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Applied Picard–Lefschetz Theory

V. A. Vassiliev Independent University of Moscow, Moscow, Russia and Steklov Mathematical Institute, Moscow, Russia
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Hardcover ISBN: 978-0-8218-2948-6
Product Code: SURV/97
List Price: $102.00 MAA Member Price:$91.80
AMS Member Price: $81.60 Electronic ISBN: 978-1-4704-1324-8 Product Code: SURV/97.E List Price:$96.00
MAA Member Price: $86.40 AMS Member Price:$76.80
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List Price: $153.00 MAA Member Price:$137.70
AMS Member Price: $122.40 Click above image for expanded view Applied Picard–Lefschetz Theory V. A. Vassiliev Independent University of Moscow, Moscow, Russia and Steklov Mathematical Institute, Moscow, Russia Available Formats:  Hardcover ISBN: 978-0-8218-2948-6 Product Code: SURV/97  List Price:$102.00 MAA Member Price: $91.80 AMS Member Price:$81.60
 Electronic ISBN: 978-1-4704-1324-8 Product Code: SURV/97.E
 List Price: $96.00 MAA Member Price:$86.40 AMS Member Price: $76.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$153.00 MAA Member Price: $137.70 AMS Member Price:$122.40
• Book Details

Mathematical Surveys and Monographs
Volume: 972002; 324 pp
MSC: Primary 14; 31; 32; 35; Secondary 33;

Many important functions of mathematical physics are defined as integrals depending on parameters. The Picard–Lefschetz theory studies how analytic and qualitative properties of such integrals (regularity, algebraicity, ramification, singular points, etc.) depend on the monodromy of corresponding integration cycles. In this book, V. A. Vassiliev presents several versions of the Picard–Lefschetz theory, including the classical local monodromy theory of singularities and complete intersections, Pham's generalized Picard–Lefschetz formulas, stratified Picard–Lefschetz theory, and also twisted versions of all these theories with applications to integrals of multivalued forms.

The author also shows how these versions of the Picard–Lefschetz theory are used in studying a variety of problems arising in many areas of mathematics and mathematical physics.

In particular, he discusses the following classes of functions:

• volume functions arising in the Archimedes–Newton problem of integrable bodies;
• Newton–Coulomb potentials;
• fundamental solutions of hyperbolic partial differential equations;
• multidimensional hypergeometric functions generalizing the classical Gauss hypergeometric integral.

The book is geared toward a broad audience of graduate students, research mathematicians and mathematical physicists interested in algebraic geometry, complex analysis, singularity theory, asymptotic methods, potential theory, and hyperbolic operators.

Graduate students, research mathematicians and mathematical physicists interested in algebraic geometry, complex analysis, singularity theory, asymptotic methods, potential theory, and hyperbolic operators.

• Chapters
• I. Local monodromy theory of isolated singularities of functions and complete intersections
• II. Stratified Picard–Lefschetz theory and monodromy of hyperplane sections
• III. Newton’s theorem on the non-integrability of ovals
• IV. Lacunas and local Petrovskiĭ condition for hyperbolic differential operators with constant coefficients
• V. Calculation of local Petrovskiĭ cycles and enumeration of local lacunas close to real singularities
• VI. Homology of local systems, twisted monodromy theory, and regularization of improper integration cycles
• VII. Analytic properties of surface potentials
• VIII. Multidimensional hypergeometric functions, their ramification, singularities, and resonances

• Reviews

• This is a book rich in ideas …

Mathematical Reviews
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• Get Permissions
Volume: 972002; 324 pp
MSC: Primary 14; 31; 32; 35; Secondary 33;

Many important functions of mathematical physics are defined as integrals depending on parameters. The Picard–Lefschetz theory studies how analytic and qualitative properties of such integrals (regularity, algebraicity, ramification, singular points, etc.) depend on the monodromy of corresponding integration cycles. In this book, V. A. Vassiliev presents several versions of the Picard–Lefschetz theory, including the classical local monodromy theory of singularities and complete intersections, Pham's generalized Picard–Lefschetz formulas, stratified Picard–Lefschetz theory, and also twisted versions of all these theories with applications to integrals of multivalued forms.

The author also shows how these versions of the Picard–Lefschetz theory are used in studying a variety of problems arising in many areas of mathematics and mathematical physics.

In particular, he discusses the following classes of functions:

• volume functions arising in the Archimedes–Newton problem of integrable bodies;
• Newton–Coulomb potentials;
• fundamental solutions of hyperbolic partial differential equations;
• multidimensional hypergeometric functions generalizing the classical Gauss hypergeometric integral.

The book is geared toward a broad audience of graduate students, research mathematicians and mathematical physicists interested in algebraic geometry, complex analysis, singularity theory, asymptotic methods, potential theory, and hyperbolic operators.

Graduate students, research mathematicians and mathematical physicists interested in algebraic geometry, complex analysis, singularity theory, asymptotic methods, potential theory, and hyperbolic operators.

• Chapters
• I. Local monodromy theory of isolated singularities of functions and complete intersections
• II. Stratified Picard–Lefschetz theory and monodromy of hyperplane sections
• III. Newton’s theorem on the non-integrability of ovals
• IV. Lacunas and local Petrovskiĭ condition for hyperbolic differential operators with constant coefficients
• V. Calculation of local Petrovskiĭ cycles and enumeration of local lacunas close to real singularities
• VI. Homology of local systems, twisted monodromy theory, and regularization of improper integration cycles
• VII. Analytic properties of surface potentials
• VIII. Multidimensional hypergeometric functions, their ramification, singularities, and resonances
• This is a book rich in ideas …

Mathematical Reviews
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