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A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a local-to-global principle: Weil's conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of Weil's conjecture, based on the geometry of the moduli stack of G-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting ℓ-adic sheaves. Using this theory, Dennis Gaitsgory and Jacob Lurie articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of G-bundles (a global object) as a tensor product of local factors.Using a version of the Grothendieck-Lefschetz trace formula, Gaitsgory and Lurie show that this product formula implies Weil's conjecture. The proof of the product formula will appear in a sequel volume.

Weil conjectures. --- Conjecture, Weil's --- Conjectures, Weil --- Tate conjectures, Weil --- -Weil-Tate conjectures --- Weil's conjecture --- Geometry, Algebraic --- Frobenius automorphism. --- G-bundles. --- Grothendieck–Lefschetz. --- Weil's conjecture. --- Weill's conjecture. --- affine group. --- algebraic geometry. --- algebraic topology. --- analogue. --- cohomology. --- continuous Künneth decomposition. --- factorization homology. --- function fields. --- global "ient stacks. --- infinity. --- local-to-global principle. --- moduli stack. --- number theory. --- rational functions. --- sheaves. --- trace formula. --- triangulated category.

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"We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness f, t)q and (f, t)p for 0 < p < q . A similar problem for the generalized K-functionals and their realizations between the couples (Lp,W p ) and (Lq,W q ) is also solved"--

Inequalities (Mathematics) --- Hardy-Littlewood method. --- Smoothness of functions. --- Approximations and expansions -- Approximations and expansions -- Multidimensional problems. --- Harmonic analysis on Euclidean spaces -- Harmonic analysis in several variables -- Multipliers. --- Real functions -- Inequalities -- Inequalities involving derivatives and differential and integral operators. --- Functional analysis -- Linear function spaces and their duals -- Sobolev spaces and other spaces of "smooth'' functions, embedding theorems, trace theorems. --- Real functions -- Functions of one variable -- Fractional derivatives and integrals. --- Approximations and expansions -- Approximations and expansions -- Inequalities in approximation (Bernstein, Jackson, Nikol'skiĭ-type inequalities). --- Real functions -- Polynomials, rational functions -- Polynomials: analytic properties, etc.. --- Approximations and expansions -- Approximations and expansions -- Approximation by polynomials.