### 张云峰

 坐标 教学 研究 简介

### 研究

• 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

 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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