Notes on global analysis
I recently received an email asking me about these notes. I replied that I
was no longer maintaining this, and should probably remove it. Then it was
pointed out to me that this page was linked on MathStackExchange and
so perhaps I should not delete it. So, I have done the opposite of deleting
it; I have added material that I have completed since this seminar ended. It
is still very incomplete, as you will see from the list of missing chapters.
This will be some sort of book some day. In the meantime I will add chapters
as they become complete.
Volume 1
Chapter 1:
Holomorphic and real analytic calculus
Chapter 2:
The Weierstrass Preparation Theorem and applications
Chapter 3:
Domains of holomorphy and notions of convexity in
Cn
Chapter 4:
Holomorphic and real analytic manifolds
Chapter 5:
Metric structures and connections
Chapter 6:
Holomorphic and real analytic jet bundles
Chapter 7:
Stein and real analytic manifolds
Chapter 8:
Topologies for spaces of maps, functions, sections, and germs
Appendix A:
General versions of the Chain Rule and the Leibniz Rule
Appendix B:
Convex analysis
Appendix C:
Topology background
Appendix D:
Distributions
Appendix E:
Operator theory
Appendix F:
Multilinear algebra
Volume 2 (what little there is, at present)
Chapter 1:
Sheaf theory
Chapter 2:
Category theory
Chapter 3:
Homological algebra
Chapter 4:
Sheaf cohomology
Chapter 5:
Coherent sheaves
Chapter 6:
Differentiable and analytic spaces
Chapter 7:
Approximation
Sources:
Several complex variables and holomorphic differential geometry:
- Jean-Pierre Demailly, Complex Analytic and Differential Geometry,
pdf
- Lars Hörmander, An Introduction to Complex Analysis in Several
Variables, Second Edition, North-Holland, Amsterdam/New York, 1973, ISBN
0-444-10523-9
- R. Michael Range, Holomorphic Functions and Integral Representations
in Several Complex Variables, number 108 in Graduate Texts in
Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1986, ISBN
0-387-96259-X
- Joseph L. Taylor, Several Complex Variables with Connections to
Algebraic Geometry and Lie Groups, number 46 in Graduate Studies in
Mathematics, American Mathematical Society, Providence, Rhode Island, 2002,
ISBN 0-8218-3178-X
Real analytic analysis and differential geometry
- Hans Grauert, On Levi's problem and the imbedding of real-analytic
manifolds, Annals of Mathematics. Second Series, 68, 450-472, 1958
- Stephen G. Krantz and Harold R. Parks, A Primer of Real Analytic
Functions, Second Edition, Birkhäuser Advanced Texts,
Birkhäuser, Boston/Basel/Stuttgart, 2002, ISBN 0-8176-4264-1
Coherent analytic sheaves
- Henri Cartan, Variétés analytiques réelles
et variétés analytiques complexes, Bulletin de la
Société Mathématique de France, 85, 77-99, 1957
- Hans Grauert and Reinhold Remmert, Coherent Analytic Sheaves,
number 265 in Grundlehren der mathematischen Wissenschaften,
Springer-Verlag, New York-Heidelberg-Berlin, 1984, ISBN 3-540-13178-7
- Taylor above