I'm a PhD candidate in mathematical physics at Cornell University, supervised by Bob Strichartz. My research revolves around harmonic analysis, stochastic processes, and statistical mechanics on infinitely ramified fractals, such as the Sierpinski carpet and its analogs in three dimensions like the Menger sponge.
In a nutshell, I study many things directly or indirectly related to the Laplacians on fractals. These include the spectra of Laplacians, heat kernels, wave operators, and spectral zeta functions, just to name a few. I'm also working on billiard dynamics and quantum fields on fractals.
During my research I both run numerical simulations (e.g. finite-element analysis) and write mathematical proofs. I started out with the former, then became more versed in rigorous mathematical analysis as my coursework and experience accumulated. In some sense, studying analysis on fractals gave me the chance to work on theoretical physics constructs which can be made rigorous, while reinforcing my knowledge about classical analysis on Euclidean spaces.
Previously I dabbled with quantum optics, and investigated the theory of sideband cooling of macroscopic objects as part of my undergraduate thesis project. It has since been widely implemented in experiments which succeeded (circa 2011) in cooling a micron-sized cantilever to its motional ground state---a prerequisite for realizing "Schrödinger's Cat" in real life.
[pdf] [arXiv] Statistical mechanics of Bose gas in Sierpinski carpets. Submitted. arXiv:1202.1274 [math-ph].
[pdf] [arXiv] Quantum theory of cavity-assisted sideband cooling of mechanical motion. Phys. Rev. Lett., 99, 093902 (2007). (with Florian Marquardt, Aashish A. Clerk and Steven M. Girvin.)
Spectral asymptotics and heat kernels on three-dimensional fractal sponges. (with Robert S. Strichartz.)
Periodic orbits in self-similar Sierpinski carpets. (with Robert G. Niemeyer.)
Gaussian free fields, occupation times, and cover times on fractals. (with Baris E. Ugurcan.)
The infamous Euler problem in Evans' PDE, 2nd ed. (MATH 6190, PDE I, L. Wahlbin [teaching]): Cornell access only.
Log-Sobolev inequality and Perelman's entropy (MATH 6130, Heat kernel analysis on Riemannian manifolds, X. Cao): Will be uploaded after I make a tight connection to the "physics" behind Perelman's entropy.
A crash course in Gaussian free fields (MATH 7770: Analysis and probability in infinite-dimensional spaces, N. Eldredge): I decided not to write up a separate note, since it wouldn't beat Scott Sheffield's anyway.
(Fall 2012) TA for MATH 4220, Applied Complex Analysis.
(Spring 2012) Grader for MATH 6220, Applied Functional Analysis. Office hours are by appointment, Malott 105.
Bose gas in Sierpinski carpets: Cornell Analysis seminar (Feb 2012), Arizona School of Analysis & Mathematical Physics (Mar 2012).
The spectrum of the Neumann Laplacian on the Level-5 standard Sierpinski carpet:
Here's the hallmark Barlow-Bass-Kumagai-Teplyaev paper that establishes the uniqueness of Brownian motion on Sierpinski carpets (Key words: Harnack inequality, Hilbert's projective metric). I thank Ben Steinhurst for helping me understand this paper through his detailed lectures in MATH 7770.
A picture of a non-self-similar Sierpinski carpet of level n, here n=4. Notice that it has nonzero area in the limit n → ∞, whereas the self-similar carpet occupies zero area.
Three standard references on analysis & probability on fractals: Barlow '96 (for probabilists); Kigami '01 (for analysts); Strichartz '06 (suitable for math undergrads).
A comprehensive review of literature on the analysis and probability on fractals, courtesy of Sasha Teplyaev. Also an updated version with more physical applications.
Towards the quantization of analysis on fractals, by Luke Rogers.
A grown-up "Cliff's Notes" version on major facts about Brownian motion on fractals, by Takashi Kumagai.
Friends & collaborators of analysis & probability on fractals: Jason Anema, Matt Begué, Matt Guay, Steve Heilman, Naotaka Kajino, Dan Kelleher, Nishu Lal, Rob Niemeyer, Ben Steinhurst, Baris Ugurcan.
Is "explosive percolation" really all that explosive (cue the O-word)? See Riordan & Warnke, Achlioptas process phase transitions are continuous.
Recently I became interested in the rigorous derivations of Gross-Pitaevskii functional and equation (a cubic NLS) for interacting Bose gas. It all started with Lieb-Seiringer-Yngvason. Starting in 2006: Erdos-Schlein-Yau, Klainerman-Machedon, Chen-Pavlovic, Kirkpatrick-Schlein-Staffilani. It's so satisfying to see a physics problem attacked using Sobolev inequalities and Strichartz estimates! And as always, a blurb from Terry Tao on this topic.
Discrete complex analysis on planar graphs, with connection to the Dirac operator and criticality in Ising model: Mercat; Kenyon; Cimasoni; Smirnov's ICM 2010 presentation (slides, full text).
Localized eigenfunctions on various irregular domains (avec beaucoup d'images!): a cute survey article by Heilman & Strichartz, and several super-accurate computational results from Trefethen & Betcke. (By the way, ever wonder about the origin of the MATLAB logo?)
Do I know/study the estimates that my advisor is famous for? No, not really. (That said, here's a primer guide.) And no, we don't know how to make such estimates on fractals yet. (But this work lays the groundwork for it.)
Quantum mechanics for mathematicians: the book, and the course.
(1+1)-dimensional Kardar-Parisi-Zhang (KPZ) equation: See surveys by Jeremy Quastel and Ivan Corwin. Simulations of TASEP from Patrik Ferrari.
Cornell Probability Summer School: 2009, 2010, 2011, 2012.
Literally hearing (and seeing) the shape of a square drum. I played this clip toward the end of a guest lecture I gave in MATH 2930. (WARNING: Be prepared to mute the last 30 seconds of the video, lest the sound drive you insane.)
Speaking of music, I've picked up piano again after a decade-long hiatus. Currently playing: Chopin's Fantasie Impromptu, Nocturne Op.9 No.2, and Andante spianato et Grande polonaise brillante (the last one is a major work in progress).
 | If I were a Springer-Verlag Graduate Text in Mathematics, I would be J.L. Doob's Measure Theory. I am different from other books on measure theory in that I accept probability theory as an essential part of measure theory. This means that many examples are taken from probability; that probabilistic concepts such as independence, Markov processes, and conditional expectations are integrated into me rather than being relegated to an appendix; that more attention is paid to the role of algebras than is customary; and that the metric defining the distance between sets as the measure of their symmetric difference is exploited more than is customary. Which Springer GTM would you be? The Springer GTM Test |
