I am excited about Math, especially things related to Analysis. I am currently working on two main research directions:

• Roots of random functions;

• The contact process and other stochastic processes on random graphs.

At the same time, I like to explore something new. Before, I studied applications of above research to real-world problems using machine learning algorithms (resulted in a paper with my students Hunter Zonnenberg and Chen Li, and collaborator Du Phan - to appear). Now, I am working on problems in optimal transport.

Currently, I work on the following projects with my students. (Updated: Spring 2024)

• 10th grader Iris Yang at the local Barrington public high-school and graduate student Phuc Lam work on the survival time of the SIRS epidemic process on star graphs. This is one of the crucial building blocks to understand SIRS on large graphs. SIRS is a good model for diseases such as Covid.

• Phuc Lam also works on the first non-universal term in the average number of real roots of the classical Kac polynomial. While the leading term is well-known to be universal, the first non-universal term is largely unknown due to the bizarre edge of the root spectrum.

• Junior students Ivan Lee and Chirag Furia work on martingale optimal transport (joint with Nhu Nguyen (URI)) - a model motivated by Math Finance. We study fundamental questions about cyclical monotonicity and duality.

• Sophomore student Anthony Wong works on SIR epidemic process on large graphs (joint with Souvik Dhara (Purdue)). We study the temporal behavior of SIR on deterministic large graphs and random graphs. SIR is a good model for diseases such as SARS (Severe Acute Respiratory Syndrome).

I have been teaching the following courses at Brown. If you are interested in getting a copy of the course materials, feel free to let me know.

• Graduate Probability I (APMA 2630)

• Graduate course: Introduction to random graphs (APMA 2812H)

• Undergraduate course on random graphs and stochastic processes on random graphs (APMA 1860)

20. Concentration inequalities for the number of real zeros of Kac polynomials (with Van Hao Can), arxiv preprint arxiv:2311.15446, 2023. [pdf][abs]  

19. Hole radii for the Kac polynomials and derivatives (with Hoi Nguyen), to appear in Stochastic Processes and their Applications. [pdf][abs]

18. Real roots of random orthogonal polynomials with exponential weights (with Yen Do, Doron Lubinsky, Hoi Nguyen, and Igor Pritsker), arxiv preprint arxiv:2212.14544, 2022. [pdf][abs]

17. Subcritical epidemics on random graphs (with Allan Sly),  arxiv preprint arxiv:2205.03551, 2022. [pdf][abs]

16. The number of limit cycles bifurcating from a randomly perturbed center (with Manjunath Krishnapur and Erik Lundberg), arxiv preprint arxiv:2112.05672, 2021. [pdf][abs]

15. Central Limit Theorem for the number of real roots of random orthogonal polynomials (with Yen Do, Hoi Nguyen and Igor Pritsker), arxiv preprint arxiv:2112.09015, 2021, to appear in Annales de l'Institut Henri Poincaré. [pdf][abs] 

14. Random orthonormal polynomials: universality and expected number of real roots (with Yen Do and Van Vu), Transactions of the American Mathematical Society 376(09), 2023, 6215-6243. [pdf][abs]

13. Random trigonometric polynomials: universality and non-universality of the variance for the number of real roots (with Yen Do and Hoi Nguyen), Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 58(03), 2022, 1460-1504. [pdf]

12. Roots of random functions: a framework for local universality (with Van Vu),  American Journal of Mathematics 144(01), 2022, 1-74. [pdf]

11. Critical value asymptotics for the contact process on random graphs (with Danny Nam and Allan Sly),  Transactions of the American Mathematical Society 375(06), 2022, 3899-3967. [pdf]

10. The full spectrum of random walks on complete finite d-ary trees (with Evita Nestoridi), Electronic Journal of Probability 26, 2021, 1-17.  [pdf]

9. Random polynomials: central limit theorems for the real roots (with Van Vu),  Duke Math Journal 170(17), 2021, 3745-3813. [pdf]

8. Survival and extinction of epidemics on random graphs with general degrees (with Shankar Bhamidi, Danny Nam and Allan Sly), The Annals of Probability 49(01), 2021, 244-286. [pdf]

7. Random eigenfunctions on flat tori: universality for the number of intersections (with Mei-Chu Chang, Hoi Nguyen and Van Vu), International Mathematics Research Notices, 2020(24), 2020, 9933-9973. [pdf]

6. On the mixing time of the Diaconis-Gangolli random walk on contingency tables over Z/qZ (with Evita Nestoridi), Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 56(2), 2020, 983–1001. [pdf]

5. Roots of random polynomials with coefficients of polynomial growth (with Yen Do and Van Vu),  The Annals of Probability 46(05), 2018, 2407–2494. [pdf]

    4. Embedding large graphs into a random graph (with Asaf Ferber and Kyle Luh), Bulletin of the London Mathematical Society 49(05), 2017, 784–797. [pdf]

3. Packing loose Hamilton cycles (with Asaf Ferber, Kyle Luh and Daniel Montealegre), Combinatorics, Probability and Computing 26(06), 2017, 839-849. [pdf]

2. On the number of real roots of random polynomials (with Hoi Nguyen and Van Vu), Communications in Contemporary Mathematics 18(04), 2016, 1550052. [pdf]

1. Anti-concentration for polynomials of independent random variables and applications in complexity theory (with Raghu Meka and Van Vu), Theory of Computing 12(11), 2016, 1-17. [pdf]