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Use this book. Many good problems. Has a wonderful collection of problems. Highly recommended. Bourbaki does everything in the utmost generality. Everything in this book is done with a mind to the student. Out of print, but can be bought online. Excellent for learning.

ISBN 13: 9783540617877

Moves slowly and gives examples, but it can be hard to see the forest for the trees. Nota Bene: Many of the journal articles in the subject are accessible. The new, expanded second edition is called Topology. Bypasses Lie algebra theory. Requires college-level mathematical maturity. Contains many exercises, as notorious as they are pedagogically sound. Highly recommended as a first book on manifolds.

Perfect for those comfortable with a fairly high level of abstraction. Elementary and very readable. Teaches in Scheme, a dialect of Lisp. Semi-direct products; central products and wreath products. Automorphism group. Horosevskiis theorem on the orders of elements in Aut G. Schur-Zassenhaus Theorem. Coprime action. Fitting theorem. Burnsides normal p-complement theorem. Focal subgroup. Frobenius actions. The Thompson subgroup. Burnsides paqb-theorem.

Permutation groups. Simple groups. Aschbacher, Finite Group Theory.

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Huppert, Endliche Gruppen I. Grudlehren der mathematischen Wissenschaften , Volume Martin Isaacs, Finite Group Theory. Graduate Studies in Mathematics , Volume American Mathematical Society. Derek J.

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Springer, Harvey E. Bertram A. Universal enveloping algebras, Poincare-Birkhoff-Witt theorem. Solvable and nilpotent Lie algebras, Engels theorem, Lies theorem. Representations of Lie algebras, Schurs lemma, representations of sl 2,C. Killing form, Cartans criteria for solvability and semisimplicity, derivations of semisimple Lie algebras.

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Cartan subalgebras, root space decomposition, Cartan subalgebras as inner product spaces. Root systems, Weyl group of a root system, Dynkin diagrams.

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Classical Lie algebras sl n,C ,so n,C ,sp n,C. Classification of root systems, irreducible root systems and complex simple Lie algebras. James E. Amalgamated free products and HNN extensions. Structure of a group acting on a tree. Ends of a group. Group actions and quasi-isometries. Hyperbolic spaces. Hyperbolic groups.

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Growth of groups. Polynomial, sub-exponential, exponential growth of groups. Gromovs theorem on groups of polynomial growth. Grigorchuk group. Subgroup growth of free groups.

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Recommended Reading Bowditch, B. Bridson, M. Ghys, E. Alexander Lubotzky, Dan Segal. Subgroup Growth. Birkhuser, Mann, A. Coordinate rings, affine coordinate transformations, discrete valuation rings, ideals with finitely many zeros, multiple points, tangent lines and local rings. The Zariski topology, morphism of varieties, algebraic function fields and dimension of varieties, rational maps. Igor R. Suggested Reading C. First order Differential operators, locally free sheaves and Vector Bundles, theorem of Frobenius. Differential operators of higher order. Integration on Manifold and adjoints of Differential operators.

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Local analysis of Elliptic operators : Schwartz space and Densities, Fourier transforms, Distributions, Sobolevs theorem, Interior regularity of Elliptic solutions, Rellichs theorem. Elliptic operators on Differentiable Manifolds, Regularity theorem, finiteness theorem, Elliptic Complexes and Laplacian. Additional Topics Pseudo-Differential operators on Manifolds. Fredholm operators and the Index of a Fredholm operator. Advanced Reading R. Dimension theory, Hilbert function of a graded module, Hilbert-Samuel polynomial of a local ring, system of parameters and multiplicity.

Stanley-Reisner rings or face rings of simplicial complexes, Hilbert series, h-vectors and f-vectors. Macaulays theorem on Hilbert functions. Shellability and Cohen-Macaulayness. Partially ordered sets, Mbius functions, Mbius inversion, Eulerian posets, Shellable posets, Poset rings.

Upper bound theorem. Free resolution of monomial ideals. Recommended Reading H.