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Joe Chen

Joe Chen

Assistant Professor of Mathematics
Mathematics, 214 McGregory Hall
p 315-228-6557

Degrees

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

Professional Experience

Postdoctoral fellow / Visiting Assistant Professor, Department of Mathematics, University of Connecticut, 2013-16.

Visiting research positions:

Research Interests

Scaling limits of particle systems on self-similar and random networks

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 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.

More details can be found on my research page.

Select Grants & Awards

Student Research

In Fall '17 I am mentoring 3 research students, which is epsilon-close to the supremum of my capability. That being said, feel free to talk to me if you wish to initiate a reading project that prepares for actual research in Spring '18.

I am happy to advise students who are interested in:
  • Probability and stochastic processes (Limit theorems for random structures, Markov chains, Brownian motions).
  • Spectral graph theory (the study of adjacency/Laplacian matrices on graphs, with applications to theoretical computer science and diffusion geometry).
  • Fractal geometry and analysis.
  • Analysis of physical and biological models via differential equations and/or probabilistic methods.
Background: Completion of linear algebra + one 300-level math course which build upon calculus (e.g. probability, DiffEq). Pluses: Training in reading and writing proofs; ablity to write computer programs.

Below is a list of undergraduate research projects which I have (co-)supervised. While I make no guarantees, many of the projects have resulted in arXiv preprints and then published in good journals.
  • Fall 2017~ (Colgate): Complex-valued graph Laplacians, random spanning forests, and sandpiles on fractal graphs.
  • Spring 2017 (Colgate): Limit shapes of cellular automata models on fractal graphs. (Paper will be available later in 2017)
  • 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.)
More details can be found in my research students page.

List of Publications

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

``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. (2017+). arXiv:1702.03376

``Regularized Laplacian determinants of self-similar fractals.''
(with Alexander Teplyaev and Konstantinos Tsougkas.) Lett. Math. Phys., Online First (2017+). arXiv:1610.10062

``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

``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).

Teaching

Fall 2017: MATH 163 A (Calculus III) and MATH 214 A/B (Linear Algebra).

I am the actuarial exam advisor for Colgate students, and can advise anyone who wishes to pursue an actuarial career on how to prepare for the Probability (P) and Financial Math (FM) exams.

For a list of classes I have taught, please visit my teaching page.