Joe Chen

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Joe Chen

Assistant Professor of Mathematics

Department/Office Information

Mathematics

Contact

(315) 228-6557

Personal website Curriculum Vitae

I do research and teaching in probability and mathematical analysis. Topics I'm passionate about include: random walks, Laplacians, card shuffling, spectral analysis, sandpiles, and central limit theorems. I thank the NSF for supporting my current endeavors.

For a 60-minute overview of what I do (circa fall ’20), watch my talk at the Colgate Science Colloquium "Mixing times of card shuffling and road traffic."

BS, Yale College, 2006; MS, Cornell University, 2010; PhD, Cornell University, 2013.

Assistant Professor of Mathematics, Colgate University, 2016-Present.
Postdoctoral fellow / Visiting Assistant Professor, Dept. of Mathematics, University of Connecticut, 2013-16.

Long-term visiting research positions (and invited minicourses)
Past grants

(The papers below have been either published, accepted, or conditionally accepted for publication. See my ARXIV PAGE for the most up-to-date list of my papers.)

2021+

Asymptotic behavior of density in the boundary-driven exclusion process on the Sierpinski gasket.
(with Patrícia Gonçalves.) Conditionally accepted by Math. Phys. Anal. Geom. (2021+) ARXIV:1904.08789

2020

Spectral decimation of the magnetic Laplacian on the Sierpinski gasket: Solving the Hofstadter-Sierpinski butterfly.
(with Ruoyu Guo '19.) Commun. Math. Phys. 380, 187-243 (2020). ARXIV:1909.05662

Laplacian growth & sandpiles on the Sierpinski gasket: limit shape universality and exact solutions.
(with Jonah Kudler-Flam '17.) Ann. Inst. Henri Poincaré Comb. Phys. Interact. 7 (2020), 585-664. ARXIV:1807.08748

Internal DLA on Sierpinski gasket graphs.
(with Wilfried Huss, Ecaterina Sava-Huss, and Alexander Teplyaev.) Chapter 7 in "Analysis and Geometry on Graphs & Manifolds," M. Keller et al. (Eds.), London Mathematical Society Lecture Notes, Cambridge University Press (2020). ARXIV:1702.04017

Fractal AC circuits and propagating waves on fractals.
(with Eric Akkermans, Gerald Dunne, Luke G. Rogers, and Alexander Teplyaev.) Chapter 18 in "Analysis, Probability, and Mathematical Physics on Fractals," P. Alonso-Ruiz et al. (Eds.), World Scientific Publishing (2020). ARXIV:1507.05682

Analysis, Probability and Mathematical Physics on Fractals. Patricia Alonso-Ruiz, Joe P. Chen, Luke G. Rogers, Robert S. Strichartz, and Alexander Teplyaev (Eds.). World Scientific Publishing (2020).

2018

From non-symmetric particle systems to non-linear PDEs on fractals.
(with Michael Hinz and Alexander Teplyaev.) To appear in the proceedings for Stochastic PDEs and Related Fields: an international conference in honor of Michael Röckner's 60th birthday. (2018). ARXIV:1702.03376

Regularized Laplacian determinants of self-similar fractals.
(with Alexander Teplyaev and Konstantinos Tsougkas.) Lett. Math. Phys. 108 (2018) 1563-1579. ARXIV:1610.10062

2017

The moving particle lemma for the exclusion process on a weighted graph.
Electron. Commun. Probab. 22 (2017), paper no. 47. ARXIV:1606.01577

Power dissipation in fractal AC circuits.
(with Luke G. Rogers, Loren Anderson, Ulysses Andrews, Antoni Brzoska, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephew Loew, and Alexander Teplyaev.). J. Phys. A: Math. Theor. 50 (2017) 325205. ARXIV:1605.03890

Wave equation on one-dimensional fractals with spectral decimation and the complex dynamics of polynomials.
(with Ulysses Andrews, Grigory Bonik, Richard W. Martin, and Alexander Teplyaev.) 
J. Fourier Anal. Appl. 23 (2017) 994--1027. ARXIV:1505.05855

Stabilization by noise of a $\mathbb{C}^2$-valued coupled system.
(with Lance Ford, Derek Kielty, Rajeshwari Majumdar, Heather McCain, Dylan O'Connell, and Fan Ny Shum.) Stoch. Dyn. 17 (2017) 1750046. ARXIV:1510.09221

2016 and prior

Singularly continuous spectrum of a self-similar Laplacian on the half-line.
(with Alexander Teplyaev.) J. Math. Phys. 57 (2016) 052104. ARXIV:1509.08875

Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces.
(with Stanislav Molchanov and Alexander Teplyaev.) J. Phys. A: Math. Theor. 48 (2015) 395203. ARXIV:1505.03923

Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs.
(with Baris E. Ugurcan.) Stoch. Proc. Appl. 125 (2015) 4632--4673. ARXIV:1307.5825

Periodic billiard orbits of self-similar Sierpinski carpets.
(with Robert G. Niemeyer.) J. Math. Anal. Appl. 416 (2014) 969--994. ARXIV:1303.4032

Quantum theory of cavity-assisted sideband cooling of mechanical motion.
(with Florian Marquardt, Aashish A. Clerk, and Steven M. Girvin.)
Phys. Rev. Lett. 99, 093902 (2007).

Interacting particle systems, (stochastic) partial differential equations, sandpiles, and fractals

My research focuses on the analysis of probability models---in particular, interacting particle systems and Laplacian growth models---on state spaces which are bounded in distance determined by electrical resistance. By studying explicit models and taking the right space-time scaling limits, I can rigorously derive (stochastic) partial differential equations or prove limit shape theorems on these spaces. These mathematical results capture various laws of nature, such as heat flow, wave propagation, charged particles in an electromagnetic field, and fluid dynamics. 

Models which I have studied include: Exclusion processes, internal diffusion-limited aggregation, rotor-router aggregation, abelian sandpiles, and Gaussian free fields. These models are often inspired from, and studied in, probability theory and statistical physics.

While the models I study may seem disparate, my approach is ultimately grounded in the study of analysis on graphs, Markov processes, and potential theory.

Some of my best mathematical results to date
  • Proved a new, optimal functional inequality for the exclusion process on any finite connected weighted graph, using the monotonicity of the Dirichlet energy under electric network reduction (a.k.a. Schur complementation). This paves the way for proving scaling limits of particle systems with microscopic interactions on non-translationally-invariant state spaces, such as trees, fractals, and random graphs.
  • Solved the magnetic Laplacian spectrum on the Sierpinski gasket under uniform magnetic field, thereby establishing the gasket version of Hofstadter's butterfly.
  • Established limit shape universality of discrete Laplacian growth models on the Sierpinski gasket, including the complete exact solution of the abelian sandpile growth problem.
  • Discovered a simple random walk model on the integer half-line which exhibits purely singularly continuous spectrum (that is, neither has point mass nor has density with respect to Lebesgue measure)---the simplest such model currently known.

More details can be found on my research page.

I work actively with postdocs and PhD students* from around the world. As indicated above, my current research focus is on the analysis of interacting particle systems on high-dimensional, rough, or random environments. An ideal collaborator would be someone who is knowledgeable in probability theory and analysis of PDEs, and has mastered the basics behind the analysis of exclusion or zero-range processes at the level of Kipnis & Landim, Scaling Limits of Interacting Particle Systems (Chapters 1-6 + Appendix A)

⚠ Note that I no longer actively work on fractals per se, other than the fact that my work tackles problems concerning interacting particle systems on rough environments such as fractals.

Current postdoc and PhD student collaborators: Chiara Franceschini (Tecnico Lisboa), Rodrigo Marinho (Tecnico Lisboa), Otavio Menezes (Purdue), Federico Sau (IST Austria).

*Please note that Colgate University is a US liberal arts college, and does not offer a PhD program.

I work with select Colgate undergraduate students on challenging problems in probability and mathematical analysis that interface with physics (statistical physics, quantum physics) or theoretical computer science (randomized algorithms). From experience, I look for students who not only have excelled in 300+-level MATH courses in the above-mentioned subjects, but also---more importantly---have the aptitude for independent problem-solving (analytical or numerical). 

⚠ I am a mathematician, and my job is to come up with potential theorems and prove them logically. Therefore I look for students whose mathematical ability is strong, whether or not they know how to program.  (Of course these two abilities are not mutually exclusive.) While I appreciate proficient programmers, the best outcomes occur with students who can contribute to theorem proving.

Below is a list of undergraduate research projects which I have (co-)supervised. Many of the projects have resulted in arXiv preprints and then published in highly ranked journals.

  • 2020~Present (Colgate): Analysis of differential privacy in discretizations of Langevin dynamics.
  • 2018~19 (Colgate): Spectral decimation of the magnetic Laplacian on the Sierpinski gasket (High Honors bachelor's thesis, expanded version ARXIV:1909.05662, published in Commun. Math. Phys.)
  • Fall 2017~Spring 2018 (Colgate): Complex-valued graph Laplacians, random spanning forests, and sandpiles on fractal graphs.
  • Spring 2017 (Colgate): Limit shapes of Laplacian growth & sandpile models on fractal graphs.  ARXIV:1807.08748 (Published in Ann. Inst. Henri Poincaré Comb. Phys. Interact.) 
  • Summer 2015 (UConn math REU): Stabilization by noise of a nonlinear coupled system with quadratic nonlinearity. ARXIV:1510.09221 (Published in Stoch. Dyn.)
  • Summer 2015 (UConn math REU): Power dissipation in fractal AC circuits. ARXIV:1605.03890 (Published in J. Phys. A: Math. Theor.)

Spring 2021

  • MATH 163: Calculus III

Fall 2020

  • MATH 163: Calculus III
  • FSEM 144: Statistics in Real Life

Spring 2020

Fall 2019

Spring 2018

  • MATH 163: Calculus III
  • MATH 377: Real Analysis I

Fall 2017

  • MATH 163: Calculus III
  • MATH 214: Linear Algebra

For more details please visit my teaching page.