Current Research Interests
My research lies at the boundary of probability theory and partial
differential equations, in the area of heat kernel analysis. I am
particularly interested in the study of heat kernels, elliptic and
parabolic PDE, and corresponding stochastic processes, in "unusual" settings
such as sub-Riemannian manifolds, infinite
dimensional spaces, and abstract structures such as Dirichlet spaces.
I have written a research statement (PDF) with a more detailed description.
My papers are published under my full name, Nathaniel
- LATEX is the
language of written mathematics. It's free too. Many, many useful
things for LATEX and TEX can be found on CTAN.
- Asymptote is a
powerful graphics language for producing accurate and beautiful
mathematical diagrams. I used to use MetaPost for such tasks,
but am now converted to Asymptote, which retains MetaPost's most
powerful concepts, and adds many more, while providing a much cleaner
and more understandable syntax.
- GAP is a very powerful
computer program for computational discrete algebra. And it's free.
- Gnuplot is a utility for
plotting functions and data. Very useful for visualizing functions
and verifying computations. It is free software.
- Maple is
a commercial computer algebra system, good for working out and
verifying algebraic manipulations and calculus-type computations.
- FriCAS is a free
computer algebra system, based on the Axiom project. IMHO it has
a much more elegant and "mathematical" design than commercial tools
like Maple or Mathematica, and its design goals are breathtaking. TeXmacs makes a convenient
- Sage is a free open-source
mathematics software system that I have recently been hearing good
things about. I plan to check it out in the near future.
- MathSciNet is an
online database of math papers. A gigantic number of papers are
available through here; it's a good place to look for research on a
topic. There are abstracts, reviews, and often links to the papers
themselves. This service is provided by the American Mathematical Society. It
requires a subscription, but your institution might have one.
- arXiv is an online archive of
preprints in mathematics, physics, computer science, and related
fields. It helps authors to make their papers available during the
(potentially lengthy) gaps between completion, acceptance, and
- MathWorld is a
comprehensive free online encyclopedia of mathematics. Just about
every major concept in mathematics can be found here. A good place to
take those questions you were too embarrassed to ask (e.g. "what the
heck is an Artinian
ring?"). The site is created and maintained by Eric
- PlanetMath is in a similar
vein, but instead of being written by one person, it's more
collaborative. Anyone can submit articles. It's not as complete, but
it is constantly growing. One advantage is that PlanetMath articles
tend to include proofs as well as theorems.