Research

Current Research Interests

My research is in the field of geometric analysis. Currently, I am studying problems related to PDEs on Lie groups, in particular subelliptic heat equations. I am also interested in applying tools from probability theory to these problems, such as infinite-dimensional Malliavin calculus.

I have written a research statement (PDF) with a more detailed description.

For general (non-mathematician) audiences, I am working on a simple introduction to the subject. (Under construction.)

Papers

• Eldredge, N. Gradient estimates for the subelliptic heat kernel on H-type groups. Submitted, J. Funct. Anal, 2009. Preprint (full text) available at http://arxiv.org/abs/0904.1781.
• Eldredge, N. Precise estimates for the subelliptic heat kernel on H-type groups. J. Math. Pures. Appl. 92 (2009), pp. 52-85. Available online at http://dx.doi.org/10.1016/j.matpur.2009.04.011 (subscription may be needed). Preprint (full text) available at http://arxiv.org/abs/0810.3218.
• Undergraduate thesis: Eldredge, N. An eigenspace approach to isotypic projections for data on binary trees. My advisor was Prof. Michael Orrison of the Department of Mathematics at Harvey Mudd College. You can read the abstract, or download the complete thesis as PDF or LaTeX source (tar.gz).

Past Presentations

• "Hypoelliptic heat kernels on Lie groups," PDE seminar, Purdue University, May 2009.
• "Subelliptic heat kernels," Oliver Club, Cornell University, February 2009.
• "Precise estimates for the subelliptic heat kernel on H-type groups," Joint Mathematical Meetings, Washington, D.C., AMS Session on Partial Differential Equations, III, January 2009. Official listing, abstract (PDF).
• "Subriemannian geometry and probability." PIMS Summer School in Probability, Vancouver, B.C., June 2008.
• "Rough paths for probabilists." Graduate Student Conference in Probability, Madison, Wisc., Apr. 2007.
• "Efficient eigenspace projections with compression." Joint work with Ruben Arenas (Harvey Mudd College) and Michael E. Orrison (Harvey Mudd College). AMS Special Session on Research in Mathematics by Undergraduates, Joint Mathematical Meetings, Baltimore, January 2003.

Links

• L^ATEX is the language of written mathematics. It's free too. Many, many useful things for L^ATEX and TEX can be found on CTAN.
• 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 interface.
• 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. (UCSD people working from off campus can use the proxy server.)
• The ISI Web of Science is another huge database for research papers throughout science, including mathematics. It requires a subscription, but your institution might have one. (UCSD people working from off campus can use the proxy server.)
• 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 publication.
• 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 Weisstein.
• 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.