(From left: Maria Dascalu '18, Ilias Stitou '19, Jonah Kudler-Flam '17, and me. Photo credit: Noam Elkies.)

**Colgate student trainees (as of Fall '17):** Quan Vu '18, Ilias Stitou '19, and Ruoyu (Tony) Guo '19.

Project theme in **Fall 2017**: Analysis of complex-valued graph Laplacians, random spanning forests, and sandpile models on fractal graphs.

**Graduated Colgate student trainee:** Jonah Kudler-Flam '17 (*Next job:* PhD Student in Theoretical Physics at University of Chicago).

**NEW!** Updates from **Fall 2017**:

- Jonah numerically studied various sandpile and aggregation models on the Sierpinski gasket, and gave the first proof that the abelian sandpile cluster on SG is an exact ball (in the graph metric).
On top of this, he and I were able to establish other related shape results.
Our paper is to be made available (hopefully) soon.

In conjunction with the recent preprints arXiv:1702.04017 and arXiv:1702.08370, our paper proves a "limit shape universality" result for Laplacian growth models on SG. - Ilias is working towards a related "limit shape universality" result for Laplacian growth models on the Vicsek tree graph, with some key differences from the SG case.
- Tony is working on the spectral analysis of vector-bundle Laplacian on SG, with applications to the "electron in a magnetic field" problem on fractal lattices.

(Summer 2017) I have presented joint works with Wilfried Huss, Ecaterina Sava-Huss, Sasha Teplyaev, and Jonah Kudler-Flam at the Institut Henri Poincare (May, poster);
the Cornell Fractals 6 conference (June);
in my scientific block course at Universität Bielefeld, Germany (July);
and at the international conference "Analysis & Geometry on Graphs & Manifolds" in Potsdam, Germany (August).

Jonah presented our joint work at the AMS Fall Western Sectional Meeting at the University of California, Riverside in November.

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?**
As of August 2017 I have a full slate of research students, and would not be able to take on more.
However, if you are interested in working with me in Spring 2018, feel free to talk to me.
We may be able to start a reading project in the fall/winter in preparation for actual research in the spring.

Back to JPC's research page

Back to JPC's homepage