Background: I am an associate professor of mathematics at the 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 (Fall 2021), I am teaching Math 3100 (Introduction to Probability) and Math 4110 (Introduction to Stochastic Processes). All materials are on UVA Collab.
Lecture notes: To be added.
Research: I work on a number of problems in analysis and probability. There are probably three main themes in my research interests.
(i) Time frequency analysis. Here my main interests are in multilinear estimates for operators with modulation invariants, a subject started since the work of L. Carleson about almost everywhere convergence of Fourier series. One of my favorite open problems in this area is the so-called nonlinear Carleson conjecture, see this blog post by T. Tao and this lecture notes of T. Tao and C. Thiele for some background details. See the recent breakthrough by A. Poltoratski. According to google scholar, this joint work with C. Thiele about outer measures is my most cited paper in this direction.
(ii) Distribution of roots for random polynomials and random functions. Here my main interests are in obtaining universality estimates for the distribution of the real roots of random polynomials and random functions (such as estimates involving expectation, variance, and asymptotic normality for the number of real roots). In this preprint, I considered the number of intersection points for random polynomial curves and deterministic algebraic curves.
(iii) Asymptotics for solutions of nonlinear integrable equations. This was the main topic of my PhD thesis, published here. Here it is well-known that there are nonlinear analogues of oscillatory integrals that can be used to represent these solutions, and it turns out they could be studied by mimicking techniques from real-variable harmonic analysis.
Grants: My research is supported in part by the National Science Foundation under the NSF grant DMS-1800855. My research was supported by the NSF during 2010-2011 under an NSF Math Institutes Postdoctoral Fellowship and partially supported during 2012-2016 by the NSF grants DMS-1201456 and DMS-1521293.
Quick links: UVA Harmonic Analysis and PDE seminar
Papers and preprints:
- Central Limit Theorem for the number of real roots of random orthogonal polynomials (with H. H. Nguyen, O. Nguyen, and I. E. Pritsker), preprint.
- Random orthonormal polynomials: local universality and expected number of real roots (with O. Nguyen and V. Vu), preprint.
- 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.
- Random trigonometric polynomials: universality and non-universality of the variance for the number of real roots (with H. Nguyen, O. Nguyen), to appear in Annales de l’Institut Henri Poincaré - Probabilités et Statistiques.
- Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applications, to appear in Electronic Journal of Probability.
- 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
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Weighted bounds for variational Fourier series (with M. Lacey), Studia Mathematica (2012), vol 211, no. 2, pp. 153--190.
- 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.
- 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.
- Weighted bounds for variational Walsh--Fourier series (with M. Lacey), Journal of Fourier Analysis and Applications (2012), vol 18, no. 6, pp. 1318--1339.
- On the convergence of lacunary Walsh-Fourier series (with M. Lacey), Bulletin London Math. Soc. (2012), vol 44, no. 2, pp. 241--254(14).
- Variational estimates for paraproducts (with C. Muscalu and C. Thiele), Rev. Mat. Iberoamericana (2012), vol 28, no. 3, pp. 857--878.
- 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 PhD thesis).