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The Stacks Project Expository Collection (SPEC) compiles expository articles in advanced algebraic geometry, intended to bring graduate students and researchers up to speed on recent developments in the geometry of algebraic spaces and algebraic stacks. The articles in the text make explicit in modern language many results, proofs, and examples that were previously only implicit, incomplete, or expressed in classical terms in the literature. Where applicable this is done by explicitly referring to the Stacks project for preliminary results. Topics include the construction and properties of important moduli problems in algebraic geometry (such as the Deligne-Mumford compactification of the moduli of curves, the Picard functor, or moduli of semistable vector bundles and sheaves), and arithmetic questions for fields and algebraic spaces.

Algebraic stacks. --- Geometry, Algebraic.

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Algebraic topology. --- Differential topology. --- Loop spaces. --- Algebraic stacks. --- Topologie algébrique. --- Topologie différentielle. --- Espaces de lacets. --- Corps algébriques.

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The first instances of deformation theory were given by Kodaira and Spencer for complex structures and by Gerstenhaber for associative algebras. Since then, deformation theory has been applied as a useful tool in the study of many other mathematical structures, and even today it plays an important role in many developments of modern mathematics. This volume collects a few self-contained and peer-reviewed papers by experts which present up-to-date research topics in algebraic and motivic topology, quantum field theory, algebraic geometry, noncommutative geometry and the deformation theory of Poisson algebras. They originate from activities at the Max-Planck-Institute for Mathematics and the Hausdorff Center for Mathematics in Bonn. Contributions by Grégory Ginot, Thomas M. Fiore and Igor Kriz, Toshiro Hiranouchi and Satoshi Mochizuki, Paulo Carrillo Rouse, Donatella Iacono and Marco Manetti, John Terilla, Anne Pichereau - Researchers in the fields of deformation theory, noncommutative geometry, algebraic topology, mathematical physics - Advanced graduate students in mathematics Dr. Hossein Abbaspour, Department of Mathematics, Université de Nantes, France. Prof. Dr. Matilde Marcolli, Department of Mathematics, California Institute of Technology, Pasadena, California, USA. Dr. Thomas Tradler, Department of Mathematics, New York City College of Technology (CUNY), New York, USA.

Algebraic stacks. --- Homology theory. --- Moduli theory. --- Homology theory --- Algebraic stacks --- Moduli theory --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Geometry --- Theory of moduli --- Stacks, Algebraic --- Cohomology theory --- Contrahomology theory --- Mathematics. --- Algebra. --- Algebraic geometry. --- Geometry. --- Algebraic Geometry. --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Algebraic topology --- Geometry, algebraic. --- Mathematical analysis --- Euclid's Elements --- Algebraic geometry

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A foundational account of a new construction in the p-adic Langlands correspondenceMotivated by the p-adic Langlands program, this book constructs stacks that algebraize Mazur’s formal deformation rings of local Galois representations. More precisely, it constructs Noetherian formal algebraic stacks over Spf Zp that parameterize étale (ϕ, Γ)-modules; the formal completions of these stacks at points in their special fibres recover the universal deformation rings of local Galois representations. These stacks are then used to show that all mod p representations of the absolute Galois group of a p-adic local field lift to characteristic zero, and indeed admit crystalline lifts. The book explicitly describes the irreducible components of the underlying reduced substacks and discusses the relationship between the geometry of these stacks and the Breuil–Mézard conjecture. Along the way, it proves a number of foundational results in p-adic Hodge theory that may be of independent interest.

MATHEMATICS / Geometry / Algebraic. --- Algebraic stacks. --- Moduli theory. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Stacks, Algebraic --- Galois representations. --- Langlands program. --- P-adic Hodge theory. --- MATHEMATICS / Geometry / Algebraic --- MATHEMATICS / Reference