By V. A. Vassiliev
Many very important features of mathematical physics are outlined as integrals reckoning on parameters. The Picard-Lefschetz thought reviews how analytic and qualitative houses of such integrals (regularity, algebraicity, ramification, singular issues, etc.) depend upon the monodromy of corresponding integration cycles. during this e-book, V. A. Vassiliev provides a number of types of the Picard-Lefschetz concept, together with the classical neighborhood monodromy conception of singularities and whole intersections, Pham's generalized Picard-Lefschetz formulation, stratified Picard-Lefschetz idea, and likewise twisted models of a majority of these theories with functions to integrals of multivalued varieties. the writer additionally indicates how those types of the Picard-Lefschetz thought are utilized in learning a number of difficulties bobbing up in lots of parts of arithmetic and mathematical physics. particularly, he discusses the next sessions of services: quantity services coming up within the Archimedes-Newton challenge of integrable our bodies; Newton-Coulomb potentials; basic options of hyperbolic partial differential equations; multidimensional hypergeometric capabilities generalizing the classical Gauss hypergeometric imperative. The ebook is aimed toward a huge viewers of graduate scholars, examine mathematicians and mathematical physicists drawn to algebraic geometry, complicated research, singularity thought, asymptotic equipment, power idea, and hyperbolic operators
Read Online or Download Applied Picard-Lefschetz theory PDF
Similar algebraic geometry books
This vintage publication includes an advent to platforms of l-adic representations, a subject of serious value in quantity idea and algebraic geometry, as mirrored via the brilliant contemporary advancements at the Taniyama-Weil conjecture and Fermat's final Theorem. The preliminary chapters are dedicated to the Abelian case (complex multiplication), the place one unearths a pleasant correspondence among the l-adic representations and the linear representations of a few algebraic teams (now referred to as Taniyama groups).
This publication resulted from studies (published in 1928 and 1932) of the Committee on Rational modifications, confirmed by means of the nationwide learn Council. the aim of the studies was once to provide a accomplished survey of the literature at the topic. every one bankruptcy is considered a separate unit that may be learn independently.
This publication is dedicated to fresh development within the learn of curves and abelian kinds. It discusses either classical features of this deep and lovely topic in addition to vital new advancements, tropical geometry and the idea of log schemes. as well as unique learn articles, this publication comprises 3 surveys dedicated to singularities of theta divisors, of compactified Jacobians of singular curves, and of ""strange duality"" between moduli areas of vector bundles on algebraic kinds
This e-book offers a self-contained, available creation to the mathematical advances and demanding situations as a result of using semidefinite programming in polynomial optimization. This speedy evolving learn sector with contributions from the varied fields of convex geometry, algebraic geometry, and optimization is called convex algebraic geometry.
- Plane Algebraic Curves
- Algebraic Geometry
- Zariski Geometries: Geometry from the Logician's Point of View
- Théorèmes de Bertini et Applications
- Mixed Automorphic Forms
- Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves (CRM Monograph Series)
Extra info for Applied Picard-Lefschetz theory
This fold is the profile that distinguishes the surface from its surroundings. In addition to metaphorical language, formulas can be used to describe the ideas of a fold. The expression y = x2 represents a parabolic cylinder with its axis lying along the z-axis in space. When this surface is projected onto the yz-plane — the plane for which x = 0 — two sheets overlap when y > 0; meanwhile, along the z-axis (x = y = 0) the surface folds onto a line. This surface is the local model of a fold just as the intersection of the pair of planes x = 0, and y = 0 is a model for a double point, and the intersection among the three planes x = 0, y = 0, and z = 0 is a model for a triple point.
Double point arcs can cross folds. 8 indicates this situation which is called a ψ-move, a double point bounce or simply a bounce. Among these names, the name ψ-move is the most precise: The critical level from one perspective resembles the letter ψ. However in the projection, the fold lines and the double points appear to become tangent. So the two sets appear to touch and bounce off each other. ” Words and terms sometimes acquire power beyond their chosen context, and it becomes difficult to disassociate the word from the idea.
To a child who grew up learning about the great new world explorers, who was fascinated with the lunar project, and who was looking for new frontiers, this mathematical world was full of the promise of excitement. It still is. Phillips’s article did not use what we now call the movie moves, but it did illustrate each stage of the eversion by using a sequence of cross-sections. One commentator says that it is not particularly easy to see how to get from one stage to another. Someone whom I know says that the illustrations in the Scientific American article contains known mistakes.
Applied Picard-Lefschetz theory by V. A. Vassiliev