In mathematics, Picard–Lefschetz theory studies the topology of a complex manifold by looking at the critical points of a holomorphic function on the...

algebraic geometry, and the theory of non-linear ordinary differential equations. He was born in Moscow, the son of Alexander Lefschetz and his wife Sarah or...

his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. To illustrate, consider a mountainous landscape...

Kähler manifolds), building on Hodge theory. The results include the Lefschetz hyperplane theorem, the hard Lefschetz theorem, and the Hodge–Riemann bilinear...

leading to more recent research interest in them. Picard–Lefschetz theory Donaldson, Simon K. (1998). "Lefschetz fibrations in symplectic geometry". Documenta...

MR 1827869, S2CID 119853564 Vassiliev, V. A. (2002), Applied Picard-Lefschetz theory, Mathematical Surveys and Monographs, vol. 97, Providence, R.I...

He made important contributions in the theory of differential equations, including work on Picard–Vessiot theory, Painlevé transcendents and his introduction...

also a professor of mathematics at MIT. Fukaya Categories and Picard-Lefschetz Theory, European Mathematical Society, 2008 "Curriculum Vitae" (PDF)....

(geometry of wavefronts), complex analysis, combinatorics, and Picard–Lefschetz theory. Vassiliev studied at the Faculty of Mathematics and Mechanics...

characteristic, and in other homology and cohomology theories. A far-reaching generalization of the hard Lefschetz theorem is given by the decomposition theorem...

\}} . Here the monodromy around 0 and 1 can be computed using Picard–Lefschetz theory, giving the monodromy around ∞ {\displaystyle \infty } by composing...

introduction to Fukaya categories. Paul Seidel, Fukaya categories and Picard-Lefschetz theory. Zurich lectures in Advanced Mathematics Fukaya, Kenji; Oh, Yong-Geun;...

ISBN 978-3-0348-8577-5. Seidel, Paul (2008). Fukaya Categories and Picard Lefschetz Theory. European Mathematical Society. ISBN 978-3037190630. Colin, Vincent;...

Symplectization based on analogies between Picard–Lefschetz theory which he interprets as the Complexified version of Morse theory and then extend them to other areas...

(2005), no. 4, 859–881. (with Ivan Smith) Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathematics. European Mathematical...

Brouwer's fixed-point theorem for "hole-free" domains can be derived from the Lefschetz fixed-point theorem. The continuous function in this theorem is not required...

are necessary because (1) reduces the technical complexity (also Picard-Lefschetz theory can be used here), (2) rigidifies the curves so that there are...

ISBN 978-0-691-08167-0. MR 0418127. Pham, Frédéric (1965), "Formules de Picard-Lefschetz généralisées et ramification des intégrales", Bulletin de la Société...

His work in complex singularity theory generalized Milnor maps into an algebraic setting and extended the Picard-Lefschetz formula beyond their general format...

(number theory) Lefschetz hyperplane theorem (algebraic topology) Leray's theorem (algebraic geometry) Manin–Drinfeld theorem (number theory) Max Noether's...

general theory was certainly both to integrate all this information, and to prove general results such as Poincaré duality and the Lefschetz fixed-point...

to Betti numbers, the Lefschetz fixed-point theorem and so on. The analogy with topology suggested that a new homological theory be set up applying within...

most important contributors to the theory of abelian functions were Riemann, Weierstrass, Frobenius, Poincaré, and Picard. The subject was very popular at...

Picard group of Y is isomorphic to Z, generated by the restriction of the line bundle O(1) on projective space. Grothendieck generalized Lefschetz's theorem...

Sciences Sup, Dunod, 2003. Les différentielles, Masson, 1996. Formules de Picard-Lefschetz généralisées et ramification des intégrales, Bulletin Societé Mathématique...

discrete invariants of a surface. The Noether-Lefschetz theorem (proved by Lefschetz) says that the Picard group of a very general surface of degree at...

program Intersection theory Intersection number Chow ring Chern class Serre's multiplicity conjectures Albanese variety Picard group Modular form Moduli...

and its endoscopic groups, and the stabilization of the Grothendieck–Lefschetz formula. None of these are possible without the fundamental lemma and...

cohomology), Nick Katz (monodromy theory, and Lefschetz pencils). Jean Giraud worked out torsor theory extensions of nonabelian cohomology there as well...

not as strong as a Kähler metric on a complex manifold, and the Hodge–Lefschetz–Dolbeault theorems on sheaf cohomology break down in every possible way...