Research

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.

Papers

• Eldredge, N. and Saloff-Coste, L. Widder's representation theorem for symmetric local Dirichlet spaces. Journal of Theoretical Probability (2013, in press).
  • Published: doi:10.1007/s10959-013-0484-1 (subscription needed).
  • Preprint: arXiv:1204.1926 (full text, free).
• Eldredge, N. Gradient estimates for the subelliptic heat kernel on H-type groups. Journal of Functional Analysis 258 (2010), pp. 504-533.
  • Published: doi:10.1016/j.jfa.2009.08.012 (subscription needed).
  • Preprint: arXiv:0904.1781 (full text, free).
• Eldredge, N. Precise estimates for the subelliptic heat kernel on H-type groups. Journal de Mathématiques Pures et Appliquées 92 (2009), pp. 52-85.
  • Published: doi:10.1016/j.matpur.2009.04.011 (subscription needed).
  • Preprint: arXiv:0810.3218.

Ph.D. Dissertation

• Eldredge, N. Hypoelliptic heat kernel inequalities on H-type groups. Ph.D. dissertation, University of California, San Diego, 2009. Advisor: Bruce Driver. You can download the PDF.

Other writings

• 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).

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.
• 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 interface.
• 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. (Cornell does.)
• 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.