Riemannian geometry and geometric Analysis      jurgen Jost

[丘成桐數學研究中心]---求真  幾何的Langlands綱領  猜想   [單值化問題]   [Einstein-Yang-Mills-Dirac theory]

[Seminar on Differential Geometry]   [Charles Morrey]   [鄭紹遠]   [李偉光] [ or]  [Rick Schoen]   [Leon Simon]  [Simon Donaldson]  [Gang Tian]

[Geometric Analysis   Peter Li]   [Simon Donaldson] 4維流形的瞬子   [Clifford Taubes]   [瞬子與Yang-Mills流中的奇異點 Alex Waldron]

幾何分析的基本哲學 : 幾何結構取決於由自身構造出的一些方程式的解。幾何分析(丘)

研究計畫

第一章 Remannian Manifolds

  1. 正則值原像定理[preimage theorem含隱函數定理]
  2. Hodler不等式   PDE的能量估計   能量泛函的Euler Lagrange方程是測地線方程[與 [哈密頓方程等價]
    [梯度估計與幾何分析]   [幾何分析]
  3. 深入了解黎曼流形上的指數映射
  4. Riemannain Metric  等距同構(isometry)
  5. Existence of Geodesics on Compact Manifolds
  6. heat flow method and the existence of Geodesics
  7. Existence of Geodesics on Complete Manifolds

第二章 Lie groups and Vector Bundles   [Bundle chart of TS^1]

  1. Vector Bundles
  2. Complex and Holomorphic Vector Bundles
  3. Integral Curves of Vector Fields : Lie Algebra
  4. Symplectic Structures
  5. Lie Groups   如何構造left invariant vector field
  6. Spin Structures

第三章 The Laplace Operator and Harmonic Differential Forms

  1. The Laplace Operator on functions    Sobolev space  (Laplacian的幾何意義  Dirichlet積分) Sobolev Space
  2. The spectrum of the Laplace operator   [Laplacian on a manifold]
  3. The Laplace Operator on Forms  (codifferential)
  4. Representing Cohomology Classes by harmonic forms
  5. The heat flow and harmonic forms
  6. 第三章習作
  7. Harnack不等式
  8.  [Rellich Embedding Theorem] [Laplacian的幾何意義]...量化了函數在該點附近的平均值與中心值得偏差
  9. Dual connection
  10. 黎曼流形的Laplace-Beltrami算子的譜(特徵值 特徵函數)決定了(例 熱方程)的[Green function]與熱核[heat kernel]
    Green function method不同領域的應用   [Sturm-Liouville 定]理  [Green function method在heat equation中的例子]

第四章 Connections and Curvature       [二十一世紀數學的挑戰]  [規範場論與微分幾何]

電磁場[Spinor1301EMF] U(1) 電磁交互作用的規範場

  1. Connections in Vector Bundles
  2. Metric connection的Leibniz法則   The Yang-Mills functional  [習作]
  3. The Levi-Civita connection    [N3401LieDerivative]
  4. Connections for Spin Structures and the Dirac Operator
  5. The Bochner Method
  6. Eigenvalue Estimates (by Li-Yau)
  7. 第四章習作

第五章 Geometry of Submanifolds

  1. [Induced connection and Second fundamental form]
  2. [Curvature of submanifolds]
  3. The volume of submanifold
  4. Minimal submanifolds
  5. 第五章習作

第六章 Geodesic and Jacobi Fields

  1. 弧長與能量的第一第二變分
  2. Jacobi Fields
  3. Conjugate points and the distance minimizing geodesics
  4. Riemannain manifolds of constant curvature
  5. Rauch comparison theorem
  6. The Hessian of the squared distance function
  7. Volume comparison
  8. Approximate Fundamental Solutions and Representation Formula
  9. The Geometry of Manifolds of Nonpositive Sectional Curvature
  10. 第六章習作

第七章 Symmetric Spaces and Kahler Manifolds

  1. Complex Projective Space
  2. Kahler Manifolds
  3. The Geometry of Symmetric Spaces
  4. Structure of Symmetric Spaces
  5. The Space SL(n,R)/SO(n,R)
  6. Symmetric Spaces of Noncompact Type:

第八章 Morse Theory and Floer Homology

  1. Aims of Morse Theory
  2. Compactness:
  3. Local Analysis:
  4. Limits of Trajectories of the Gradient Flow
  5. The Morse-Smale-Floer Condition:
  6. Orientations and Z-homology
  7. Homotopies
  8. Graph Flows
  9. Orientations
  10. The Morse Inequality
  11. The Palais-Smale Condition and the Existence of closed Geodesics

第九章 Hamonic Maps between Riemannian Manifolds

  1. Harmonic Maps (Definition and Formula):The Bochner Technique

第十章 Harmonic Maps from Riemann Surfaces

第11章 Variational problems from Quntum Field Theory

  1. The Ginzburg-Landau Functional
  2. The Seiberg-Witten Functional
  3. Dirac-Harmonic Maps