Yen Do

Background: I am an associate professor at UVA Math Department. I graduated from UCLA in 2010 and was a postdoc at Georgia Tech and Yale, here are the links to my math genealogy page, my CV, and my Google scholar page.

Teaching: This semester (Spring 2024), I am teaching Math 4110 (Intro to Stochastic Processes) and Math 4250 (Differential Equations and Dynamical Systems). All materials are on UVA Canvas.

Lecture notes: Introduction to Probability (Math 3100, Fall 2021)

Research: I work on several problems in analysis and probability. There are probably three main themes in my research.

(i) Time-frequency analysis. I am interested in multilinear estimates for operators with modulation invariants, a subject started since the work of L. Carleson about almost everywhere convergence of Fourier series.  

(ii) Distribution of roots for random polynomials and random functions. I am interested in the distribution of the real roots of random polynomials and random functions (such as estimates involving expectation, variance, asymptotic normality, law of large numbers for the number of real roots). 

(iii) Asymptotics for solutions of nonlinear integrable equations. This was the main topic of my Ph.D. thesis, published here. I am interested in long-time asymptotics for nonlinear analogs of oscillatory integrals that can be used to represent the solutions of nonlinear PDEs.

Grants: My research is partly supported by the National Science Foundation (DMS-1800855, DMS-1521293, and DMS-1201456.).

Graduate students: Nhan Nguyen (2019-2023), and Mark Lewers (2017-2020).

Quick links: UVA Harmonic Analysis and PDE seminar

Papers and preprints:

1. A strong law of large numbers for real roots of random polynomials, preprint.

2. Real roots of random polynomials with coefficients of polynomial growth: asymptotics of the variance (with Nhan D.V. Nguyen), preprint.

3. Real roots of random orthogonal polynomials with exponential weights (with D. Lubinsky, H. H. Nguyen, O. Nguyen, and I. E. Pritsker), preprint.
4. Central Limit Theorem for the number of real roots of random orthogonal polynomials (with H. H. Nguyen, O. Nguyen, and I. E. Pritsker), Annales de L'Institut Henri Poincaré, Probabilités et Statistiques, to appear.
5. Random orthonormal polynomials: local universality and expected number of real roots (with O. Nguyen and V. Vu), to appear in Transactions of the AMS.
6. Generalized Carleson embeddings into weighted outer measure spaces (with M. Lewers), Journal of Mathematical Analysis and Applications, Volume 506, Issue 2, 15 February 2022, 125698.
7. Random trigonometric polynomials: universality and non-universality of the variance for the number of real roots (with H. Nguyen, O. Nguyen), Annales de L'Institut Henri Poincaré, Probabilités et Statistiques 58(03), 2022, 1460--1504.
8. Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applications, Electron. J. Probab. 26 (2021), article no. 144, 1--45.
9. Oscillations and integrability of the vorticity in the 3D NS flows (with A. Farhat, Z. Grujic, L. Xu), Indiana University Mathematics Journal 69 (2020), 1559--1578
10. Central limit theorems for the real zeros of Weyl polynomials (with V. H. Vu), American Journal of Mathematics (2020), vol. 142, issue 2, pp. 1327--1369.
11. Positive sparse domination of variational Carleson operators (with F. Di Plinio and G. Uraltsev), Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XVIII (2018), 1443--1458.
12. Variational estimates for the bilinear iterated Fourier integral (with C. Muscalu and C. Thiele), Journal of Functional Analysis (2017), vol. 272, issue 5, pp. 2176--2233.
13. Roots of random polynomials with coefficients having polynomial growth (with O. Nguyen and V. H. Vu), Annals of Probability, 2018, Vol. 46, no. 5, 2407--2494.
14. Variation-norm and fluctuation estimates for ergodic bilinear averages (with R. Oberlin and E. A. Palsson), Indiana U. J. Math. (2017), vol. 66, issue 1, 55--99.
15. Real roots of random polynomials: expectation and repulsion (with H. Nguyen and V. H. Vu), Proc. London Math. Soc. (3) 111 (2015), no. 6, 1231--1260.
16. Lp theory for outer measures and two themes of Lennart Carleson united (with C. Thiele), Bulletin AMS (N.S.), 52 (2015), no. 2, pp. 249--296.
17. An operator van der Corput estimate arising from oscillatory Riemann--Hilbert problems (with P. T. Gressman), J. Funct. Anal. (2014), vol. 267, issue 12, pp. 4775--4805.
18. Weighted bounds for variational Fourier series (with M. Lacey), Studia Mathematica (2012), vol 211, no. 2, pp. 153--190.
19. The spectrum of random kernel matrices: universality results for rough and varying kernels (with V. H. Vu), Random Matrices: Theory and Applications, vol 02, issue 03, July 2013.
20. Variational bounds for a dyadic model of the bilinear Hilbert transform (with R. Oberlin and E. A. Palsson), Illinois Journal of Math. (2013), vol 57, no. 1, 105--119.
21. Weighted bounds for variational Walsh--Fourier series (with M. Lacey), Journal of Fourier Analysis and Applications (2012), vol 18, no. 6, pp. 1318--1339.
22. On the convergence of lacunary Walsh-Fourier series (with M. Lacey), Bulletin London Math. Soc. (2012), vol 44, no. 2, pp. 241--254(14).
23. Variational estimates for paraproducts (with C. Muscalu and C. Thiele), Rev. Mat. Iberoamericana (2012), vol 28, no. 3, pp. 857--878.
24. A nonlinear stationary phase method for oscillatory Riemann--Hilbert problems, Int. Math. Res. Notices, vol 2011, issue 12, pp. 2650--2765.
    (this paper is essentially the same as my Ph.D. thesis).