Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book studies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q. In general, these varieties are not compact. The intersection cohomology of the Shimura variety associated to a reductive group G carries commuting actions of the absolute Galois group of the reflex field and of the group G(Af) of finite adelic points of G. The second action can be studied on the set of complex points of the Shimura variety. In this book, Sophie Morel identifies the Galois action--at good places--on the G(Af)-isotypical components of the cohomology. Morel uses the method developed by Langlands, Ihara, and Kottwitz, which is to compare the Grothendieck-Lefschetz fixed point formula and the Arthur-Selberg trace formula. The first problem, that of applying the fixed point formula to the intersection cohomology, is geometric in nature and is the object of the first chapter, which builds on Morel's previous work. She then turns to the group-theoretical problem of comparing these results with the trace formula, when G is a unitary group over Q. Applications are then given. In particular, the Galois representation on a G(Af)-isotypical component of the cohomology is identified at almost all places, modulo a non-explicit multiplicity. Morel also gives some results on base change from unitary groups to general linear groups.

Shimura varieties. --- Homology theory. --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Varieties, Shimura --- Arithmetical algebraic geometry --- Accuracy and precision. --- Adjoint. --- Algebraic closure. --- Archimedean property. --- Automorphism. --- Base change map. --- Base change. --- Calculation. --- Clay Mathematics Institute. --- Coefficient. --- Compact element. --- Compact space. --- Comparison theorem. --- Conjecture. --- Connected space. --- Connectedness. --- Constant term. --- Corollary. --- Duality (mathematics). --- Existential quantification. --- Exterior algebra. --- Finite field. --- Finite set. --- Fundamental lemma (Langlands program). --- Galois group. --- General linear group. --- Haar measure. --- Hecke algebra. --- Homomorphism. --- L-function. --- Logarithm. --- Mathematical induction. --- Mathematician. --- Maximal compact subgroup. --- Maximal ideal. --- Morphism. --- Neighbourhood (mathematics). --- Open set. --- Parabolic induction. --- Permutation. --- Prime number. --- Ramanujan–Petersson conjecture. --- Reductive group. --- Ring (mathematics). --- Scientific notation. --- Shimura variety. --- Simply connected space. --- Special case. --- Sub"ient. --- Subalgebra. --- Subgroup. --- Symplectic group. --- Theorem. --- Trace formula. --- Unitary group. --- Weyl group.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the well-known Yamabe type. They involve Schrödinger operators on the left hand side and a critical nonlinearity on the right hand side. A significant development in the study of such equations occurred in the 1980's. It was discovered that the sequence splits into a solution of the limit equation--a finite sum of bubbles--and a rest that converges strongly to zero in the Sobolev space consisting of square integrable functions whose gradient is also square integrable. This splitting is known as the integral theory for blow-up. In this book, the authors develop the pointwise theory for blow-up. They introduce new ideas and methods that lead to sharp pointwise estimates. These estimates have important applications when dealing with sharp constant problems (a case where the energy is minimal) and compactness results (a case where the energy is arbitrarily large). The authors carefully and thoroughly describe pointwise behavior when the energy is arbitrary. Intended to be as self-contained as possible, this accessible book will interest graduate students and researchers in a range of mathematical fields.

Calculus of variations. --- Differential equations, Nonlinear. --- Geometry, Riemannian. --- Riemann geometry --- Riemannian geometry --- Generalized spaces --- Geometry, Non-Euclidean --- Semi-Riemannian geometry --- Nonlinear differential equations --- Nonlinear theories --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Asymptotic analysis. --- Cayley–Hamilton theorem. --- Contradiction. --- Curvature. --- Diffeomorphism. --- Differentiable manifold. --- Equation. --- Estimation. --- Euclidean space. --- Laplace's equation. --- Maximum principle. --- Nonlinear system. --- Polynomial. --- Princeton University Press. --- Result. --- Ricci curvature. --- Riemannian geometry. --- Riemannian manifold. --- Simply connected space. --- Sphere theorem (3-manifolds). --- Stone's theorem. --- Submanifold. --- Subsequence. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Unit sphere.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The description for this book, Knot Groups. Annals of Mathematics Studies. (AM-56), Volume 56, will be forthcoming.

Topology --- 512 --- Algebra --- 512 Algebra --- Knot theory. --- Knots (Topology) --- Low-dimensional topology --- Abelian group. --- Alexander duality. --- Alexander polynomial. --- Algebraic theory. --- Algorithm. --- Analytic continuation. --- Associative property. --- Automorphism. --- Axiom. --- Bijection. --- Binary relation. --- Calculation. --- Central series. --- Characterization (mathematics). --- Cobordism. --- Coefficient. --- Cohomology. --- Combinatorics. --- Commutator subgroup. --- Complete theory. --- Computation. --- Conjugacy class. --- Conjugate element (field theory). --- Connected space. --- Connectedness. --- Coprime integers. --- Coset. --- Covering space. --- Curve. --- Cyclic group. --- Dehn's lemma. --- Determinant. --- Diagonalization. --- Diagram (category theory). --- Dimension. --- Direct product. --- Equivalence class. --- Equivalence relation. --- Euclidean space. --- Euler characteristic. --- Existential quantification. --- Fiber bundle. --- Finite group. --- Finitely generated module. --- Frattini subgroup. --- Free abelian group. --- Fundamental group. --- Geometry. --- Group ring. --- Group theory. --- Group with operators. --- Hausdorff space. --- Homeomorphism. --- Homology (mathematics). --- Homomorphism. --- Homotopy group. --- Homotopy. --- Identity matrix. --- Inner automorphism. --- Interior (topology). --- Intersection number (graph theory). --- Knot group. --- Linear combination. --- Manifold. --- Mathematical induction. --- Monomorphism. --- Morphism. --- Morse theory. --- Natural transformation. --- Non-abelian group. --- Normal subgroup. --- Orientability. --- Permutation. --- Polynomial. --- Presentation of a group. --- Principal ideal domain. --- Principal ideal. --- Root of unity. --- Semigroup. --- Simplicial complex. --- Simply connected space. --- Special case. --- Square matrix. --- Subgroup. --- Subset. --- Summation. --- Theorem. --- Three-dimensional space (mathematics). --- Topological space. --- Topology. --- Torus knot. --- Transfinite number. --- Trefoil knot. --- Trichotomy (mathematics). --- Trivial group. --- Triviality (mathematics). --- Two-dimensional space. --- Unit vector. --- Wreath product.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book proves an analogue of William Thurston's celebrated hyperbolic Dehn surgery theorem in the context of complex hyperbolic discrete groups, and then derives two main geometric consequences from it. The first is the construction of large numbers of closed real hyperbolic 3-manifolds which bound complex hyperbolic orbifolds--the only known examples of closed manifolds that simultaneously have these two kinds of geometric structures. The second is a complete understanding of the structure of complex hyperbolic reflection triangle groups in cases where the angle is small. In an accessible and straightforward manner, Richard Evan Schwartz also presents a large amount of useful information on complex hyperbolic geometry and discrete groups. Schwartz relies on elementary proofs and avoids "ations of preexisting technical material as much as possible. For this reason, this book will benefit graduate students seeking entry into this emerging area of research, as well as researchers in allied fields such as Kleinian groups and CR geometry.

CR submanifolds. --- Dehn surgery (Topology). --- Three-manifolds (Topology). --- CR submanifolds --- Dehn surgery (Topology) --- Three-manifolds (Topology) --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Cauchy-Riemann submanifolds --- Submanifolds, CR --- Low-dimensional topology --- Topological manifolds --- Surgery (Topology) --- Manifolds (Mathematics) --- Arc (geometry). --- Automorphism. --- Ball (mathematics). --- Bijection. --- Bump function. --- CR manifold. --- Calculation. --- Canonical basis. --- Cartesian product. --- Clifford torus. --- Combinatorics. --- Compact space. --- Conjugacy class. --- Connected space. --- Contact geometry. --- Convex cone. --- Convex hull. --- Coprime integers. --- Coset. --- Covering space. --- Dehn surgery. --- Dense set. --- Diagram (category theory). --- Diameter. --- Diffeomorphism. --- Differential geometry of surfaces. --- Discrete group. --- Double coset. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence class. --- Equivalence relation. --- Euclidean distance. --- Four-dimensional space. --- Function (mathematics). --- Fundamental domain. --- Geometry and topology. --- Geometry. --- Harmonic function. --- Hexagonal tiling. --- Holonomy. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Horosphere. --- Hyperbolic 3-manifold. --- Hyperbolic Dehn surgery. --- Hyperbolic geometry. --- Hyperbolic manifold. --- Hyperbolic space. --- Hyperbolic triangle. --- Hypersurface. --- I0. --- Ideal triangle. --- Intermediate value theorem. --- Intersection (set theory). --- Isometry group. --- Isometry. --- Limit point. --- Limit set. --- Manifold. --- Mathematical induction. --- Metric space. --- Möbius transformation. --- Parameter. --- Parity (mathematics). --- Partial derivative. --- Partition of unity. --- Permutation. --- Polyhedron. --- Projection (linear algebra). --- Projectivization. --- Quotient space (topology). --- R-factor (crystallography). --- Real projective space. --- Right angle. --- Sard's theorem. --- Seifert fiber space. --- Set (mathematics). --- Siegel domain. --- Simply connected space. --- Solid torus. --- Special case. --- Sphere. --- Stereographic projection. --- Subgroup. --- Subsequence. --- Subset. --- Tangent space. --- Tangent vector. --- Tetrahedron. --- Theorem. --- Topology. --- Torus. --- Transversality (mathematics). --- Triangle group. --- Union (set theory). --- Unit disk. --- Unit sphere. --- Unit tangent bundle.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattés map has been made more inclusive, and the écalle-Voronin theory of parabolic points is described. The résidu itératif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated. Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.

Functions of complex variables --- Holomorphic mappings --- Riemann surfaces --- Fonctions d'une variable complexe --- Applications holomorphes --- Riemann, surfaces de --- Holomorphic mappings. --- Mappings, Holomorphic --- Functions of complex variables. --- Riemann surfaces. --- Surfaces, Riemann --- Functions --- Functions of several complex variables --- Mappings (Mathematics) --- Complex variables --- Elliptic functions --- Functions of real variables --- Absolute value. --- Addition. --- Algebraic equation. --- Attractor. --- Automorphism. --- Beltrami equation. --- Blaschke product. --- Boundary (topology). --- Branched covering. --- Coefficient. --- Compact Riemann surface. --- Compact space. --- Complex analysis. --- Complex number. --- Complex plane. --- Computation. --- Connected component (graph theory). --- Connected space. --- Constant function. --- Continued fraction. --- Continuous function. --- Coordinate system. --- Corollary. --- Covering space. --- Cross-ratio. --- Derivative. --- Diagram (category theory). --- Diameter. --- Diffeomorphism. --- Differentiable manifold. --- Disjoint sets. --- Disjoint union. --- Disk (mathematics). --- Division by zero. --- Equation. --- Euler characteristic. --- Existential quantification. --- Exponential map (Lie theory). --- Fundamental group. --- Harmonic function. --- Holomorphic function. --- Homeomorphism. --- Hyperbolic geometry. --- Inequality (mathematics). --- Integer. --- Inverse function. --- Irrational rotation. --- Iteration. --- Jordan curve theorem. --- Julia set. --- Lebesgue measure. --- Lecture. --- Limit point. --- Line segment. --- Linear map. --- Linearization. --- Mandelbrot set. --- Mathematical analysis. --- Maximum modulus principle. --- Metric space. --- Monotonic function. --- Montel's theorem. --- Normal family. --- Open set. --- Orbifold. --- Parameter space. --- Parameter. --- Periodic point. --- Point at infinity. --- Polynomial. --- Power series. --- Proper map. --- Quadratic function. --- Rational approximation. --- Rational function. --- Rational number. --- Real number. --- Riemann sphere. --- Riemann surface. --- Root of unity. --- Rotation number. --- Schwarz lemma. --- Scientific notation. --- Sequence. --- Simply connected space. --- Special case. --- Subgroup. --- Subsequence. --- Subset. --- Summation. --- Tangent space. --- Theorem. --- Topological space. --- Topology. --- Uniform convergence. --- Uniformization theorem. --- Unit circle. --- Unit disk. --- Upper half-plane. --- Winding number.