I am interested in a lot of problems related to Harmonic analysis, ranging from Euclidean harmonic (Fourier) analysis to incidence geometry and to harmonic analysis on locally symmetric spaces.

For Euclidean harmonic analysis, currently I am most interested in restriction type problems: In R^n, let M be a subset in the frequency space with a natural measure dσ, e.g. the unit sphere with the surface measure. We can ask how f behaves if the support of the Fourier transform of f is contained in M. In particular, if we know some information of the Fourier transform of f as well as that |f| is roughly H on a set X, can we bound H from above? Restriction problems and decoupling theory for paraboloids are two such subjects that I am very much interested in. 

Recently I am also interested in additive combinatorics for locally compact groups. One fundamental question there is: Given a locally compact unimodular group and two compact subsets X and Y, what can you say about the measure of XY when the measures of X and Y are given (perhaps with some other constraints)? These results have already found many applications in e.g. the theory of approximate groups.

I am also interested in the harmonic analysis on locally symmetric spaces, especially on those that are noncompact with finite volume and higher rank. The book on Eisenstein Series by Langlands established the spectral theory of such spaces. Alternatively, Harish-Chandra's book Automorphic Forms on Semisimple Lie Groups and Moeglin and Waldspurger's book Spectral Decomposition and Eisenstein Series serve together as a very good reference. I am also very much interested in the trace formula, especially in its analytic aspects. The truncation operator plays a crucial role when the locally symmetric space is not compact.

Some key words of my research: Fourier restriction estimates, decoupling, reversed square function estimates, local smoothing, Schrödinger maximal estimates, refined Strichartz estimates, Kakeya, the Fractal Uncertainty Principle, polynomial methods, polynomial partitioning, Falconer's distance set problem, Parsell-Vinogradov Systems, Translation-Dilation-Invariant Systems, weighted restriction estimates, oscillatory integrals, Brascamp-Lieb inequalities, Loomis-Whitney inequalities, incidence geometry, multijoints, the Furstenberg set problem, locally symmetric spaces, spectral decomposition, the truncation operator, the Brunn-Minkowski inequality, Tarry's problem, the Bochner-Riesz problem, Hilbert irreducibility theorem, the Selberg sieve, number field counting, Polynomial Wolff Axioms, Hörmander type operators, Kakeya compression.

Publication and Preprints:

Most of my papers can be found on this arXiv page.

Research articles:

[39] (With Xiumin Du, Jianhui Li and Hong Wang) $L^p$ weighted Fourier restriction estimates, arXiv:2404.10951

[38] (With Ciprian Demeter) On the $N$-set occupancy problem, arXiv:2403.10678

[37] (With Alex Iosevich) A distinction between the paraboloid and the sphere in weighted restriction, arXiv:2312.12779, submitted

[36] (With Shaoming Guo, Alex Iosevich and Pavel Zorin-Kranich) $L^p$ integrability of functions with Fourier support on a smooth space curve, arxiv:2311.11529, submitted

[35] (with Song Dai, Liuwei Gong and Shaoming Guo) Oscillatory integral operators on manifolds and related Kakeya and Nikodym problems, arXiv:2310.20122, submitted

[34] (With Xiumin Du, Yumeng Ou and Kevin Ren) Weighted refined decoupling estimates and application to Falconer distance set problem, arXiv:2309.04501, submitted

[33] (With Xiumin Du, Yumeng Ou and Kevin Ren) New improvement to Falconer distance set problem in higher dimensions, arXiv:2309.04103, submitted

[32] (with Yifan Jing and Chieu-Minh Tran) Measure doubling of small sets in $SO(3, \mathbb{R})$, arXiv:2304.09619, submitted

[31] (with Xiumin Du, Yumeng Ou and Hong Wang) On a free Schrödinger solution studied by Barceló--Bennett--Carbery--Ruiz--Vilela, arXiv:2303.10563, to appear in a Contemporary Mathematics volume

[30] (with Shaoming Guo and Hong Wang) A dichotomy for Hörmander-type oscillatory integral operators, arXiv:2210.05851, submitted

[29] (with Ziming Shi) Sobolev Differentiability Properties of Logarithmic Modulus of Real Analytic Functions, arXiv:2205.02159, submitted

[28] (with Theresa C. Anderson, Ayla Gafni, Kevin Hughes, Robert J. Lemke Oliver, David Lowry-Duda, Frank Thorne and Jiuya Wang) Improved bounds on number fields of small degree, arXiv:2204.01651, to appear in Discrete Analysis.

[27] (with Michael Law, Ya'acov Ritov and Ziwei Zhu) Rank-Constrained Least-Squares: Prediction and Inference, arXiv:2111.14287, submitted

[26] (with Jinpeng An, Yifan Jing and Chieu-Minh Tran) On the small measure expansion phenomenon in connected noncompact nonabelian groups, arXiv:2111.05236, submitted

[25] (with Theresa C. Anderson, Ayla Gafni, Robert J. Lemke Oliver, David Lowry-Duda and George Shakan) Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants, IMRN (3) (2023): 2188-2214.

[24] (with Xiumin Du and Yumeng Ou) On the multiparameter Falconer distance problem, Trans. Amer. Math. Soc. 375 (7) (2022): 4979-5010

[23] (with Shaoming Guo, Changkeun Oh, Hong Wang and Shukun Wu) The Bochner-Riesz problem: an old approach revisited, arxiv:2104.11188, submitted

[22] (with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich) A stationary set method for estimating oscillatory integrals, arxiv:2103.08844, to appear in Journal of the European Mathematical Society.

[21] (with Yifan Jing and Chieu-Minh Tran) A nonabelian Brunn--Minkowski inequality, Geometric and Functional Analysis (2023): 1-53.

[20] (with Shaoming Guo, Changkeun Oh and Pavel Zorin-Kranich) Decoupling inequalities for quadratic forms, Duke Mathematical Journal 172(2) (2023): 387-445.

[19] (with Xiumin Du, Alex Iosevich, Yumeng Ou and Hong Wang) An improved result for Falconer's distance set problem in even dimensions, Math. Ann. 380 (3-4) (2021): 1215-1231.

[18] (with Xinliang An) Polynomial Blow-Up Upper Bounds for the Einstein-Scalar Field System Under Spherical Symmetry, Commun. Math. Phys. 376 (2020): 1671–1704.

[17] (with Larry Guth and Hong Wang) A sharp square function estimate for the cone in R^3, Ann. of Math. 192 (2) (2020): 551-581

[16] (with Jonathan Hickman and Keith Rogers) Improved bounds for the Kakeya maximal conjecture in higher dimensions, Amer. J. Math. 144(6) (2022): 1511-1560.

[15] (with Xiumin Du, Jongchon Kim and Hong Wang) Lower bounds for estimates of the Schrödinger maximal function, Math. Res. Lett. 27(3) (2020): 687-692 

[14] (with Xiumin Du) Sharp L^2 estimates of Schrödinger maximal function in higher dimensions, Ann. of Math. 189(3) (2019): 837-861 

[13] (with Shaoming Guo) On integer solutions of Parsell-Vinogradov systems, Invent. Math. 218(1) (2019): 1-81.

[12] (with Xiumin Du, Larry Guth and Xiaochun Li) Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimate, Forum of Math., Sigma 6 (2018), e14, 18 pp.

[11] (with Xiumin Du, Larry Guth, Yumeng Ou, Hong Wang and Bobby Wilson) Weighted restriction estimates and application to Falconer distance set problem, Amer. J. Math. 143 (1) (2021): 175-211

[10] (with Long Jin) Fractal Uncertainty Principle with explicit exponent, Math. Ann. 376(3) (2020): 1031-1057 

[9] A proof of the Multijoints Conjecture and Carbery's generalization, Journal of the European Mathematical Society 22(8) (2020): 2405-2417

[8] The endpoint perturbed Brascamp-Lieb inequality with examples, Anal. PDE 11(3) (2018): 555-581

[7] (with Brandon Hanson and Robert C. Vaughan) The least number with prescribed Legendre symbols, J. Number Theory 179 (2017): 3-16

[6] (with Noam Solomon) Highly incidental patterns on a quadratic hypersurface in R^4, Discrete Mathematics 340 (4) (2017): 585-590

[5] Polynomials with dense zero sets and discrete models of the Kakeya conjecture and the Furstenberg set problem, Selecta Mathematica (2016), 1-18 

[4] On configurations where the Loomis-Whitney inequality is nearly sharp and applications to the Furstenberg set problem,  Mathematika 61(1) (2015), 145-161

[3] (with Hong Wang and Ben Yang) Bounds of incidences between points and algebraic curves, arXiv: 1308.0861 

[2] On sharp local turns of planar polynomials, Math. Z. 277 (3) (2014), 1105-1112

[1] On the number of ordinary circles determined by n points, Discrete Comput. Geom. 46(2) (2011), 205-211 

Survey articles:

[1] The Brascamp-Lieb inequality and its influence on Fourier analysis, in The Physics and Mathematics of Elliott Lieb, Vol. 2, 585-628