Gradient Flow in Metric Spaces -- Curves and Gradients in Metric Spaces -- Existence of Curves of Maximal Slope and their Variational Approximation -- Proofs of the Convergence Theorems -- Uniqueness, Generation of Contraction Semigroups, Error Estimates -- Notation -- Gradient Flow in the Space of Probability Measures -- Preliminary Results on Measure Theory -- The Optimal Transportation Probl…
Free Resolutions and Hilbert Functions -- First Examples of Free Resolutions -- Points in ?2 -- Castelnuovo-Mumford Regularity -- The Regularity of Projective Curves -- Linear Series and 1-Generic Matrices -- Linear Complexes and the Linear Syzygy Theorem -- Curves of High Degree -- Clifford Index and Canonical Embedding.Algebraic Geometry often seems very abstract, but in fact it is full of co…
Sheaves and all that -- The category of differential triads -- Lie sheaves of groups -- Principal sheaves -- Vector and associated sheaves -- Connections on principal sheaves -- Connections on vector and associated sheaves -- Curvature -- Chern-Weil theory -- Applications and further examples.
Preface -- Part I Subspaces and Sequences in Banach Spaces: Disposition of Subspaces -- Sequences in Normed Spaces -- Isomorphism, Isometries and Embeddings -- Spaces of Universal Disposition -- Bounded Approximation Properties -- Part II On the Geometry of Müntz Sequences: Coefficient Estimates and the Müntz Theorem -- Classification and Elementary Properties of Müntz Sequences -- More on t…
Beauville surfaces without real structures -- Couniformization of curves over number fields -- On the V-filtration of -modules -- Hecke orbits on Siegel modular varieties -- Ax-Kochen-Eršov Theorems for p-adic integrals and motivic integration -- Nested sets and Jeffrey-Kirwan residues -- Counting extensions of function fields with bounded discriminant and specified Galois group -- Classical a…
Introductory Chapter -- Laplace Operators on Riemannian Manifolds -- Lagrangian Formalism on Riemannian Manifolds -- Harmonic Maps from a Lagrangian Viewpoint -- Conservation Theorems -- Hamiltonian Formalism -- Hamilton-Jacobi Theory -- Minimal Hypersurfaces -- Radially Symmetric Spaces -- Fundamental Solutions for Heat Operators with Potentials -- Fundamental Solutions for Elliptic Operators …
Frobenius Splitting: General Theory -- Frobenius Splitting of Schubert Varieties -- Cohomology and Geometry of Schubert Varieties -- Canonical Splitting and Good Filtration -- Cotangent Bundles of Flag Varieties -- Equivariant Embeddings of Reductive Groups -- Hilbert Schemes of Points on Surfaces.The theory of Frobenius splittings has made a significant impact in the study of the geometry of f…
Straightedge and compass -- Euclid’s approach to geometry -- Coordinates -- Vectors and Euclidean spaces -- Perspective -- Projective planes -- Transformations -- Non-Euclidean geometry.For two millennia the right way to teach geometry was the Euclidean approach, and in many respects, this is still the case. But in the 1950s the cry "Down with triangles!" was heard in France and new geometry …
Modular Forms, Elliptic Curves, and Modular Curves -- Modular Curves as Riemann Surfaces -- Dimension Formulas -- Eisenstein Series -- Hecke Operators -- Jacobians and Abelian Varieties -- Modular Curves as Algebraic Curves -- The Eichler-Shimura Relation and L-functions -- Galois Representations. .This book introduces the theory of modular forms with an eye toward the Modularity Theorem: All r…
Infinite Galois Theory and Profinite Groups -- Valuations and Linear Disjointness -- Algebraic Function Fields of One Variable -- The Riemann Hypothesis for Function Fields -- Plane Curves -- The Chebotarev Density Theorem -- Ultraproducts -- Decision Procedures -- Algebraically Closed Fields -- Elements of Algebraic Geometry -- Pseudo Algebraically Closed Fields -- Hilbertian Fields -- The Cla…