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Yun-Feng Zhang

D. of Mathematics - U. of Connecticut
CONTACT
TEACHING
RESEARCH
ABOUT

About


Welcome to my mathematics page. I am employed as an Assistant Research Professor at the University of Connecticut. I will move to Peking University as a TAL Assistant Professor starting in Fall 2021. I received my Ph.D. from UCLA in 2018 under the advisement of Rowan Killip and Monica Visan.



Research


I am interested in the fields of analysis and representation theory, and in particular harmonic analysis on semisimple Lie groups and symmetric spaces and its applications. My recent focus is on Fourier restriction type estimates such as Strichartz estimates for dispersive equations and Lp bounds of Laplacian eigenfunctions on globally symmetric spaces. This circle of problems already bear a rich theory in the Euclidean setting, and are further challenges when considered on locally symmetric spaces. My papers:

  • Improved Fourier restriction estimates on compact globally symmetric spaces, in preparation.
  • On Fourier restriction type problems on compact Lie groups. [pdf] [arXiv]
    In this article, we obtain new results for Fourier restriction type problems on compact Lie groups. We first provide a sharp form of $\small L^p$ estimates of irreducible characters in terms of their Laplace-Beltrami eigenvalue and as a consequence provide some sharp $\small L^p$ estimates of joint eigenfunctions for the ring of invariant differential operators. Then we improve upon the previous range of exponent for scale-invariant Strichartz estimates for the Schrödinger equation, and prove $\small L^p$ bounds of Laplace-Beltrami eigenfunctions in terms of their eigenvalue matching the known bounds on tori. The main novelties in our approach consist of a barycentric-semiclassical subdivision of the Weyl alcove and sharp $\small L^p$ estimates on each component of this subdivision of some weight functions coming out of the Weyl denominator.

  • Schrödinger equations on compact globally symmetric spaces.
    The Journal of Geometric Analysis (2021), 42 pp. [pdf] [doi]
    In this article, we establish scale-invariant Strichartz estimates for the Schrödinger equation on arbitrary compact globally symmetric spaces and some bilinear Strichartz estimates on products of rank-one spaces. As applications, we provide local well-posedness results for nonlinear Schrödinger equations on such spaces in both subcritical and critical regularities.

  • Strichartz estimates for the Schrödinger equation on products of odd-dimensional spheres.
    Nonlinear Analysis 199 (2020), 21 pp. [pdf] [doi]
    We prove certain Strichartz estimates which are scale-invariant up to an $\small \varepsilon$-loss on any product of odd-dimensional spheres. Some partial results toward such Strichartz estimates on a general compact globally symmetric space are also given, including a kernel estimate sharp up to an $\small \varepsilon$-loss near rational times and near corners of a maximal torus.

  • Strichartz estimates for the Schrödinger flow on compact Lie groups.
    Analysis & PDE 13 (2020), no. 4, 1173–1219. [pdf] [doi]
    We establish scale-invariant Strichartz estimates for the Schrödinger flow on any compact Lie group equipped with canonical metrics. The highlights of this paper include an estimate for some Weyl type sums defined on rational lattices, the different decompositions of the Schrödinger kernel determined by how close the points inside the maximal torus are to the cell walls, and an application of the BGG-Demazure operators or Harish-Chandra’s integral formula to the estimate of the difference between characters.

  • PhD thesis: Strichartz estimates for the Schrödinger flow on compact symmetric spaces. [pdf]




Teaching


I am teaching Partial Differential Equations (Math 3435) remotely in Spring 2021. Previous teaching:

As instructor:
Fall 2020:   Math 3435 (Partial Differential Equations), two classes, UConn.
Spring 2020:   Math 2360Q (Axiomatic Geometry), two classes, UConn.
Fall 2019:   Math 3146 (Introduction to Complex Variables), two classes, UConn.
Spring 2019:   Math 3435 (Partial Differential Equations), two classes, UConn.
Fall 2018:   Math 1152Q (Honors Calculus II), UConn.
  Math 2130Q (Honors Multivariable Calculus), UConn.
Summer 2017:   Math 3B (Calculus for the Life Sciences), UCLA.
As assistant:
Spring 2018:   Math 170B (Probability Theory), UCLA.
Winter 2018:   Math 117 (Algebra for Applications), UCLA.
Fall 2017:   Math 131A (Analysis), UCLA.
Spring 2017:   Math 170B (Probability), UCLA.
Winter 2017:   Math 170A (Probability), UCLA.
  Math 170B (Probability), UCLA.
Fall 2016:   Math 31A (Calculus), UCLA.
Winter 2016:   Math 131A (Analysis), UCLA.
  Math 170B (Probability), UCLA.
Fall 2015:   Math 131A (Analysis), UCLA.
  Math 134 (Linear and Nonlinear ODEs), UCLA.
Summer 2015:   Math 167 (Game Theory), UCLA.
Spring 2015:   Math 134 (Linear and Nonlinear ODEs), UCLA.
  Math 136 (Partial Differential Equations), UCLA.
Winter 2015:   Math 61 (Discrete Mathematics), UCLA.
  Math 170A (Probability), UCLA.
Fall 2014:   Math 1 (Precalculus), UCLA.
  Math 3A (Calculus for Life Sciences), UCLA.
Summer 2014:   Math 115A (Linear Algebra), UCLA.
Spring 2014:   Math 120B (Differential Geometry), UCLA.
  Math 135 (Ordinary Differential Equations), UCLA.
Winter 2014:   Math 134 (Linear and Nonlinear ODEs), UCLA.
  Math 135 (Ordinary Differential Equations), UCLA.
Fall 2013:   Math 31B (Calculus), UCLA.
Spring 2013:   Math 132 (Complex Analysis), UCLA.
Winter 2013:   Math 33B (Differential Equations), UCLA.
Fall 2012:   Math 1 (Precalculus), UCLA.




Contact


Address: Department of Mathematics
University of Connecticut
341 Mansfield Road U1009
Storrs, CT 06269-1009
Office: Monteith Building 409
Email: yunfeng.zhang@uconn.edu
Assistant: Dr. Mengmeng Zhang
(mmzhang@bnu.edu.cn)






Lie Groups and/or Analysis Seminars (*temporary)


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Last updated: April 10 2021