- The 1st non-constant Neumann Laplacian eigenfunction on the level-5 Sierpinski carpet (my 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, as well as the relevant (stochastic) PDEs, on self-similar and random geometries. The overarching goal is to prove limit theorems that describe various laws of nature that take place in these nonsmooth spaces, such as heat flow, wave propagation, charged particles in an electromagnetic field, and fluid dynamics.

From the mathematics point of view, I carry out two interconnected lines of research, one **analytic** and the other **probabilistic**.

- On the one hand, I analyze variants of the Laplace operator \(\Delta\) on fractal spaces, focusing on their spectral properties, as well as analyzing the solvability and regularity of the (S)PDEs they generate.
- On the other hand, I study the scaling limits (law of large numbers, central limit theorem) of random processes involving many interacting particles driven by Laplacian-like operators, representing models of magnetism or crystal growth.

- (Non-equilibrium)
**statistical mechanics**models which are many-particle extensions of the (simple) random walk process; **Quantum mechanics**models of a single particle moving under the influence of a (possibly random) scalar or vector potential.

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: October 10, 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 (2017+)

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**

*Electron. Commun. Probab.* **22** (2017), paper no. 47.

I prove in this paper a Sobolev-type 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**

Submitted

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.

**NEW!** PDF An hour-long talk on the moving particle lemma and local ergodicity in the exclusion process, delivered at the CUNY Probability Seminar.

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.

PDF A 20-minute talk on semilinear evolution equations on resistance spaces, delivered at the Northeast Analysis Network conference in Syracuse.

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

Submitted

(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.

PDF A 20-minute talk on limit shape universality of Laplacian growth models on SG, delivered at the conference "Analysis & Geometry on Graphs & Manifolds" at Universität Potsdam, Germany. (Open in Adobe Acrobat to run the simulations.)

◇ **Power dissipation in fractal AC circuits**,
with Luke G. Rogers, Loren Anderson, Ulysses Andrews, Antoni Brzoska, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen Loew, and Alexander Teplyaev. (2015 UConn math REU fractals group)

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

We give a "fractal spin" on the classic infinite ladder circuit discussed in the Feynman Lectures on Physics.
For the Feynman-Sierpinski ladder circuit (pictured) 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.

◆

◇

◆

◆

◆

◆

◇

◆

◆

◆

As of September 2017, my research is supported in part by the Simons Foundation (via a Collaboration Grant for Mathematicians, 2017-2022) and the Research Council of Colgate University.

Back to JPC's homepage