Arithmetic and Geometry by Clemens Fuchs
Arithmetic and Geometry presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings height and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach.
The first course, taught by Peter Scholze, contains his recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces.
The second course, taught by Umberto Zannier, addresses the famous Pell equation—not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians.
The third course, taught by Shou-Wu Zhang, originates in the Chowla–Selberg formula, which was taken up by Gross and Zagier to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Zhang, X. Yuan, and Wei Zhang prove the Gross–Zagier formula on Shimura curves and verify the Colmez conjecture on average.
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Seminar on the Atiyah-Singer Index Theorem
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Moments, Monodromy, and Perversity
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The Master Equation and the Convergence Problem in Mean Field Games
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Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations
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Lectures on Curves on an Algebraic Surface
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Intermittent Convex Integration for the 3D Euler Equations
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Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik
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The Einstein-Klein-Gordon Coupled System
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A Hierarchy of Turing Degrees
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Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems
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Dynamics in One Complex Variable
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Elementary Differential Topology
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An Introduction to G-Functions
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Some Problems of Unlikely Intersections in Arithmetic and Geometry
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The Norm Residue Theorem in Motivic Cohomology
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The Structure of Groups with a Quasiconvex Hierarchy
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Lectures on P-Adic L-Functions
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Curvature and Betti Numbers
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Radically Elementary Probability Theory
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Period Spaces for p-divisible Groups
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Exponential Sums and Differential Equations
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Functional Operators, Volume 2
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Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom
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The Decomposition of Global Conformal Invariants
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Classification of Pseudo-reductive Groups
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Weyl Group Multiple Dirichlet Series
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Green's Function Estimates for Lattice Schroedinger Operators and Applications. (AM-158)
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Automata Studies
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Stationary Processes and Prediction Theory
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Arithmetic Moduli of Elliptic Curves
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Flows on Homogeneous Spaces
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Seminar on Singularities of Solutions of Linear Partial Differential Equations
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Functional Operators, Volume 1
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Contributions to Fourier Analysis
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Unipotent Ideals and Harish-Chandra Bimodules
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Hoelder Continuous Euler Flows in Three Dimensions with Compact Support in Time
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Non-Archimedean Tame Topology and Stably Dominated Types
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The p-adic Simpson Correspondence
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The Mathematics of Shock Reflection-Diffraction and von Neumann's Conjectures
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Contributions to the Theory of Games, Volume IV
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Seminar on Micro-Local Analysis
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Contributions to the Theory of Games, Volume II
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Euler Systems
| SKU | Unavailable |
| ISBN 13 | 9780691193779 |
| ISBN 10 | 0691193770 |
| Title | Arithmetic and Geometry |
| Author | Clemens Fuchs |
| Series | Annals Of Mathematics Studies |
| Condition | Unavailable |
| Binding Type | Paperback |
| Publisher | Princeton University Press |
| Year published | 2019-10-08 |
| Number of pages | 186 |
| Cover note | Book picture is for illustrative purposes only, actual binding, cover or edition may vary. |
| Note | Unavailable |