Academic Profile
My Curriculum Vitae (PDF) is usually no more
than a year out of date. You can also read my News and Merit Reports: 2009, 2010, 2011.
- Short bio, for the Canada/USA Mathcamp (December 2008):
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/.
- NSERC Research Proposal, (PDF,
2007).
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? [...]
- Summary of Proposal for Public
Release, (HTML, 2007).
[...] 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.
- Summary of Proposal, (HTML,
2007).
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.
- Research Statement (PDF,
2005).
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.
- Teaching Philosophy
(PDF, 2005).
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.
- Research
Proposal (HTML, 2002).
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.
- Research Plan
(HTML, 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
[structure, pictorial simplicity, computer
assisted mathematics and weblications]