants are polynomial,
Mathematical Research Letters 2 (1995) 239-246.
- (With S. Garoufalidis) On the Melvin-Morton-Rozansky conjecture,
Inventiones Mathematica 125 (1996) 103-133.
- Lie algebras and the Four Color Theorem,
Combinatorica 17-1 (1997) 43-52.
- (With A. Stoimenow) The fundamental theorem of Vassiliev invariants,
in Proc. of the Århus Conf. Geometry and physics,
(J. E. Andersen, J. Dupont, H. Pedersen, and A. Swann, eds.), lecture
notes in pure and applied mathematics 184 (1997) 101-134,
Marcel Dekker, New-York.
- On Associators and the Grothendieck-Teichmuller Group I,
Selecta Mathematica, New Series 4 (1998) 183-212.
- (With S. Garoufalidis, L. Rozansky and D. P. Thurston)
The Århus integral of rational homology 3-spheres I: A highly
non trivial flat connection on ,
Selecta Mathematica, to appear.
- (With M. Bar-Hillel, G. Kalai and B. D. McKay)
Solving the Bible Code Puzzle,
Statistical Science 14-2 (1999) 150-173.
- (With R. Lawrence) A Rational Surgery Formula for the LMO Invariant,
Israel Journal of Mathematics, to appear.
- On Khovanov's categorification of the Jones polynomial.
- (With D. P. Thurston) Algebraic Structures on Knotted Objects and Universal Finite Type
- Other papers by Dror Bar-Natan.