- The 1st non-constant Neumann Laplacian eigenfunction on the level-5 Sierpinski carpet (JPC finite element work, 2010).
- A supercritical percolation cluster on the square lattice.
- Internal diffusion-limited aggregation on the Sierpinski gasket (GIF made from Python simulations written by Jonah Kudler-Flam, 2017).
- Abelian sandpiles on the Sierpinski gasket (same credit as above).

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.

PDF My research in 1 slide. (September '16, presented at the "meet-and-greet" session during Colgate's NASC divisional colloquium.)

PDF 5-page description of my research, pitched at the general scientific audience. (Written in January '17 and updated in February '17. It is NOT up-to-date. An updated statement will appear in Fall '17.)Last updated: August 1, 2017

◆ **From non-symmetric particle systems to non-linear PDEs on fractals (a review)** *(with Michael Hinz and Alexander Teplyaev)*

To appear in the proceedings for the 2016 conference *Stochastic Partial Differential Equations & Related Fields* in honor of Michael Röckner's 60th birthday, Bielefeld

This is a summary of a series of four papers concerning the hydrodynamic limit of the boundary-driven exclusion process on a resistance space (the Sierpinski gasket being the model space).

◆ **Part I: The moving particle lemma for the exclusion process on a weighted graph**

Conditionally accepted by *Electron. Commun. Probab.*

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.

PDF A 15-minute presentation on this work, targeted towards undergraduates.

◆ **Part II: Local ergodicity in the exclusion process on an infinite weighted graph**

In this paper I prove, on every strongly recurrent weighted graph, the coarse-graining arguments needed to pass from the microscopic observables (in the exclusion process) to the corresponding macroscopic averages.
The two-blocks estimate is based on the moving particle lemma established in Part I.

PDF A 30-minute talk on local ergodicity and the hydrodynamic program, delivered at the University of Edinburgh.

Parts III & IV, concerning the hydrodynamic limit theorems and existence/uniqueness/regularity of solutions to semilinear PDEs on SG, are being written in collaboration with Michael Hinz and Alexander Teplyaev.

◆ **Internal DLA on Sierpinski gasket graphs** *(with Wilfried Huss, Ecaterina Sava-Huss, and Alexander Teplyaev)*

(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 this GIF animation.)

See also the shape theorem for divisible sandpiles on SG by Huss and Sava-Huss.

In an upcoming paper with Jonah Kudler-Flam, we show a universal limit shape for four cellular automata models on SG, and identify the precise order of fluctuations in each model.

**NEW!** Presentation at Universität Potsdam: PDF Slides (open in Adobe Acrobat to run the simulations).

◇ **Power dissipation on the Feynman-Sierpinski ladder circuit and the Hanoi tower circuits** *(with 10 other coauthors, from the 2015 UConn math REU fractals group)*

*J. Phys. A: Math. Theor.* **50** 325205 (2017) DOI

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.

A refinement of our results (concerning the energy measure) on the F-S ladder circuit was attained recently by Patricia Alonso-Ruiz.

◆

◆

◆

During the academic year, I often attend the analysis seminar and the probability seminar at the Cornell math department.

May-June 2017: Research stay at Institut Henri Poincaré, Paris. Trimester program "Stochastic Dynamics out of Equilibrium." (Contributed poster, photo diary, Paris math doodles)

June 2017: 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals, Cornell University. (Conference co-organizer, tutorial lecturer)

July 2017: International Workshop on BSDEs, SPDEs and their Applications, University of Edinburgh. (Invited talk at special session, see book of abstracts)

July-August 2017: Research stay at Universität Bielefeld, Germany, sponsored by the Bielefeld-Seoul IRTG 2235. (Invited scientific block course lecturer)

July-August 2017: Analysis & Geometry on Graphs and Manifolds, Universität Potsdam, Germany. (Contributed 30-minute talk, see book of abstracts)

September 2017: Northeast Analysis Network Conference, Syracuse University.

JPC's current and recent past research has been supported in part by the National Science Foundation (DMS-1262929), the Research Council of Colgate University (discretionary grant, student wage grant, and a major grant for 2017-2018), and starting in September 2017, a Simons Foundation Collaboration Grant for Mathematicians (2017-2022).

Back to JPC's homepage