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This collection brings together influential papers by mathematicians exploring the research frontiers of topology, one of the most important developments of modern mathematics. The papers cover a wide range of topological specialties, including tools for the analysis of group actions on manifolds, calculations of algebraic K-theory, a result on analytic structures on Lie group actions, a presentation of the significance of Dirac operators in smoothing theory, a discussion of the stable topology of 4-manifolds, an answer to the famous question about symmetries of simply connected manifolds, and a fresh perspective on the topological classification of linear transformations. The contributors include A. Adem, A. H. Assadi, M. Bökstedt, S. E. Cappell, R. Charney, M. W. Davis, P. J. Eccles, M. H. Freedman, I. Hambleton, J. C. Hausmann, S. Illman, G. Katz, M. Kreck, W. Lück, I. Madsen, R. J. Milgram, J. Morava, E. K. Pedersen, V. Puppe, F. Quinn, A. Ranicki, J. L. Shaneson, D. Sullivan, P. Teichner, Z. Wang, and S. Weinberger.

Topology --- Adjunction (field theory). --- Algebraic cycle. --- Algebraic topology. --- Analytic function. --- Automorphism. --- Base change. --- Basis (linear algebra). --- Borel conjecture. --- Characteristic class. --- Circle group. --- Classifying space. --- Cobordism. --- Codimension. --- Cohomology ring. --- Cohomology. --- Combinatorial group theory. --- Commutative diagram. --- Commutative property. --- Compactification (mathematics). --- Coxeter group. --- Cyclic group. --- Cyclic homology. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential topology. --- Dimension (vector space). --- Dirac operator. --- Discrete valuation ring. --- Divisor (algebraic geometry). --- Elliptic operator. --- Equivariant K-theory. --- Equivariant cohomology. --- Equivariant map. --- Exterior (topology). --- Exterior algebra. --- Fiber bundle. --- Fibration. --- Fundamental group. --- Gauss map. --- Geometrization conjecture. --- Group algebra. --- H-cobordism. --- Homeomorphism. --- Homotopy fiber. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Identity matrix. --- Inclusion map. --- Intersection form (4-manifold). --- Isomorphism class. --- J-homomorphism. --- Knot theory. --- L-theory. --- Lens space. --- Lie algebra. --- Lie group. --- Linear algebra. --- Mapping cone (homological algebra). --- Mapping cone (topology). --- Marriage theorem. --- Metric space. --- Moduli space. --- Motivic cohomology. --- Neighbourhood (mathematics). --- Operator norm. --- Pushout (category theory). --- Quasi-isometry. --- Quotient space (topology). --- Real projective space. --- Regularization (mathematics). --- Representation theory. --- Riemann surface. --- Riemannian manifold. --- Set (mathematics). --- Sheaf (mathematics). --- Sign (mathematics). --- Simplicial complex. --- Stable homotopy theory. --- Subgroup. --- Submanifold. --- Submersion (mathematics). --- Subset. --- Support (mathematics). --- Sylow theorems. --- Tangent space. --- Theorem. --- Topological K-theory. --- Topological group. --- Topological manifold. --- Topological space. --- Topology. --- Torsion sheaf. --- Transversality (mathematics). --- Unification (computer science). --- Vector bundle. --- Whitehead torsion. --- Zariski topology. --- Zorn's lemma.

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This volume investigates the interplay between the classical theory of automorphic forms and the modern theory of representations of adele groups. Interpreting important recent contributions of Jacquet and Langlands, the author presents new and previously inaccessible results, and systematically develops explicit consequences and connections with the classical theory. The underlying theme is the decomposition of the regular representation of the adele group of GL(2). A detailed proof of the celebrated trace formula of Selberg is included, with a discussion of the possible range of applicability of this formula. Throughout the work the author emphasizes new examples and problems that remain open within the general theory.TABLE OF CONTENTS: 1. The Classical Theory 2. Automorphic Forms and the Decomposition of L2(PSL(2,R) 3. Automorphic Forms as Functions on the Adele Group of GL(2) 4. The Representations of GL(2) over Local and Global Fields 5. Cusp Forms and Representations of the Adele Group of GL(2) 6. Hecke Theory for GL(2) 7. The Construction of a Special Class of Automorphic Forms 8. Eisenstein Series and the Continuous Spectrum 9. The Trace Formula for GL(2) 10. Automorphic Forms on a Quaternion Algebr?

Number theory --- Representations of groups --- Linear algebraic groups --- Adeles --- Representations of groups. --- Automorphic forms. --- Linear algebraic groups. --- Adeles. --- Nombres, Théorie des --- Formes automorphes --- Automorphic forms --- Algebraic fields --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Automorphic functions --- Forms (Mathematics) --- Group representation (Mathematics) --- Groups, Representation theory of --- Nombres, Théorie des. --- Abelian extension. --- Abelian group. --- Absolute value. --- Addition. --- Additive group. --- Algebraic group. --- Algebraic number field. --- Algebraic number theory. --- Analytic continuation. --- Analytic function. --- Arbitrarily large. --- Automorphic form. --- Cartan subgroup. --- Class field theory. --- Complex space. --- Congruence subgroup. --- Conjugacy class. --- Coprime integers. --- Cusp form. --- Differential equation. --- Dimension (vector space). --- Direct integral. --- Direct sum. --- Division algebra. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Euler product. --- Existential quantification. --- Exponential function. --- Factorization. --- Finite field. --- Formal power series. --- Fourier series. --- Fourier transform. --- Fuchsian group. --- Function (mathematics). --- Function space. --- Functional equation. --- Fundamental unit (number theory). --- Galois extension. --- Global field. --- Group algebra. --- Group representation. --- Haar measure. --- Harish-Chandra. --- Hecke L-function. --- Hilbert space. --- Homomorphism. --- Induced representation. --- Infinite product. --- Inner automorphism. --- Integer. --- Invariant measure. --- Invariant subspace. --- Irreducible representation. --- L-function. --- Lie algebra. --- Linear map. --- Matrix coefficient. --- Mellin transform. --- Meromorphic function. --- Modular form. --- P-adic number. --- Poisson summation formula. --- Prime ideal. --- Prime number. --- Principal series representation. --- Projective representation. --- Quadratic field. --- Quadratic form. --- Quaternion algebra. --- Quaternion. --- Real number. --- Regular representation. --- Representation theory. --- Ring (mathematics). --- Ring of integers. --- Scientific notation. --- Selberg trace formula. --- Simple algebra. --- Square-integrable function. --- Sub"ient. --- Subgroup. --- Summation. --- Theorem. --- Theory. --- Theta function. --- Topological group. --- Topology. --- Trace formula. --- Trivial representation. --- Uniqueness theorem. --- Unitary operator. --- Unitary representation. --- Universal enveloping algebra. --- Upper half-plane. --- Variable (mathematics). --- Vector space. --- Weil group. --- Nombres, Théorie des