51·çÁ÷

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

Associate Professor of Mathematics, 51·çÁ÷, 2021-.
Assistant Professor of Mathematics, 51·çÁ÷, 2016-2021.
Postdoctoral fellow / Visiting Assistant Professor, Dept. of Mathematics, University of Connecticut, 2013-16.

Long-term visiting research positions (and invited minicourses)

•  (January-February 2019: ); June-July 2019)
•  (July-August 2017, September-November 2018): Scientific block course lecturer (5x 90-min): , )
•  (October-November 2018; May 2019 as NWO STAR visitor, co-sponsoring institution: Universiteit Leiden)
•  (May-June 2017)
•  (May-August 2012)

Past grants

• Netherlands Organisation for Scientific Research "Stochastics--Theoretical & Applied Research (STAR)" visitor grant (2019) 
• Co-PI,  conference grant "6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals" (2017)
• 51·çÁ÷ Research Council Picker Research Fellowship (2019~2020)
• 51·çÁ÷ Research Council Major Grant (2017~2018)

(The papers below have been either published, accepted, or conditionally accepted for publication. See my  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.) Math. Phys. Anal. Geom. 24, article no. 24 (2021). 

Higher order hydrodynamics and equilibrium fluctuations of interacting particle systems.
(with Federico Sau.) Markov Processes Relat. Fields 27, 339-380 (2021). 

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). 

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. 

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). 

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). 

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). 

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

2017

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

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. 

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. 

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. 

2016 and prior

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

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. 

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

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

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).

• Invited session speaker "Fluctuations of Interacting Particle Systems," at the 41st Stochastic Processes & their Applications conference, Northwestern University (2019) 
• Invited hour-long colloquium or seminar talks at: Brown, UChicago, CCNY, CUNY Graduate Center, Colby, Cornell, TU Delft, Düsseldorf, TU Graz, Leiden, Técnico Lisboa, CRM Montreal, SUNY Albany.

Quantitative convergence of interacting particle systems and of machine learning algorithms

My research focuses on the analysis of probability models---in particular,  and ---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 ()  or prove  on . 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: , , , , and . 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 , , and .

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 .

I work with select 51·çÁ÷ 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). 

âš  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.

• 2020~22 (51·çÁ÷): Analysis of differential privacy in discretizations of Langevin dynamics.
• 2018~19 (51·çÁ÷): Spectral decimation of the magnetic Laplacian on the Sierpinski gasket (High Honors bachelor's thesis, expanded version , published in Commun. Math. Phys.)
• Fall 2017~Spring 2018 (51·çÁ÷): Complex-valued graph Laplacians, random spanning forests, and sandpiles on fractal graphs.
• Spring 2017 (51·çÁ÷): Limit shapes of Laplacian growth & sandpile models on fractal graphs.   (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.  (Published in Stoch. Dyn.)
• Summer 2015 (UConn math REU): Power dissipation in fractal AC circuits.  (Published in J. Phys. A: Math. Theor.)

•  
• The 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals (June 13-17, 2017)

Since 2016 I have taught the following courses at 51·çÁ÷:

• CORE 143S: Introduction to Statistics
• FSEM 144: Statistics in Real Life
• MATH 162: Calculus II
• MATH 163: Calculus III
• MATH 214: Linear Algebra
• MATH 313: Functions of a Complex Variable
• MATH 316: Probability
• MATH 377: Real Analysis I
• MATH 417: Brownian Motion & Stochastic Calculus
• MATH 487: Real Analysis II

For more details please visit .