By V. A. Vassiliev

Many very important services of mathematical physics are outlined as integrals reckoning on parameters. The Picard-Lefschetz conception experiences how analytic and qualitative homes of such integrals (regularity, algebraicity, ramification, singular issues, etc.) depend upon the monodromy of corresponding integration cycles. during this e-book, V. A. Vassiliev offers a number of models of the Picard-Lefschetz idea, together with the classical neighborhood monodromy concept of singularities and whole intersections, Pham's generalized Picard-Lefschetz formulation, stratified Picard-Lefschetz thought, and in addition twisted types of these types of theories with purposes to integrals of multivalued varieties. the writer additionally exhibits how those models of the Picard-Lefschetz conception are utilized in learning a number of difficulties bobbing up in lots of parts of arithmetic and mathematical physics. specifically, 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 features generalizing the classical Gauss hypergeometric crucial. The booklet is aimed at a vast viewers of graduate scholars, learn mathematicians and mathematical physicists attracted to algebraic geometry, advanced research, singularity conception, asymptotic equipment, capability thought, and hyperbolic operators

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We say that two oriented simple loops in the plane are coherent if they have the same rotation number. Let Si and Sk be two Seifert circles in the system of Seifert circles of a link diagram D. Assume that Sk is inside Si and their orientations are not coherent. 10. 11). 2 BRAID PRESENTATIONS Fig. 12). This new circle may intersect some connecting arcs. 13. 14. 15. This deformation of the system of Seifert circles is called a concentric deformation of type I. Fig. 10 Fig. 13 Fig. 11 Fig. 14 Fig.

And vi(i = 1,2) of KnB~ are the overbridges and the underbridges of K, respectively. We assume that K meets 52 in four points A, B, C, and D, where the initial point and the terminal point of Wi are A and B, respectively, the initial point and the terminal point of W2 are C and D, respectively, the initial point of Vi is B, and the initial point of V2 is D. e. geodesic) lines in 52, and each of the under bridges p( Vi) and p( V2) intersects the overbridges transversally and alternately. 1. Thus Band D are numbered 0, and A and C are numbered a.

3 The minimal number of Seifert circles of all diagrams of a given link is equal to the braid index of the link. 4 Show that there are infinitely many link types of braid index 2. Next, we discuss a necessary and sufficient condition for two closed braids to belong to the same link type. Let Bn be the n-string braid group. For any two integers m, n with m < n, we consider that Bm C Bn by identifying each generator lTi E Bm with lTi E Bn (i = 1, ... , m - 1). Set B = {(b,n) I bE Bn,n = 1,2,3, ...