“…We discuss how we systematically understand these surfaces and how the singular behavior of the rational function can be incorporated into an Ansatz using techniques from algebraic geometry. To perform the numerical sampling, we make use of $p$-adic numbers, a number-theoretical field that can be considered a cousin of finite fields. …”
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“…They have engendered remarkable discoveries concerning non-linear differential equations and algebraic geometry, and have produced profound insights in both condensed matter physics and quantum fieldtheory. …”
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“…Apart from a historical presentation of his life and interaction with the classical Italian school of algebraic geometry, the volume presents surveys and original research papers on the mathematics he studied. …”
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“…The book features lecture notes, survey articles, and research notes - based on workshops held in Berlin, Essen, and Madrid - explaining the significance of these and related objects (such as double affine Hecke algebras and affine Springer fibers) in representation theory (e.g., the theory of symmetric polynomials), arithmetic geometry (e.g., the fundamental lemma in the Langlands program), and algebraic geometry (e.g., affine flag manifolds as parameter spaces for principal bundles). …”
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“…While such a general question cannot have a complete answer, it offers a good occasion for reviewing various techniques in l-adic cohomology and group representations, presented in a context that is appealing to specialists in number theory and algebraic geometry. Along with covering open problems, the text examines the size and congruence properties of NX(p) and describes the ways in which it is computed, by closed formulae a…”
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“…This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. …”
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“…Inspired by results from modern number theory and algebraic geometry, a lot of progress has recently been made regarding the computation of multi-loop integrals and scattering amplitudes. …”
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“…The reader is expected to be familiar with the basic concepts of algebraic geometry, algebraic number theory and the theory of automorphic representation.…”
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“…This book is of interest to students as well as experts in the area of real algebraic geometry, quadratic forms, orderings, valuations, lattice ordered groups and rings, and in model theory. …”
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“…This CIME course focused on complex Monge-Ampére equations, applications of pluripotential theory to Kahler geometry and algebraic geometry and to holomorphic dynamics. The contributions provide an extensive description of the theory and its very recent developments, starting from basic introductory materials and concluding with open questions in current research.…”
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“…Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valuation theory, the notion of a completion has to be replaced by that of the so-called Henselization. …”
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“…One of the most exciting new subjects in Algebraic Number Theory and Arithmetic Algebraic Geometry is the theory of Euler systems. Euler systems are special collections of cohomology classes attached to p-adic Galois representations. …”
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“…Numerous well-presented and important papers from the conference are gathered in the proceedings for the purpose of pointing directions for useful future research in diverse areas of mathematics including algebraic geometry, analysis, commutative algebra, complex analysis, discrete mathematics, dynamical systems, number theory and topology. …”
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“…It can serve as an introduction to contemporary mathematics as a whole as it develops background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry. It is unique among textbooks on Riemann surfaces in including an introduction to Teichmüller theory. …”
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“…As a result, curves provide a natural introduction to algebraic geometry. In this book, the authors also bring out aspects of curves that are unique to them and emphasize connections with algebra. …”
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“…The book is written from the viewpoint of algebraic geometry and includes the theory of quadratic forms over fields of characteristic two, with proofs that are characteristic independent whenever possible. …”
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“…It has found profound applications in other areas of mathematics, notably in algebraic geometry and in singularity theory. Since Wilkie's results on the o-minimality of the expansion of the reals by the exponential function, and most recently even by all Pfaffian functions, the study of o-minimal expansions of the reals has become a fascinating topic. …”
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“…Related topics include the theory of algorithms, convex integer polyhedra, Morse inequalities, real algebraic geometry, statistical physics, and algebraic number theory. …”
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“…In particular, the book promotes applications to robotics --and control in general-- of new optimization techniques based on recent results from real algebraic geometry.…”
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“…This volume is the third of three in a series surveying the theory of theta functions which play a central role in the fields of complex analysis, algebraic geometry, number theory and most recently particle physics. …”
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