My research focuses on the analysis of probability models on self-similar and random graphs. The overarching goal is to prove limit theorems that describe various laws of nature that take place in fractal or random geometries, such as heat flow, wave propagation, charged particles in an electromagnetic field, and fluid dynamics.
I carry out two interconnected lines of research, one analytic and the other probabilistic. On the one hand, I analyze the eigenvalues and eigenfunctions of variants of the Laplace operator \(\Delta\) on fractal spaces. On the other hand, I study the scaling limits of random processes involving many interacting particles driven by Laplacian-like operators, representing models of magnetism or crystal growth.
My research in 1 PDF slide. Here is a 5-page description of my research, pitched at the general scientific audience.
All of my research papers can be found on the arXiv.
NEW! Spherical asymptotics in the shape of internal DLA on Sierpinski gasket graphs. (An informal 1-sentence summary of this paper: "You can grow a crystal on a fractal, and yet the crystal will be round almost surely." Illustrated by the above animation.)
Power dissipation on the Feynman-Sierpinski ladder circuit and the Hanoi tower circuits. We give a "fractal spin" on the classic infinite ladder circuit discussed in the Feynman Lectures on Physics. For the F-S ladder we can rigorously prove the convergence of the effective impedances using the dynamics of Möbius transformations. This is just one example of how undergraduates can contribute to mathematical research. A refinement of our results (concerning the energy measure) on the F-S ladder circuit was attained recently by Patricia Alonso-Ruiz.
NEW! From non-symmetric particle systems to non-linear PDEs on fractals (a review). This is a summary of a series of preprints concerning the hydrodynamic limit of the boundary-driven exclusion process on a resistance space, which will appear in the near future. The starting point is:
The moving particle lemma for the exclusion process on a weighted graph.
I prove in this paper an energy inequality for the exclusion process, which is analogous to Thomson's (or Dirichlet's) principle for random walks / electric networks.
It builds upon the marvelous "octopus inequality" proved by Caputo, Liggett, and Richthammer in 2009.
Current Colgate student trainees: Jonah Kudler-Flam '17 and Ilias Stitou '19.
In Spring 2017, Jonah and Ilias are working on numerical simulations of aggregation models on fractal graphs, in an attempt to better understand the (fluctuations of the) limit shapes of the aggregate clusters. Their projects build upon the spherical shape theorem for internal DLA on the graphical Sierpinski gasket, proved recently by Huss, Sava-Huss, Teplyaev, and myself. Source codes will be made public upon the conclusion of the projects.
I'm always looking for bright, self-motivated, and mathematically inclined students to do research with me. Contrary to what you might have heard, it does not take a genius to do math research. With the right skill sets and persistence, anyone can contribute to the solution of an open mathematical problem.
For the past few years I have led projects that address open math questions via numerical experiments: By writing and running programs that numerically simulate the given problem, one produces results or pictures that suggest the existence of certain patterns, and then aims to justify their existence through rigorous proofs. Depending on your background and training, you may choose to focus on the theory, the numerics, or both.
My research problems are drawn from the intersection of probability, statistical physics, and theoretical computer science. In particular I like studying random (drunkard's) walks and the associated (discrete) calculus on various graphs, and then discover the appropriate limit theorems under a suitable space-time scaling. (Why take limits? Read about the infinite monkey theorem, and maybe you'll be convinced why taking limits is a good idea.)
My philosophy: I take a hands-on approach to supervising research. Roughly speaking, I meet with research students once a week, to discuss any issues they have come across and to debrief them on the necessary theory. Beyond the weekly meetings, it is up to the students to organize their own schedules to make progress. Usually I have some intuition about what the expected results may be, but I am also ready to be proven wrong if you can demonstrate accurate numerics (or even better, supported by proofs)! If you produce good results, I will encourage you to give conference presentations, and then (possibly) write a research paper (with me or on your own). (That way you'll have something to put on your resumé when applying for grad schools or jobs!)
Doing research is a highly NONLINEAR process: expect bruises and bumps (not literally, but figuratively) along the way. But don't easily give up: work hard, be persistently creative, and chances are that you will experience that moment of eureka!
Suggested preparation: I expect some level of mathematical maturity developed through coursework in math. Thus a (concurrent) background in linear algebra (MATH 214) is a bare minimum. Ideally I would look for someone who has further taken a course on mathematical proofs (MATH 250) and/or mathematical computations (MATH 260), plus a 300-level course that uses calculus actively, such as differential equations (MATH 308) and probability (MATH 316). A solid background in physics or CS is also desirable. (Extensive programming experience is a real plus!)
How to get involved? I anticipate hiring 2 students in each of Fall 2017 and Spring 2018 to work on projects TBA. (Unfortunately I'm not hiring in Summer 2017 due to my own travel schedule.) Look for the job posting in the Student Employment section of the Colgate portal approximately 4 weeks prior to the start of the semester. Apply through the portal; be sure to describe your previous course and research experience. Successful candidates will be notified shortly before the start of the semester. (You can choose to work with me for either $$$ or credit through independent study, but not both.)
I am also happy to provide informal advice on attacking problems coming from mathematical analysis (real & complex), probability & stochastic processes, and mathematical physics (quantum mechanics & statistical mechanics).
During the academic year, I often attend the analysis seminar and the probability seminar at the Cornell math department.
March 2017: Seminar on Stochastic Processes, University of Virginia. (Contributed talk)
April 2017: Hudson River Undergraduate Mathematics Conference, Westfield State University (Westfield, MA). (Contributed talk)
April 2017: Finger Lakes Probability Seminar, Syracuse University. (Contributed talk)
May-June 2017: Research stay at Institut Henri Poincaré, Paris. Thematic quarter program "Stochastic Dynamics out of Equilibrium."
June 2017: 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals, Cornell University. (Conference co-organizer)
June 2017: Dynamics, Aging and Universality in Complex Systems, in honor of Gérard Ben Arous’ 60th birthday, Courant Institute, NYU.
July 2017: International Workshop on BSDEs, SPDEs and their Applications, University of Edinburgh. (Invited talk at special session)
July-August 2017: Research stay at Bielefeld University, Germany, sponsored in part by the Bielefeld-Seoul IRTG 2235. (Planned lectures TBA)
July-August 2017: Analysis & Geometry on Graphs and Manifolds, University of Potsdam, Germany.
September 2017: Northeast Analysis Network Conference, Syracuse University.
June 13-17, 2017: Bob Strichartz, Sasha Teplyaev, Luke Rogers and I are organizing the 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals.
March 19-20, 2016: Bob Strichartz, Sasha Teplyaev, Luke Rogers and I organized a special session at the AMS Spring Eastern Sectional Meeting at SUNY Stony Brook.
From Spring 2015 to Spring 2016, Matthew Badger and I co-organized the UConn Analysis & Probability Seminar, Fridays 3:15-4:15p.
In Summer 2015, I was one of the research mentors for the UConn math REU program, working mainly with the fractals group and the stochastic control group. I also organized the summer S.I.G.M.A. seminar (Fridays 12:15p~1:15p).
On July 28, 2015, Luke Rogers and I co-organized the 3rd Northeast Mathematics Undergraduate Research Mini-Symposium.
JPC's current and recent past research has been supported in part by the National Science Foundation (DMS-1262929) and the Research Council of Colgate University (discretionary grant, student wage grant, and a major grant for 2017~18).