
Curriculum of the First Year CoursesAll Ph.D. students are required to take (or place out of) these 600level courses, although it is possible to take the 500level 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 AlgebraPrerequisitesGroups, 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, noncommutative rings  matrix ring. Readings: Lang, Chapters 5,6,9,10 (para. 14), 17. Jacobson, Part I, Chap. 2, Part II, Chap. 4 (para. 16), 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. 13), Part II, Chap. 3,5 (para. 13). 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. 14); Jacobson, Part I, Chap. 4, Part II, Chap. 8. General references: Lang  Algebra; Jacobson  Basic Algebra I, II; AtiyahMacdonald  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; ZariskiSamuel, Commutative algebra (Vol. 1). The First Year Curriculum in AnalysisPrerequisitesAxiomatic 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 termbyterm. 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, Analysis I (Math 608)The first twothirds of the semester concerns conplex analysis: analyticity, Cauchy theory, meromorphic functions, isolated singularities, analytic continuation, Runge's theorem, dbar equation, MittlagLeffler theorem, harmonic and subharmonic functions, Riemann mapping theorem, Fourier transform from the analytic perspective. The last third of the semester provides an introduction to real analysis: Weierstrass approximation, Lebesgue measure in Euclidean spaces, Borel measures and convergence theorems, C^0 and the RieszMarkov theorem, L^p spaces, Fubini theorem.Course Content, Analysis II (Math 609)The first third of the semester continues the study of real analysis begun in Math 608. Topics will include: general measure theory, outer measures and Cartheodory construction, Hausdorff measures, RadonNikodym theorem, Fubini's theorem, Hilbert space and L^2theory of the Fourier transform. The last twothirds of the semester concerns functional analysis: normed linear spaces, convexity, the HahnBanach Theorem, duality for Banach spaces, weak convergence, bounded linear operators, Baire category theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, compact operators, Fredholm theory, interpolation theorems, L^p theory for the Fourier transform.The First Year Curriculum in GeometryTopologyPrerequisitesBasic familiarity with pointset (general) topology: metric spaces, topological spaces, separation axioms, compactness, completeness.Course Content, GeometryTopology 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", PrenticeHall, 1974. Course content, GeometryTopology 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.
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