By V. A. Vassiliev
Many very important services of mathematical physics are outlined as integrals looking on parameters. The Picard-Lefschetz concept stories how analytic and qualitative houses of such integrals (regularity, algebraicity, ramification, singular issues, etc.) rely on the monodromy of corresponding integration cycles. during this ebook, V. A. Vassiliev provides numerous types of the Picard-Lefschetz thought, together with the classical neighborhood monodromy concept of singularities and entire intersections, Pham's generalized Picard-Lefschetz formulation, stratified Picard-Lefschetz concept, and in addition twisted types of these kind of theories with functions to integrals of multivalued varieties. the writer additionally indicates how those types of the Picard-Lefschetz thought are utilized in learning quite a few difficulties coming up in lots of components of arithmetic and mathematical physics. particularly, he discusses the next sessions of capabilities: quantity capabilities bobbing up within the Archimedes-Newton challenge of integrable our bodies; Newton-Coulomb potentials; primary suggestions of hyperbolic partial differential equations; multidimensional hypergeometric features generalizing the classical Gauss hypergeometric vital. The ebook is aimed at a huge viewers of graduate scholars, examine mathematicians and mathematical physicists attracted to algebraic geometry, advanced research, singularity thought, asymptotic equipment, capability thought, and hyperbolic operators
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Extra info for Applied Picard-Lefschetz theory
J 1 g-r 1 «> . 00) = i=l ^ 5;Q + J R -goo e Jac C-Gj 1 1 L J (because it's the image of a point in Z ) . If g-r is even, then D € (Jac C-0) + er„ D € (Jac C-0) + er „ \ ; if ^"^ is odd .. QED IR^,.. ,Kg__^,oo^ So, we take one copy of Z for each T, and we glue them together according to their identification as subsets of the Jacobian; we have to see that this glueing satisfies the conditions to give the atlas of a variety. _ Here - in Jac C-0 is a difference of sets, but + in (Jac C-0)+e_ means translation of a set by a point using the group law on Jac C.
33 Lemma 2 . 5 . * U | ( J a c C - 0) + e^^ = Jac C or Proof. n T (e + e ) =
V^ - UW = 2g I a^(U^,V ,W^)t°'. ,W. ,V. ,R ) c (C^'^. ) and 1 (U^,Vj) are inverse of one another. 3. 2 the the Zariski given by the equations s^ = -S2/ t^ = t^ (s. ,t. ,S2,t2) are coordinates. in >C if Then everything is tied together in: The equations ^Qr'*'/^2a 9^^^^^^^ ^ prime ideal CC[U. ,V . 3 will consist of 2 steps. 1. ,V^,WQ,. ,W 2 2 Starting with any solution U,V,W to the equation f-V = UW (with prescribed degrees) we will show that the vector space of triples U,V,W (deg U,V <^ v-1, deg W <^ 2g-v) such that f-(V+eV)^ has dimension = (U+eU)(W+eW) mod e^ (*) v .
Applied Picard-Lefschetz theory by V. A. Vassiliev