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. 

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.

Publication and Preprints:

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

Research articles:

[42] (With Shaoming Guo, Larry Guth, Arian Nadjimzadah and Minxing Shen) Curved Kakeya problems and the projective geometry of paths, arXiv:2607.27629, submitted

[41] (With Hannah Cairo) Power loss for the Mizohata-Takeuchi conjecture on $C^k$ convex hypersurfaces, arXiv:2512.08064, submitted

[40] (With Wencai Liu, Matthew Powell, Yiding Max Tang, Xueyin Wang and Justin Zhou) Semi-algebraic discrepancy estimates for multi-frequency shift sequences with applications to quantum dynamics, Journal of Differential Equations, 2026, 464: 114196.

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

[38] (With Ciprian Demeter) On the $N$-set occupancy problem, Results in Mathematics. 2025, 80(8): 239.

[37] (With Alex Iosevich) A distinction between the paraboloid and the sphere in weighted restriction, Selecta Mathematica. 2026, 32(3): 49.

[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, Cambridge Journal of Mathematics 12 (4) (2024): 937-1015.

[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, in the Collection “Recent Developments in Harmonic Analysis and its Applications” in Contemporary Mathematics series, 2024 (792): 95-99.

[30] (with Shaoming Guo and Hong Wang) A dichotomy for Hörmander-type oscillatory integral operators, Invent. Math. 238 (2) (2024): 503-584.

[29] (with Ziming Shi) Sobolev Differentiability Properties of Logarithmic Modulus of Real Analytic Functions, Journal of Functional Analysis. 2025, 288(6): 110804.

[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, published 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, Peking Mathematical Journal, 2025, 8(2): 201-270.

[22] (with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich) A stationary set method for estimating oscillatory integrals, Journal of the European Mathematical Society, 2024, 28(5): 2067-2100.

[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:

[2] The Role of Some Geometric Objects in Fourier Restriction Theory, in Proceedings of the International Congress of Mathematicians 2026, 801-821

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