Lectures on the Geometry of Manifolds最新文献

FRONT MATTER 前页

Lectures on the Geometry of Manifolds Pub Date : 2020-10-15 DOI: 10.1142/9789811214820_fmatter

引用次数: 0

Manifolds 集合管

Lectures on the Geometry of Manifolds Pub Date : 2020-10-15 DOI: 10.1142/9789811214820_0001

引用次数: 0

Dirac Operators 狄拉克算子

Lectures on the Geometry of Manifolds Pub Date : 2020-10-15 DOI: 10.1142/9789814261012_0010

Peter Hochs

引用次数: 0

Characteristic Classes 特性类

Lectures on the Geometry of Manifolds Pub Date : 2020-10-15 DOI: 10.1142/9789811214820_0008

A. Ranicki

{"title":"Characteristic Classes","authors":"A. Ranicki","doi":"10.1142/9789811214820_0008","DOIUrl":"https://doi.org/10.1142/9789811214820_0008","url":null,"abstract":"The goal of this lecture notes is to introduce to Characteristic Classes. This is an important tool of the contemporary mathematics, indispensable to work in geometry and topology, and also useful in number theory. Classical roots of characteristic classes overlap: the Euler characteristic, indices of vector fields and the Poincaré–Hopf theorem, Plücker formulas for plane curves, the Euler characteristic of the Milnor fibre, Riemann–Roch and Riemann–Hurwitz theorems for curves and Schubert calculus. The present approach to the characteristic classes treats them as elements of the cohomology rings and their analogues. We shall discuss the Chern classes of complex vector bundles, characteristic classes of real vector bundles, various characteristic classes of singular analytic varieties. The fundamental theorems on characteristic classes will be proven, in particular, the Grothendieck– –Hirzebruch–Riemann–Roch theorem. Characteristic classes mark out the place where many domains of the contemporary mathematics meet: geometry, topology, singularities, representation theory, algebra and combinatorics. As for what concerns these last three domains, we shall discuss the basic properties of Schur functions. To the memory of my Father (1928–2012 )","PeriodicalId":141790,"journal":{"name":"Lectures on the Geometry of Manifolds","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121554119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The Fundamental Group and Covering Spaces 基本群与覆盖空间

Lectures on the Geometry of Manifolds Pub Date : 2020-10-15 DOI: 10.1142/9789811214820_0006

V. Cheltsov, Edinburgh, A. Ranicki

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BACK MATTER 回到问题

Lectures on the Geometry of Manifolds Pub Date : 2020-10-15 DOI: 10.1142/9789811214820_bmatter

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Cohomology 上同调

Lectures on the Geometry of Manifolds Pub Date : 2020-10-15 DOI: 10.1142/9789811214820_0007

Colloqu Ium, Mathemat Icum

引用次数: 13

Calculus on Manifolds 流形微积分

Lectures on the Geometry of Manifolds Pub Date : 2018-01-03 DOI: 10.1201/B22209-9

M. Blennow

引用次数: 0

Spectral Geometry 谱几何

Lectures on the Geometry of Manifolds Pub Date : 2017-12-16 DOI: 10.1142/9789814460057_0001

B. Iochum

{"title":"Spectral Geometry","authors":"B. Iochum","doi":"10.1142/9789814460057_0001","DOIUrl":"https://doi.org/10.1142/9789814460057_0001","url":null,"abstract":"The goal of these lectures is to present the few fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have defined an action based on spectral considerations, the so-called spectral action. The idea is to review the necessary tools which are behind this spectral action to be able to compute it first in the case of Riemannian manifolds (Einstein–Hilbert action). Then, all primary objects defined for manifolds will be generalized to reach the level of noncommutative geometry via spectral triples, with the concrete analysis of the noncommutative torus which is a deformation of the ordinary one. The basics of different ingredients will be presented and studied like, Dirac operators, heat equation asymptotics, zeta functions and then, how to get within the framework of operators on Hilbert spaces, the notion of noncommutative residue, Dixmier trace, pseudodifferential operators etc. These notions are appropriate in noncommutative geometry to tackle the case where the space is swapped with an algebra like for instance the noncommutative torus. Its non-compact generalization, namely the Moyal plane, is also investigated. Update: 2017, December 15 1 ar X iv :1 71 2. 05 94 5v 1 [ m at hph ] 1 6 D ec 2 01 7 Motivations: Let us first expose few motivations from physics to study noncommutative geometry which is by essence a spectral geometry. Of course, precise mathematical definitions and results will be given in the other sections. The notion of spectrum is quite important in physics, for instance in classical mechanics, the Fourier spectrum is essential to understand vibrations or the light spectrum in electromagnetism. The notion of spectral theory is also important in functional analysis, where the spectral theorem tells us that any selfadjoint operator A can be seen as an integral over its spectral measure A = ∫ a∈Sp(a) a dPa if Sp(A) is the spectrum of A. This is of course essential in the axiomatic formulation of quantum mechanics, especially in the Heisenberg picture where the tools are the observables namely are selfadjoint operators. But this notion is also useful in geometry. In special relativity, we consider fields ψ(~x) for ~x ∈ R4 and the electric and magnetic fields E, B ∈ Function(M = R4,R3). Einstein introduced in 1915 the gravitational field and the equation of motion of matter. But a problem appeared: what are the physical meaning of coordinates x and equations of fields? Assume the general covariance of field equation. If gμν(x) or the tetradfield eμ(x) is a solution (where I is a local inertial reference frame), then, for any diffeomorphism φ of M which is active or passive (i.e. change of coordinates), e′I ν (x) = ∂x μ ∂φ(x)ν e I μ(x) is also a solution. As a consequence, when relativity became general, the points disappeared and it remained only fields on fields in the sense that there is no fields on a gi","PeriodicalId":141790,"journal":{"name":"Lectures on the Geometry of Manifolds","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124922589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 12

Elements of the Calculus of Variations 变分法的基本原理

Lectures on the Geometry of Manifolds Pub Date : 2009-12-01 DOI: 10.1142/9789814289573_0003

A. Georgescu, L. Palese

引用次数: 0