The Third Edition · Kunming

第三届云师 数学前沿讲习班

YNNU Advanced Workshop in Mathematics III

  • 2026 年 7 月 20 日至 8 月 5 日
  • 云南师范大学数学学院
  • 昆明，中国
了解讲习班 查看日程
01
About the Workshop

会议简介

第三届云师数学前沿讲习班旨在为数学不同领域的研究者搭建开放而深入的交流平台， 通过短期课程、前沿报告与自由讨论，促进学术思想的传播、交汇与合作。

Scope 讲习班面向高校教师、青年学者及研究生。主题涵盖表示论、微分几何、几何拓扑、 调和分析、偏微分方程、数论、数学物理及相关方向。

第三届云师数学前沿讲习班参会人员合影
第三届云师数学前沿讲习班合影
02
Mini-courses

短期课程

Mini-course 01 01
课程已公布

Classification of solutions to nonlinear elliptic partial differential equations—a vector field approach

授课教师 麻希南
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Course Abstract

课程摘要

The energy method is a powerful tool for establishing a priori estimates for elliptic partial differential equations; from another perspective, it is the vector field method. Inspired by geometric problems such as the Bochner technique and the Obata method, the vector field method for elliptic PDEs has been applied to various equations since the 1970s, such as the applications to second-order elliptic equations by Gidas-Spruck and Serrin-Zou, as well as our recent new insights into sub-elliptic equations on the Heisenberg group and a class of fourth-order elliptic equations. We will give a brief review of the history and techniques of the related problems, and elaborate in detail on their applications to extremal functions and best constants for the relevant second-order and fourth-order Sobolev inequalities.

Mini-course 02 02
课程已公布

Yamabe 型流

Yamabe Type Flow

授课教师 严泽田

本课程介绍 Yamabe 型流中的长期收敛与冒泡分析。 首先以 Brendle 的 Yamabe 流收敛定理为主线，讲解能量单调性、 Lojasiewicz 型不等式、浓度紧性及严格低于球面能量的测试函数构造。 随后讨论 Paneitz 算子与 Q-曲率流，重点介绍非局部流的收敛机制、 高阶 bubble 修正、谱分解与误差估计。课程旨在说明如何结合几何分析、 变分方法和精细测试函数构造，克服临界共形问题中的非紧性。

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Lecture Map

讲义提要

  1. 01
    Yamabe 流的基本结构

    从归一化 Yamabe 流的演化方程出发，讨论体积保持、 能量单调性、有限时间估计与长期存在。

  2. 02
    Brendle 的长期收敛框架

    说明序列型 Lojasiewicz 估计如何升级为全时间估计， 并导出曲率误差的可积性、无集中性与光滑收敛。

  3. 03
    浓度紧性与测试 bubble

    结合 Struwe 分解、bubble 参数分离与 Green 函数尾部， 构造能量严格低于球面值的测试函数。

  4. 04
    Paneitz 算子与非局部 Q-曲率流

    介绍共形协变结构、Gursky–Malchiodi 非局部流、 正性与能量衰减，以及临界 Paneitz–Sobolev 商的 Palais–Smale 结构。

  5. 05
    高阶修正与收敛机制

    比较低维与高维 bubble 构造，讲解修正项、谱分解、 强制性与误差估计如何共同排除冒泡并导出收敛。

Mini-course 03 03
课程已公布

流形上的微积分

授课教师 张秉宇

本课程介绍如何将欧式空间上的微积分推广到一种常见的弯曲空间——微分流形上。

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Recommended Reading

参考书目

  1. M. Spivak, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. W. A. Benjamin, 1965.
  2. M. P. do Carmo, Riemannian Geometry. Birkhäuser, 1992.
  3. 陈维桓，《微分流形初步》，北京大学出版社。
Mini-course 04 04
课程已公布

双曲空间及其推广上的薛定谔方程

Schrödinger Equations on Hyperbolic Spaces and Beyond

授课教师 张鸿伟

本课程介绍实双曲空间上的调和分析及其在薛定谔方程中的应用： 从演化算子卷积核的逐点估计与色散估计出发，建立 Strichartz 估计并讨论半线性问题； 还将延伸至高秩非紧对称空间及部分局部对称空间。

Prerequisites · 先修要求

建议具备经典 Fourier 分析、群论与偏微分方程基础； 熟悉 Lie 群和 Lie 代数有助于理解第 4、5 部分。

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Course Outline

课程内容

  1. 01
    实双曲空间上的调和分析

    介绍实双曲空间的几何与代数结构、相应的 Fourier 分析、 球函数及 Harish–Chandra c-函数。

  2. 02
    卷积核的逐点估计

    运用 Fourier 分析建立薛定谔演化算子卷积核的逐点估计。

  3. 03
    色散估计、Strichartz 估计及应用

    结合逐点核估计与 Kunze–Stein 卷积估计证明色散估计， 再通过标准的 TT* 论证建立 Strichartz 估计， 并应用于半线性薛定谔方程的适定性与散射。

  4. 04
    高秩非紧对称空间 选讲

    介绍高秩空间的结构、高维 Weyl 室带来的困难， 以及重心分解方法与处理 Harish–Chandra c-函数的基本思路。

  5. 05
    局部对称空间 选讲

    讨论离散子群及其对算子谱的影响，并介绍这一背景下 Strichartz 估计的相关结果。

完整参考文献（11 项）
Schrödinger Equations in Euclidean Spaces
  1. T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003, xiv+323 pp.
  2. T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, vol. 106, American Mathematical Society, Providence, RI, 2006, xvi+373 pp.
Background of Hyperbolic / Symmetric Spaces
  1. P. B. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996, vii+449 pp.
  2. A. W. Knapp, Lie Groups Beyond an Introduction, second edition, Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002, xviii+812 pp.
Harmonic Analysis on Hyperbolic / Symmetric Spaces
  1. W. O. Bray, Aspects of Harmonic Analysis on Real Hyperbolic Space, Lecture Notes in Pure and Applied Mathematics, vol. 157, Marcel Dekker, Inc., New York, 1994, 77–102.
  2. S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions , Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI, 2000, xxii+667 pp.
Research Papers
  1. J.-P. Anker and V. Pierfelice, Nonlinear Schrödinger equation on real hyperbolic spaces, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 5, 1853–1869.
  2. J.-P. Anker and H.-W. Zhang, Wave equation on general Riemannian non-compact symmetric spaces , Amer. J. Math. 146 (2024), no. 4, 983–1031.
  3. N. Burq, C. Guillarmou, and A. Hassell, Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics , Geom. Funct. Anal. 20 (2010), no. 3, 627–656.
  4. A. Fotiadis, N. Mandouvalos, and M. Marias, Schrödinger equations on locally symmetric spaces, Math. Ann. 371 (2018), no. 3–4, 1351–1374.
  5. A. D. Ionescu and G. Staffilani, Semilinear Schrödinger flows on hyperbolic spaces: scattering H1 , Math. Ann. 345 (2009), no. 1, 133–158.
03
Invited Lectures

学术报告

Invited Talk 01 01

Speaker · 报告人 孟龙

Approximate eigenfunctions for some aperiodic crystals

Abstract · 摘要

The electronic structure of aperiodic crystals poses a major challenge in quantum condensed matter physics. While periodic systems benefit from Bloch's theorem, aperiodic materials are typically studied via tight-binding models or mesoscopic effective Hamiltonians due to the loss of translation invariance. Extending such analyses to fully atomic-scale Hamiltonians, however, introduces substantial difficulties.

In this talk, I will present a rigorous mathematical framework to address this problem. Under explicit and easily verified assumptions, we derive sufficient conditions for the emergence of approximately localized eigenfunctions in aperiodic systems. Moreover, we establish a direct connection between these approximate eigenfunctions and their effective Hamiltonian counterparts, and we observe some new terms never explained in physics. Finally, I will illustrate our findings with applications to the Landau–Schrödinger, Landau–Dirac, and quantum harmonic oscillator effective Hamiltonians.

Invited Talk 02 02

Speaker · 报告人 曲华迪

Spin number of symplectic diffeomorphisms and applications

Abstract · 摘要

In this talk, we discuss Ruelle rotation invariants for symplectic diffeomorphisms and their applications to periodic-orbit forcing. For a compact symplectic manifold admitting global symplectic trivializations, we introduce a spin number that measures the average rotational component of a tangent vector under the dynamics. The spin number is well defined almost everywhere by the Birkhoff ergodic theorem; its integration over the volume form is known as the Ruelle invariant.

In the three-dimensional contact manifold case, this invariant has found significant use in recent progress in ECH theory and contact geometry. We establish a formula describing the change of the Ruelle invariant under a change of trivialization and give applications of this quantity to area-preserving surface diffeomorphisms, especially in the case of disk diffeomorphisms.

Invited Talk 03 03

Speaker · 报告人 王云翔

Sharp Lp-Estimates for Wave Equation on ax+b Groups

Abstract · 摘要

In this talk we discuss the Lp-estimates for wave equation on the n-dimensional ax+b group G := ℝ+ ⋉ ℝn−1 . Let dρ be the right Haar measure and L be the positive definite distinguished Laplacian on G. Let u = u(t, ·) be the solution to

∂ttu + Lu = 0, u(0, ·) = f, ∂tu(0, ·) = 0.

We show that for t ∈ ℝ ∖ {0}, α ∈ ℝ and 1 < p < ∞,

‖u(t, ·)‖Lp(dρ) ≲p (1 + |t|)2|1/p−1/2| ‖f‖Lpα(dρ)

holds if and only if

α ≥ (n−1)|1/p−1/2|.

This gives an endpoint result conjectured by Müller and Thiele.

04
Programme

课程与报告日程

Preliminary Programme

初步日程

2026 年 7 月 27 日 — 8 月 4 日 课程与报告的时间和教室已按当前安排列出。

  • 午休 12:15–14:30
  • 地点 汇学1-102 / 406

以下日程已按日期分组，空白时段不显示。

日期 09:00–10:30 10:45–12:15 14:30–15:30 15:30–16:30
星期一 张秉宇 406 教室 张秉宇 406 教室
星期二 张秉宇 406 教室 张秉宇 406 教室
星期四 麻希南 汇学1-102 王云翔 406 教室 张鸿伟 406 教室 张鸿伟 406 教室
星期五 麻希南 汇学1-102 AI 课程 406 教室 张鸿伟 406 教室 张鸿伟 406 教室
星期六 麻希南 汇学1-102 张秉宇 406 教室 张鸿伟 406 教室 张鸿伟 406 教室
星期日 麻希南 汇学1-102 严泽田 406 教室 严泽田 汇学1-102
星期一 严泽田 406 教室 孟龙 406 教室 严泽田 406 教室 严泽田 406 教室
星期二 严泽田 406 教室 曲华迪 406 教室 严泽田 406 教室 严泽田 406 教室

注：以上为初步日程，安排如有调整，以后续通知为准。

05
Visitor Information

会务信息

Date 2026 年 7 月 20 日
至 8 月 5 日 20 July — 5 August 2026
Venue 云南师范大学
数学学院 School of Mathematics, YNNU
City 昆明
中国 Kunming, China
06
Contact

联系方式

如有关于讲习班的问题，请通过电子邮件联系。

yangzhipeng326@163.com