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张云峰

康涅狄格大学数学系
坐标
教学
研究
简介

简介


欢迎来到我的数学页面. 我目前在康涅狄格大学工作 (Assistant Research Professor). 在2021年秋季学期我将去北京大学工作 (TAL Assistant Professor). 我在2018年从加州大学洛杉矶分校获得博士学位, 我的导师是 Rowan KillipMonica Visan.



研究


我对分析和表示论感兴趣, 尤其半单李群和对称空间上的调和分析及其应用. 我最近在做整体对称空间上所谓傅里叶限制类型的估计, 比如关于色散方程的Strichartz估计和Laplace特征函数的Lebesgue范数估计. 这些问题在欧氏空间上已然具有丰富的理论, 而在局部对称空间上考虑则会更加有趣. 我的论文如下:

  • 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.





教学


目前在2021春季学期, 我在线上教偏微分方程 (Math 3435). 之前的主讲教学记录如下:

2020秋:   Math 3435 (Partial Differential Equations), UConn.
2020春:   Math 2360Q (Axiomatic Geometry), UConn.
2019秋:   Math 3146 (Introduction to Complex Variables), UConn.
2019春:   Math 3435 (Partial Differential Equations), UConn.
2018秋:   Math 1152Q (Honors Calculus II), UConn.
  Math 2130Q (Honors Multivariable Calculus), UConn.
2017夏:   Math 3B (Calculus for the Life Sciences), UCLA.




坐标


地址 : Department of Mathematics
University of Connecticut
341 Mansfield Road U1009
Storrs, CT 06269-1009
办公室 : Monteith Building 409
电邮 : yunfeng.zhang@uconn.edu
助理 : Dr. Mengmeng Zhang
(mmzhang@bnu.edu.cn)






李群和/或分析研讨会 (*临时)


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最后更新: April 15 2021