研究Research

研究RESEARCH

几何 · 分析 · 偏微分方程Geometry · Analysis · Partial Differential Equations

我目前的兴趣主要集中在微分几何、几何分析、椭圆型偏微分方程与凸几何的交汇处。 My current interests lie mainly at the interface of differential geometry, geometric analysis, elliptic PDEs, and convex geometry.

01

研究方向Research Interests

01

微分几何与几何分析Differential Geometry & Geometric Analysis

微分几何Differential Geometry 极小曲面Minimal Surfaces 几何分析Geometric Analysis 黎曼曲面Riemann Surfaces 共形几何Conformal Geometry Hodge 理论观点Hodge-theoretic ideas 几何 PDEGeometric PDE

这是我目前兴趣的核心——几何与分析交汇之处，涉及曲率、共形几何，以及支配它们的偏微分方程。其中，极小曲面是我反复回归的主题。

This is the core of my current interests — the place where geometry and analysis meet, through curvature, conformal geometry, and the partial differential equations that govern them. Of these, Minimal Surfaces is the theme I keep returning to.

02

椭圆型偏微分方程Elliptic Partial Differential Equations

非线性椭圆方程nonlinear elliptic equations 先验估计a priori estimates 极大值原理maximum principles Harnack 不等式Harnack inequalities blow-up 分析blow-up analysis Sobolev 方法Sobolev methods Moser 迭代Moser iteration 变分方法variational methods

这是一个研究兴趣，也是正在进行的学习。我正在通过仔细阅读来学习椭圆理论的经典技巧——先验估计、极大值原理、Harnack 不等式和 blow-up 分析，而非新的结果。

A research interest and an ongoing study. I am learning the classical techniques of elliptic theory — a priori estimates, maximum principles, Harnack inequalities, and blow-up analysis — through careful reading rather than new results.

03

凸几何与 Kakeya 型问题Convex Geometry & Kakeya-type Problems

凸几何Convex Geometry Kakeya 问题Kakeya Problem 旋转Rotations 几何优化Geometric Optimization

这是一个我正在探索的方向。考虑平面集合 \(S \subset \mathbb{R}^2\) 以及连续平移函数 \(x : [0, 2\pi) \to \mathbb{R}^2\)。我感兴趣的是包含所有旋转平移副本的集合 \(A\)， \[ R_\theta S + x(\theta). \]

A direction I am currently exploring. Consider a planar set \(S \subset \mathbb{R}^2\) and a continuous translation function \(x : [0, 2\pi) \to \mathbb{R}^2\). I am interested in a set \(A\) that contains all the rotated translated copies \[ R_\theta S + x(\theta). \]

我目前正在思考的问题包括：

Questions I am currently thinking about include:

  • 最小可能面积
  • minimal possible area
  • 直径
  • diameter
  • 周长
  • perimeter
  • 最优构型的几何结构
  • geometric structure of optimal configurations
  • 对原集合 \(S\) 的依赖
  • dependence on the original set \(S\)
  • \(S\) 收敛时的行为
  • behavior under convergence of \(S\)
  • 线段型构型与旋转对称构型之间的过渡
  • the transition between line-segment-type configurations and rotationally symmetric configurations

持续探索中Ongoing Exploration

02

当前项目Current Projects

这些是我正在积极研究的问题和方向——不是已完成的成果，而是我正在尝试理解的问题。 These are problems and directions I am actively working through — not completed results, but questions I am trying to understand.

01

凸 Kakeya 集合Convex Kakeya Sets

持续探索中Ongoing Exploration
凸几何Convex Geometry Kakeya 问题Kakeya Problem 旋转Rotations 几何优化Geometric Optimization

我正在探索 Kakeya 问题的一个凸几何变体：一个集合必须包含给定平面集合 \(S\) 的所有连续平移旋转副本。我感兴趣的是面积、直径和周长等几何量，以及最优构型的稳定性与极限行为。

I am exploring a convex-geometric variant of the Kakeya problem in which a set must contain continuously translated rotated copies of a prescribed planar set \(S\). I am interested in geometric quantities such as area, diameter and perimeter, as well as stability and limiting behavior of optimal configurations.

02

极小曲面与几何分析Minimal Surfaces & Geometric Analysis

当前兴趣 · 持续学习中Current Interest · Ongoing Study
极小曲面Minimal Surfaces 微分几何Differential Geometry 几何分析Geometric Analysis 黎曼曲面Riemann Surfaces

我感兴趣的是极小曲面，它作为微分几何、复分析与椭圆型偏微分方程的交汇点。

I am interested in minimal surfaces as a meeting point of differential geometry, complex analysis and elliptic PDEs.

  • 经典极小曲面理论
  • classical minimal surface theory
  • Weierstrass 表示
  • Weierstrass representation
  • 共形参数化
  • conformal parametrizations
  • 曲率
  • curvature
  • 调和坐标函数
  • harmonic coordinate functions
  • 与黎曼曲面的关系
  • relations with Riemann surfaces
03

非线性椭圆方程Nonlinear Elliptic Equations

阅读 · 研究学习Reading · Research Study
Harnack 不等式Harnack inequalities blow-up 分析blow-up analysis 先验估计a priori estimates 正解positive solutions

这与我对 Gidas–Spruck 等经典工作的长期阅读有关。我正在通过分析正解、先验估计和 blow-up 论证来学习非线性椭圆 PDE 的经典技巧。

This is tied to a long-running reading of classical work such as Gidas–Spruck. I am studying classical techniques in nonlinear elliptic PDE through the analysis of positive solutions, a priori estimates and blow-up arguments.

  • Harnack 不等式
  • Harnack inequalities
  • 正解的局部与全局行为
  • local and global behavior of positive solutions
  • blow-up 分析
  • blow-up analysis
  • 尺度论证
  • scaling arguments
  • 先验估计
  • a priori estimates
  • Sobolev 不等式
  • Sobolev inequalities
  • 共形变换
  • conformal transformations
  • 非线性椭圆方程
  • nonlinear elliptic equations

《Global and Local Behavior of Positive Solutions of Nonlinear Elliptic Equations》笔记 Global and Local Behavior of Positive Solutions of Nonlinear Elliptic Equations 笔记 →

04

球面上的 Poisson 方程Poisson Equation on the Sphere

探索 · 阐述性项目Exploration · Expository Project
椭圆 PDEElliptic PDE 球面分析Spherical Analysis 正则性Regularity

我正在研究球面上的 Poisson 方程， \[\Delta_{S^2} u = f,\] 目标是尽可能纯粹地从分析的角度理解它。

I am working through the Poisson equation on the sphere, \[\Delta_{S^2} u = f,\] with the aim of understanding it as purely analytically as possible.

  • 弱形式
  • weak formulation
  • Lax–Milgram 定理
  • Lax–Milgram theorem
  • Poincaré 不等式
  • Poincaré inequality
  • \(H^1\) 估计
  • \(H^1\) estimates
  • 正则性
  • regularity
  • Moser 迭代
  • Moser iteration
  • Green 函数
  • Green functions

主要目标是先理解其分析结构，再引入谱分解——之后再引入球谐函数与谱理论作为另一种观点。

The main goal is to understand the analytic structure first, before introducing the spectral decomposition — later bringing in spherical harmonics and spectral theory as another point of view.

05

谱几何Spectral Geometry

探索中Exploring
拉普拉斯特征值Laplacian eigenvalues 区域的几何Geometry of domains 谱不等式Spectral inequalities

这是一个探索性方向，而非主要项目。我感兴趣于：

An exploratory direction, rather than a main project. I am interested in:

  • 拉普拉斯特征值
  • Laplacian eigenvalues
  • 区域的几何
  • geometry of domains
  • 谱不等式
  • spectral inequalities
  • 几何与谱之间的关系
  • relationships between geometry and spectrum
  • 解析证明与计算机辅助证明
  • analytic proofs versus computer-assisted proofs
03

阅读与背景Reading & Background

我正在系统学习的内容，但我不将其视为独立的研究项目。 Things I am studying systematically, but which I do not treat as independent research projects.

微分流形Differential Manifolds

  • 单位分解
  • partitions of unity
  • Sard 定理
  • Sard theorem
  • Whitney 嵌入定理
  • Whitney embedding theorem
  • 向量场与流
  • vector fields and flows
  • Frobenius 定理
  • Frobenius theorem
  • 李群与李代数
  • Lie groups and Lie algebras
  • 微分形式
  • differential forms
  • Stokes 定理
  • Stokes theorem
  • de Rham 上同调
  • de Rham cohomology
  • 度理论
  • degree theory
  • Poincaré 对偶
  • Poincaré duality
  • Chern–Weil 理论入门
  • introductory Chern–Weil theory

几何分析Geometric Analysis

  • 极小曲面
  • minimal surfaces
  • 黎曼曲面
  • Riemann surfaces
  • Hodge 理论
  • Hodge theory
  • 椭圆 PDE
  • elliptic PDE

数学物理Mathematical Physics

长期兴趣 · 正在学习Long-term interest · currently learning
  • 量子力学
  • Quantum Mechanics
  • 量子场论
  • Quantum Field Theory
  • 规范理论
  • Gauge Theory
  • Yang–Mills 理论
  • Yang–Mills Theory
  • 弦论
  • String Theory
写作Writing

相关笔记Related Notes