Penn Math
Department of Mathematics

Curriculum of the First Year Courses

There are introductory graduate courses in Algebra, Analysis, and Geometry-Topology. Each is offered in two versions: a masters level version and a Ph.D. level version. The masters level algebra course is Math 502/503, and the Ph.D. level course is Math 602/603. In analysis, the corresponding numbers are 508/509 and 608/(609 or 610). In geometry-topology, the corresponding numbers are 500/501 and 600/601.

All Ph.D. students are required to take (or place out of) these 600-level courses, although it is possible to take the 500-level courses first if this seems appropriate.

In addition, all Ph.D. students are required to take a year of algebraic topology, Math 618/619. Below are descriptions of Math 602/603, 608/(609 or 610), and 600/601. See the full list of course descriptions for information about other courses.

The First Year Curriculum in Algebra

Prerequisites

Groups, normal subgroups and conjugacy classes, finite groups of order 12.

Readings: Lang, Algebra, Chapter 1; Jacobson, Basic Algebra, Part I, chapter 1.

Rings, polynomial rings in one variable, unique factorization, non-commutative rings - matrix ring.

Readings: Lang, Chapters 5,6,9,10 (para. 1-4), 17. Jacobson, Part I, Chap. 2, Part II, Chap. 4 (para. 1-6), Chap. 7. Symmetric and Hermitian matrices, spectral theorem, quadratic forms and signature.

Readings: Lang, Chap. 3,13,14,15,16. Jacobson, Part I, Chap. 3,6 (para. 1-3), Part II, Chap. 3,5 (para. 1-3).

Definition of a field, field of fractions of an integral domain.

Course content, Algebra I,II (Math 602, 603)

Groups: Sylow's theorem and its applications, finite abelian groups, nilpotent and solvable groups.

Rings: commutative noetherian rings, Hilbert basis theorem, prime and maximal ideals and localizations, primary decomposition, integral extensions and normal rings, Dedekind domains, Eisenstein irreducibility criteria, group ring, semisimple rings and Wedderburn's theorem.

Modules: tensor product, symmetric and exterior algebras and induced maps, exact functors, projective and injective modules, finitely generated modules over a Principal Ideal Domain with application to canonical forms of a matrix over a field, elementary theory of group representations.

Field extensions and Galois theory: separable and inseparable extensions, norm and trace, algebraic and transcendental extensions, transcendence basis, algebraic closure, fundamental theorem of Galois theory, solvability of equations, cyclotomic extensions and explicit computations of Galois groups.

Readings: Lang, Chap. 7,8,10 (para. 1-4); Jacobson, Part I, Chap. 4, Part II, Chap. 8.

General references: Lang - Algebra; Jacobson - Basic Algebra I, II; Atiyah-Macdonald - Introduction to commutative algebra; Kaplansky - Fields and Rings; Artin - Galois Theory; van der Waerden - Modern Algebra; Kaplansky - Commutative rings; Serre - Linear representations of finite groups; Zariski-Samuel, Commutative algebra (Vol. 1).

The First Year Curriculum in Analysis

Prerequisites

Axiomatic development of the real number system, especially the completeness axiom; Abstract metric spaces, open and closed sets, completeness, compactness; Continuous functions from one metric space to another, uniform continuity; Continuous functions on a compact metric space have compact image and are uniformly continuous; Pointwise and uniform convergence of sequences and series of functions; continuity of a uniform limit of continuous functions. Differentiation: mean value theorem, Taylor's theorem and Taylor's series, partial differentiation and total differentiability of functions of several variables.
Riemann integration: definition and elementary properties, fundamental theorem of calculus. Interchange of limit operations, of order of partial differentiation, integration of spaces term-by-term. Implicit function theorem. Fourier analysis up to pointwise convergence for piecewise smooth functions. Use of Fourier analysis to solve heat and vibration equations. Differential equations, solution of common forms. Complex numbers, power series and Fourier series (an undergraduate course in complex analysis would be helpful).

Readings: Except for the material on Fourier analysis, the above is all in Rosenlicht's "Introduction to Analysis", Rudin's "Principles of Mathematical Analysis", Boyce and de Prima's "Elementary Differential Equations" and many other books.

Course Content, Complex Analysis (Math 608)

Rapid development of Cauchy theory, the maximum principle and subharmonic functions, basic harmonic function theory, conformal mapping and the Dirichlet Problem. Representations in terms infinite series and products and connections to the theory of entire functions. Meromorphic functions and the Mittag-Leffler and Weierstrass problems. Other topics as time permits (Second order linear O.D.E.'s, asymptotics, prime number theorem, Picard theorems, elliptic functions, basic sheaf theory, Riemann surfaces, fluid flow problems.)

Students have a choice of choosing Math 609 or Math 610 or both.

Course Content, Real Analysis (Math 609)

Construction and properties of Lebesgue measures in Euclidean space, Borel measures and convergence theorems. Elementary function spaces. Some general measure theory, including the Caratheodory construction of measures from outer measures, the Radon-Nikodym theorem, the Fubini theorem, and Hausdorff measure. Stone Weierstrass theorem. Elements of classical Harmonic analysis: the Fourier transform on basic function spaces, the Hilbert and Cauchy transforms.

Course Content, Functional Analysis (Math 610)

Convexity and the Hahn Banach Theorem. Hilbert Spaces, Banach Spaces, and examples: Sobolev spaces, Holder spaces. The uniform boundedness principle, Baire category theorem, bounded operators, open m\ apping theorem, closed graph theorem and applications. The concepts of duality and dual spaces. The Riesz theory of compact operators and Fredholm theory. Functional calculus and elementary Spectral Theory. Interpolation theorems. Applications to partial differential equations and approximation theory.

The First Year Curriculum in Geometry-Topology

Prerequisites

Basic familiarity with point-set (general) topology: metric spaces, topological spaces, separation axioms, compactness, completeness.

Course Content, Geometry-Topology I (Math 600)

Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions.

Readings: M. Spivak, "A Comprehensive Introduction to Differential Geometry", vol.I, 2nd edition. Publish or Perish, 1979. Supplementary: V. Guillemin & A. Pollack, "Differential topology", Prentice-Hall, 1974.

Course content, Geometry-Topology II (Math 601)

Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces. Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology. Brouwer fixed point theorem, CW complexes, cup and cap products, Poincare duality, Kunneth and universal coefficient theorems, Alexander duality, Lefschetz fixed point theorem.

Readings: M.J. Greenberg & J. Harper, "algebraic Topology, a first course". Math Lecture Note Series, vol.58. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. Supplementary: A. Hatcher, "Algebraic topology", Cambridge University Press, 2002.

General references:
1. I. Singer - J. Thorpe, Lecture notes on elementary topology and geometry.
2. J. Kelley, General topology (look at the exercises).
3. J. Djugundji, Topology.
4. J. Munkres, Topology: a first course.
5. F. Warner, Foundations of differentiable manifolds and Lie groups (first half).
6. M. Spivak, A comprehensive introduction to differential geometry, Vol. 1.
7. H. Flanders, Differential forms, with applications to the physical sciences.
8. W. Fleming, Functions of several variables.
9. G. Simmons, Introduction to topology and modern analysis.
10. F. Croom, Basic concepts of algebraic topology (good for fundamental groups and homology).
11. W. Massey, Algebraic topology, an introduction (good for fundamental groups).
12. C.T. Wall, A geometric introduction to topology.
13. M.A. Armstrong, Basic topology.
14. J. Milnor, Topology from the differentiable viewpoint.
15. Greenberg, Lectures on algebraic topology (first two chapters are good for fundamental groups and covering spaces).
16. Bourbaki, General topology.
17. Guillemin and Pollack, Differential topology (good exercises).
18. Y. Matsushima, Differentiable manifolds.

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