Table of contents

CONTENTS § 1. Introduction § 2. The action of Hecke operators on non-homogeneous theta-series § 3. Siegel measures and indices of groups of units § 4. Multiplication of weighted sums of representations by Hecke elements § 5. Factorization of integral representations of integers by quadratic forms § 6. Composition of the representation masses for integral representations § 7. Concluding remarks References

CONTENTS I. Introduction II. Topology of Grassmann manifolds 1. Local coordinates 2. The cell decomposition and basic topological characteristics 3. Plücker coordinates III. Riemannian geometry of Grassmann manifolds: geometric approach 1. The metric and angles between planes 2. Curvature tensor, sectional curvature, closed geodesics, the limit set 3. More about Plücker embeddings 4. G⁺(2,n) as a Kähler manifold IV. Grassmann manifolds as symmetric spaces 1. The structure of a symmetric space 2. Totally geodesic and totally umbilical submanifolds 3. Standard embeddings of Grassmann manifolds in Euclidean space 4. Generalization of Grassmann manifolds V. Grassmann image. Intrinsic geometry 1. Induced metric. Homothety 2. Volume of the Grassmann image 3. Grassmann image of minimal surfaces 4. Harmonicity of the Grassmann map VI. Extrinsic geometry of the Grassmann image 1. Curvature of a Grassmann manifold along the Grassmann image of a surface 2. Reconstruction of a surface from the Grassmann image 3. Second fundamental form of the Grassmann image. Surfaces with totally geodesic and totally umbilical Grassmann image VII. Notes References

CONTENTS Introduction § 1. Semigroups of weighted translation operators § 2. Algebras of weighted translation operators § 3. Weighted translation operators and the spectral theory of linear extensions § 4. Weighted translation operators and the multiplicative ergodic theorem References

CONTENTS Introduction Chapter I. Classification of simply-connected topological four-dimensional manifolds § 1. Intersection forms of four-dimensional manifolds § 2. The Pontryagin-Whitehead and Novikov-Wall theorems § 3. Handle body theory. The h-cobordism theorem § 4. Casson handles § 5. Geometric control theorem. Casson handle design § 6. Freedman's theorem and its corollaries § 7. Classification of simply-connected topological four-dimensional manifolds Chapter II. Geometric methods § 1. Gauge fields and Donaldson's theorem § 2. Construction of cobordism § 3. Further results § 4. Floer homology References