解析几何学家与代数几何学家通常研究相同的几何结构，但运用不同的方法。虽然这种对偶方法在解决问题方面取得了令人瞩目的成功，但代数和分析之间的语言差异对于学习几何的学生和研究人员来说也是一个困难，特别是复几何学。
PCMI(Park City Mathematics Institute)计划旨在通过深入浅出的语言来介绍解析几何与代数几何中的一些新进展，从而部分解决这种语言鸿沟问题。暑期学校的一个焦点是乘子理想，这是目前解析几何与代数几何两个领域都广泛关注的课题。
杰弗里·麦克尼尔、米尔恰·穆斯塔塔著的《解析几何与代数几何--相同问题不同方法(英文版)(精)/美国数学会经典影印系列》源于以解析几何和代数几何为主题的PCMI暑期学校的一系列讲座。该系列讲座旨在介绍解析几何和代数几何中最新进展背后所运用的高级技巧。讲座包含了许多说明性的例子、详细的计算和关于提出主题的新观点，以便增强非专业人士对这些材料的理解。

Preface
Jeffery McNeal and Mircea Mustata
Introduction
Bo Berndtsson
An Introduction to Things
Introduction
Lecture 1. The one-dimensional case
1.1. The 0-equation in one variable
1.2. An alternative proof of the basic identity
1.3. An application: Inequalities of Brunn-Minkowski type
1.4. Regularity -- a disclaimer
Lecture 2. Functional analytic interlude
2.1. Dual formulation of the problem
Lecture 3. The -equation on a complex manifold
3.1. Metrics
3.2. Norms of forms
3.3. Line bundles
3.4. Calculation of the adjoint and the basic identity
3.5. The main existence theorem and L2-estimate for compact manifolds
3.6. Complete Kahler manifolds
Lecture 4. The Bergman kernel
4.1. Generalities
4.2. Bergman kernels associated to complex line bundles
Lecture 5. Singular metrics and the Kawamata-Viehweg vanishing theorem
5.1. The Demailly-Nadel vanishing theorem
5.2. The Kodaira embedding theorem
5.3. The Kawamata-Viehweg vanishing theorem
Lecture 6. Adjunction and extension from divisors
6.1. Adjunction and the currents defined by divisors
6.2. The Ohsawa-Takegoshi extension theorem
Lecture 7. Deformational invariance of plurigenera
7.1. Extension of pluricanonical forms
Bibliography
John P. D'Angelo
Real and Complex Geometry meet the Cauchy-Riemann Equations
Preface
Lecture 1. Background material
1. Complex linear algebra
2. Differential forms
3. Solving the Cauchy-Riemann equations
Lecture 2. Complex varieties in real hypersurfaces
1. Degenerate critical points of smooth functions
2. Hermitian symmetry and polarization
3. Holomorphic decomposition
4. Real analytic hypersurfaces and subvarieties
5. Complex varieties, local algebra, and multiplicities
Lecture 3. Pseudoconvexity, the Levi form, and points of finite type
1. Euclidean convexity
2. The Levi form
3. Higher order commutators
4. Points of finite type
5. Commutative algebra
6. A return to finite type
7. The set of finite type points is open
Lecture 4. Kohn's algorithm for subelliptic multipliers
1. Introduction
2. Subelliptic estimates
3. Kohn's algorithm
4. Kohn's algorithm for holomorphic and formal germs
5. Failure of effectiveness for Kohn's algorithm
6. Triangular systems
7. Additional remarks
Lecture 5. Connections with partial differential equations
1. Finite type conditions
2. Local regularity for
3. Hypoellipticity, global regularity, and compactness
4. An introduction to L2-estimates
Lecture 6. Positivity conditions
1. Introduction
2. The classes Pk
3. Intermediate conditions
4. The global Canchy-Schwarz inequality
5. A complicated example
6. Stabilization in the bihomogeneous polynomial case
7. Squared norms and proper mappings between balls
8. Holomorphic line bundles
……