By V.I. Arnold, Alexander Varchenko, S.M. Gusein-Zade

... there's not anything so enchanting, so grandiose, not anything that stuns or captivates the human soul really lots as a primary path in a technology. After the 1st 5 - 6 lectures one already holds the brightest hopes, already sees oneself as a seeker after fact. I too have wholeheartedly pursued technology passionately, as one might a loved girl. i used to be a slave, and sought no different sunlight in my existence. Day and evening I stuffed myself, bending my again, ruining myself over my books; I wept whilst I beheld others exploiting technology fot own achieve. yet i used to be no longer lengthy enthralled. in fact each technology has a starting, yet by no means an finish - they cross on for ever like periodic fractions. Zoology, for instance, has came across thirty-five thousand kinds of existence ... A. P. Chekhov. "On the line" during this booklet a commence is made to the "zoology" of the singularities of differentiable maps. This conception is a tender department of study which at present occupies a crucial position in arithmetic; it's the crossroads of paths major from very summary corners of arithmetic (such as algebraic and differential geometry and topology, Lie teams and algebras, advanced manifolds, commutative algebra and so on) to the main utilized parts (such as differential equations and dynamical platforms, optimum regulate, the idea of bifurcations and catastrophes, short-wave and saddle-point asymptotics and geometrical and wave optics).

By Michèle Audin

The cloth and references during this prolonged moment version of "The Topology of Torus activities on Symplectic Manifolds", released as quantity ninety three during this sequence in 1991, were up to date. Symplectic manifolds and torus activities are investigated, with a variety of examples of torus activities, for example on a few moduli areas. even supposing the ebook remains to be established on convexity effects, it comprises even more fabric, particularly plenty of new examples and routines.

By O I Mokhov

This overview provides the differential-geometric concept of homogeneous constructions (mainly Poisson and symplectic structures)on loop areas of soft manifolds, their normal generalizations and functions in mathematical physics and box idea.

By Jacob T Schwartz

By An-min Li

During this monograph, the interaction among geometry and partial differential equations (PDEs) is of specific curiosity. It provides a selfcontained creation to analyze within the final decade referring to worldwide difficulties within the idea of submanifolds, resulting in a few varieties of Monge-Ampère equations.

From the methodical standpoint, it introduces the answer of yes Monge-Ampère equations through geometric modeling thoughts. the following geometric modeling capability the perfect selection of a normalization and its triggered geometry on a hypersurface outlined by means of an area strongly convex international graph. For a greater realizing of the modeling recommendations, the authors supply a selfcontained precis of relative hypersurface conception, they derive very important PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). touching on modeling suggestions, emphasis is on conscientiously based proofs and exemplary comparisons among assorted modelings.

By Cao F.

By Dusa McDuff and Dietmar Salamon

J$-holomorphic curves revolutionized the research of symplectic geometry while Gromov first brought them in 1985. via quantum cohomology, those curves are actually associated with the various most fun new rules in mathematical physics. This publication offers the 1st coherent and entire account of the idea of $J$-holomorphic curves, the main points of that are shortly scattered in numerous learn papers. the 1st half the ebook is an expository account of the sphere, explaining the most technical elements. McDuff and Salamon supply whole proofs of Gromov's compactness theorem for spheres and of the life of the Gromov-Witten invariants. the second one 1/2 the e-book makes a speciality of the definition of quantum cohomology. The authors identify that this multiplication exists, and provides a brand new evidence of the Ruan-Tian outcome that's associative on applicable manifolds. They then describe the Givental-Kim calculation of the quantum cohomology of flag manifolds, resulting in quantum Chern periods and Witten's calculation for Grassmannians, which pertains to the Verlinde algebra. The Dubrovin connection, Gromov-Witten power on quantum cohomology, and curve counting formulation also are mentioned. The publication closes with an summary of connections to Floer idea

By Sean Dineen

The Schwarz lemma is without doubt one of the least difficult ends up in complicated research that seize the stress of holomorphic features. This self-contained quantity presents a radical evaluate of the topic; it assumes no wisdom of intrinsic metrics and goals for the most effects, introducing notation, secondary ideas, and strategies as precious. appropriate for complicated undergraduates and graduate scholars of arithmetic, the two-part remedy covers simple concept and functions.
Starting with an exploration of the topic when it comes to holomorphic and subharmonic capabilities, the remedy proves a Schwarz lemma for plurisubharmonic capabilities and discusses the elemental houses of the Poincaré distance and the Schwarz-Pick platforms of pseudodistances. extra subject matters comprise hyperbolic manifolds, distinctive domain names, pseudometrics outlined utilizing the (complex) eco-friendly functionality, holomorphic curvature, and the algebraic metric of Harris. the second one half explores fastened element theorems and the analytic Radon-Nikodym estate.

By Leon Simon

The purpose of those lecture notes is to provide an primarily self-contained creation to the elemental regularity concept for strength minimizing maps, together with fresh advancements about the constitution of the singular set and asymptotics on method of the singular set. really good wisdom in partial differential equations or the geometric calculus of diversifications is no longer required. as an alternative, a superb basic historical past in mathematical research will be enough guidance.

By J.L. Koszul, S. Ramanan