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.

Associate Professor of Mathematics, Colgate University, 2021-.

Assistant Professor of Mathematics, Colgate University, 2016-2021.

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

## Long-term visiting research positions (and invited minicourses)

- Instituto Superior Técnico, Lisboa (January-February 2019:
*Invited minicourse (6x 90-min);*June-July 2019) - Universität Bielefeld (July-August 2017, September-November 2018):
*Scientific block course lecturer (5x 90-min):*2017, 2018) - Technische Universiteit Delft (October-November 2018; May 2019 as NWO STAR visitor, co-sponsoring institution: Universiteit Leiden)
- Institut Henri Poincaré, Paris (May-June 2017)
- Hausdorff Research Institute for Mathematics, Bonn (May-August 2012)

## Past grants

- Simons Foundation Collaboration Grant for Mathematicians (2017~2019)
- Netherlands Organisation for Scientific Research "Stochastics--Theoretical & Applied Research (STAR)" visitor grant (2019)
- Co-PI, NSF DMS-1700187 conference grant "6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals" (2017)
- Colgate University Research Council Picker Research Fellowship (2019~2020)
- Colgate University Research Council Major Grant (2017~2018)

(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.) *Math. Phys. Anal. Geom. ***24**, article no. 24 (2021). ARXIV:1904.08789

**Higher order hydrodynamics and equilibrium fluctuations of interacting particle systems.**

(with Federico Sau.) *Markov Processes Relat. Fields* **27**, 339-380 (2021). ARXIV:2008.13403

## 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).

**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, **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 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.

**2020~22 (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.)

- AMS Central & Western Sectional Meeting, Special Session on Geometry, Analysis, Dynamics and Mathematical Physics on Fractal Spaces (March 2019)
- Joint Mathematics Meetings, AMS Special Session on Differential Equations on Fractals (January 2019)
- 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 Colgate:

- 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 my teaching page.