Applied Picard-Lefschetz theory by V. A. Vassiliev PDF

By V. A. Vassiliev

ISBN-10: 0821829483

ISBN-13: 9780821829486

ISBN-10: 3419872682

ISBN-13: 9783419872680

ISBN-10: 3819371311

ISBN-13: 9783819371318

ISBN-10: 4019901481

ISBN-13: 9784019901480

ISBN-10: 4419884894

ISBN-13: 9784419884895

ISBN-10: 8419623873

ISBN-13: 9788419623874

Many very important features of mathematical physics are outlined as integrals looking on parameters. The Picard-Lefschetz conception experiences how analytic and qualitative houses of such integrals (regularity, algebraicity, ramification, singular issues, etc.) rely on the monodromy of corresponding integration cycles. during this e-book, V. A. Vassiliev offers a number of models of the Picard-Lefschetz conception, together with the classical neighborhood monodromy concept of singularities and entire intersections, Pham's generalized Picard-Lefschetz formulation, stratified Picard-Lefschetz idea, and in addition twisted models of a majority of these theories with functions to integrals of multivalued types. the writer additionally indicates how those types of the Picard-Lefschetz conception are utilized in learning a number of difficulties bobbing up in lots of components of arithmetic and mathematical physics. particularly, he discusses the next periods of services: quantity capabilities coming up within the Archimedes-Newton challenge of integrable our bodies; Newton-Coulomb potentials; basic ideas of hyperbolic partial differential equations; multidimensional hypergeometric features generalizing the classical Gauss hypergeometric crucial. The e-book is aimed toward a wide viewers of graduate scholars, learn mathematicians and mathematical physicists drawn to algebraic geometry, complicated research, singularity conception, asymptotic tools, capability conception, and hyperbolic operators

Show description

Read or Download Applied Picard-Lefschetz theory PDF

Similar algebraic geometry books

Complex Geometry - download pdf or read online

Provides the lawsuits of a global convention on complicated geometry and similar themes, held in commemoration of the fiftieth anniversary of Osaka college, Osaka, Japan. The textual content specializes in the CR invariants, hyperbolic geometry, Yamabe-type difficulties, and harmonic maps.

Explicit Birational Geometry of 3-folds - download pdf or read online

One of many major achievements of algebraic geometry during the last two decades is the paintings of Mori and others extending minimum versions and the Enriques-Kodaira type to 3-folds. This built-in suite of papers facilities round functions of Mori conception to birational geometry. 4 of the papers (those by way of Pukhlikov, Fletcher, Corti, and the lengthy joint paper via Corti, Pukhlikov and Reid) figure out intimately the speculation of birational tension of Fano 3-folds.

Singularities I: Algebraic and Analytic Aspects: by Jean-Paul Brasselet, Jose Luis Cisneros-molina, David PDF

This can be the 1st a part of the lawsuits of the assembly 'School and Workshop at the Geometry and Topology of Singularities', held in Cuernavaca, Mexico, from January eighth to twenty sixth of 2007, in party of the sixtieth Birthday of Le Dung Trang. This quantity includes fourteen state-of-the-art learn articles on algebraic and analytic elements of singularities of areas and maps.

Extra resources for Applied Picard-Lefschetz theory

Sample text

18. Barycentric coordinates and volume ratios. D1 D0 w D2 v0 v1 of another simplex t. Let s = v0v1 · · · vk and t = w0w1 · · · ws. If we express points of s in terms of the (unique) barycentric coordinates with respect to its vertices, then f induces a well-defined map f :sÆ t defined by ˆ k Ê k f Á Â a i v i ˜ = Â a if (v i ). ¯ i =0 Ë i =0 Definition. The map |f| is called the map from s to t induced by the vertex map f. In Chapter 6 we shall see that the map f is a special case of what is called a simplicial map between simplicial complexes and |f| is the induced map on their underlying spaces.

If {w0, w1, . . , wj} Õ {v0, v1, . . , vk}, then t = w0w1 · · · wj is called a j-dimensional face of s and we shall write t ՞ s. 17 shows some examples of simplices and shows that our use of the term “k-dimensional” is justified. 16. A convex linear polyhedron X. 17. Some simplices. simplex. In general, Rn contains at most n-dimensional simplices because it is not possible to find j linearly independent points in Rn for j > n + 1. Also, a simplex depends only on the set of vertices and not on their ordering.

Next, we show that S belongs to every convex set C containing the points v0, v1, . . , and vk. The case k = 0 is trivial. Assume that k ≥ 1 and that the statement has been proved for all values smaller than k. Let k w =  ai v i i =0 belong to S. Since not all ai can be zero, we may assume without loss of generality that a0 π 0. The case a0 = 1 is trivial, and so assume that a0 < 1. Thus we can write ˆ Ê k ai w = a 0 v 0 + (1 - a 0 )Á  vi ˜. Ë i =1 1 - a 0 ¯ But k a  1 - ia0 = i =1 1 1 - a0 k  ai i =1 1 (1 - a 0 ) = 1 - a0 =1 and 0 £ ai/(1 - a0) £ 1.

Download PDF sample

Applied Picard-Lefschetz theory by V. A. Vassiliev

by Jeff

Rated 4.42 of 5 – based on 35 votes