Academic Profile

I believe math is too deep. Rather than making it deeper, a better use of my time would be to make some deep ends easier and more accessible. I believe math is too abstract, or at least appears to be too abstract, for much of what may be computed hardly ever is. Thus, whenever I can, I code. Yet I have sinned a few times and written on deep math that was not accompanied with programs. I usually work on knot theory and its surprising relationship with algebra, geometry and quantum field theory. I got my Ph.D. at Princeton, did time at Harvard, Hebrew U., Berkeley and MSRI, and I now work at the University of Toronto. Practically everything I've ever done (including even this paragraph) can be found somewhere on my website, starting at http://www.math.toronto.edu/~drorbn/.

I'll start with the question, and then try to put some meaning into it.
Question. Sometimes a bit of algebra turns out to be a bit of topology, in disguise. Is that true for the theory of quantum groups? [...]

[...] Number theory just seems to be related to everything.
Likewise, though on a smaller scale, many knot theorists such as myself care little about shoelaces, yet care a lot about the unexpected ways by which the study of knotted shoelaces is intricately and deeply related to such a priori remote subjects as 3-dimensional manifolds, hyperbolic geometry, quantum field theory, differential geometry, Lie theory and representation theory, quantum algebra, combinatorics, homological algebra and sophisticated algorithmics.

What about Khovanov homology? I made significant contributions to the highly fashionable subject of Khovanov homology (in fact, while Khovanov is definitely the father of the field, I share the credit for making it fashionable...). Yet at the moment I don't feel mature enough to study this topic any further. I'd rather "categorify" knot invariants only after I properly understand the "algebra" on which they ought to be defined (in the sense of my first project above). And how can I even start categorifying other aspects of the theory of quantum groups, when in my opinion this theory in itself is so poorly understood (at least in the sense of my second project)? With luck, at the end of this grant period I will be ready to return to Khovanov homology and categorification in general.

We consider computers to be outside of our field rather than a part of it, hence most of us know nothing about them. We (as a group) are happy when an undergrad writes a program to compute something for us; this done, we are happy to forget the program and use only the results. Hence a coherent uniform framework for mathematical computation does not yet exist. So every time I try to compute some complicated homology, I have to teach my computer linearity, Gaussian elimination, and tensor products practically from scratch. [...] for most mathematicians and most students of mathematics the entry barriers are way too high, their education is largely irrelevant and they get no credit for the effort. Hence so many math papers describe what amounts to be an algorithm, and so few actually implement it.

I'm a horrible student and a horrible listener. It's very difficult for me to pay attention in talks and classes; for the least reasons my mind wanders and I'm totally lost. And when I'm lost, I'm lost for good, for usually I haven't the will power to concentrate again and pick up from the last I followed.

My current computer program for computing Khovanov homology is extremely inefficient and the main reason for that is inherently mathematical - as it is, Khovanov's chain complex is just too big. An indication for that is the fact that the rank of the homology is invariably much smaller than the dimensions of the spaces of chains involved. I believe I can do a lot better by mixing some homological algebra and some sophisticated programming, and I hope to do so sometime over the grant period.

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 [structure, pictorial simplicity, computer assisted mathematics and weblications]