List of recommended books for Ph229: Advanced Mathematical Methods of
B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Modern Geometry - Methods and
Applications," Parts 1 and 2, Springer-Verlag (1984/85). An accessible
introduction to topology, differential geometry, Lie groups and algebras,
with illustrations from mechanics, relativity, electromagnetism, and Yang-Mills
theory. Mathematical pre-requisites are kept to a minimum, which makes
it suitable for physics students.
V.G. Boltyanskii and V.A. Efremovich, "Intuitive Combinatorial Topology,"
Springer (2001). If you have had no prior exposure to topology, this is
a good place to start. The topics include graphs, classical surfaces,
map-coloring problems, knots, the fundamental group, some notions of homology,
and fiber bundles. The level of discussion is elementary.The style is very
informal and intuitive, rather than rigorous, with lots of pictures.
J.L. Kelley, "General Topology," Springer-Verlag (1975). A nice book on
general topology with many stimulating problems. We will only need some
material from Chapters 1, 3, and 5.
V.A. Vassiliev, "Introduction to Topology," American Mathematical Society
(2001). A concise introduction to basic algebraic topology. Covers almost
exactly the same material as I intend to cover in the fall term. If I knew
about this book earlier, I would have chosen it as a required textbook
(for the fall term).
W.S. Massey, "A basic course in algebraic topology," Springer-Verlag (1991).
An introduction to fundamental groups, covering spaces, homology
and other basic algebraic topology. Some familiarity with general topology
J.J. Rotman, "An Introduction to Algebraic Topology," Springer-Verlag (1998).
Covers the same topics as Massey's book, plus some homotopy theory. The
level of exposition is slightly more advanced than in Massey, but the author
makes every effort to motivate formal developments, and overall I found
it very readable.
R. Bott and L.W. Tu, "Differential Forms in Algebraic Topology," Springer-Verlag
(1982). A classic. An introduction to algebraic topology using de Rham
theory as a starting point. Covers more advanced topics such as spectral
sequences, the Postnikov approximation, basics of rational homotopy theory,
etc. The last chapter is a very nice introduction to characteristic classes.
Not an easy reading, but a diligent student will be rewarded by a substantial
illumination into a very beautiful field of mathematics.
J.W. Milnor and J. Stasheff, "Characteristic Classes," Princeton University
Press (1974). A beautiful exposition of the theory of characteristic classes
from an axiomatic point of view. A gem of a book.
F.W. Warner, "Foundations of Differentiable Manifolds and Lie Groups,"
Springer-Verlag (1983). A very clear and concise exposition of calculus
on manifolds and much more. Includes such topics as Frobenius theorem,
the relation between Lie groups and Lie algebras, de Rham theory, foundations
of sheaf theory, elliptic operators and the Hodge theorem. Highly recommended.
M. Postnikov, "Smooth manifolds," Mir (1989). An in-depth introduction
to manifolds, differential forms, and Cech-de Rham theory, including spectral
sequences. Roughly covers the same ground as Warner's chapters 1,2,4, and
5, but the style is less formal (i.e. more motivation is provided). Also
includes classic differential geometry of curves and surfaces, basics of
general topology, and such classic results as Brouwer's fixed-point theorem,
Sard's theorem, and Whitney's embedding theorem.
R. L. Bishop and R. J. Crittenden, "Geometry of Manifolds" (2nd edition),
American Mathematical Society (2001). An excellent introduction to differential
geometry, including Riemannian geometry and the theory of connections on
principal and vector bundles. Highly recommended.
W. D. Curtis and F. R. Miller, "Differential Manifolds and Theoretical
Physics," Academic Press (1985). This is an introduction to differential
geometry and Lie groups aimed at physicists. Besides the standard mathematical
material, various topics of interest to physics students are covered, such
as special relativity, Hamiltonian mechanics, Marsden-Weinstein reduction,
Yang-Mills theory, and a brief discussion of geometric quantization.
S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," vol.
I, Wiley (1996). A classic exposition of differential geometry. This book
is substantially more advanced than the course.
V. I. Arnold, "Mathematical Methods of Classical Mechanics" (2nd edition),
Springer (1989). An excellent blend of physics (classical mechanics) and
mathematics (differential and especially symplectic geometry). Highly recommended.
V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics" (2nd
edition), Cambridge University Press (1990). Chapter I explains the
role of symplectic geometry in various physics problems. Chapter II is
devoted to Hamiltonian actions of Lie groups and the geometry of the moment
map. Chapter III discusses the interaction of particles with Yang-Mills
fields from the point of view of symplectic geometry. Chapters IV and V
cover completely integrable systems and symplectic homogeneous spaces,
A. Cannas da Silva and A. Weinstein, "Geometric Models for Noncommutative
Algebras," American Mathematical Society (1999). This unique monograph,
a first of its kind, describes inter-relations between noncommutative geometry
and more classical topics, such as symplectic and Poisson geometry. The
authors managed to squeeze an amazing quantity of material into a slim
volume, without sacrificing the clarity of exposition. Highly recommended.
R.O. Wells, "Differential Analysis on Complex Manifolds" (2nd edition),
Springer-Verlag (1980). A good introduction to complex and Kahler manifolds.
Covers sheaf theory, holomorphic vector bundles and their characteristic
classes, elliptic operators and Hodge theory on Kahler manifolds,
and Kodaira's vanishing and embedding theorems. Concise, but clear.
I. R. Shafarevich, "Basic Algebraic Geometry," Springer-Verlag (1994).
Probably the best introduction to algebraic geometry available.
P. Griffiths and J. Harris, "Principles of Algebraic Geometry," Wiley (1994).
Another classic. An introduction to algebraic geometry from the point of
view of complex analysis. This book is a must for anyone who wants to really
learn what algebraic geometry is about.
F. Zheng, "Complex Differential Geometry," American Mathematical
R. Miranda, "Algebraic curves and Riemann surfaces," American Mathematical
V. V. Shokurov, "Riemann surfaces and algebraic curves," in: Algebraic
Geometry I, Encyclopaedia of Mathematical Sciences, vol. 23. This
is a review, not a textbook.
V. I. Danilov, "Algebraic varieties and schemes," in: Algebraic
Geometry I, and "Cohomology of algebraic varieties," in: Algebraic Geometry
II, Encyclopaedia of Mathematical Sciences, vols. 23 and 35. This
is a excellent review of the methods and results of modern algebraic geometry.