Ph.D. Mathematics Preliminary Exam Syllabi

     

    Syllabus for the Preliminary Examination in Algebra

    It is assumed that students know the basic material from an undergraduate course in linear algebra and an undergraduate abstract algebra course.

    1. Groups:     Finite groups, including Sylow theorems, p-groups, direct products and sums, semi-direct products, permutation groups, simple groups, finite Abelian groups; infinite groups, including normal and composition series, solvable and nilpotent groups, Jordan-Holder theorem, free groups.
     
    References:  Goldhaber  Ehrlich, Ch. I except 14; Hungerford, Ch. I, II; Rotman, Ch. I-VI, VII (first three sections).

    2. Rings and modules:     Unique factorization domains, principal ideal domains, modules over principal ideal domains (including finitely generated Abelian groups), canonical forms of matrices (including Jordan form and rational canonical form), free and projective modules, tensor products, exact sequences, Wedderburn-Artin theorem, Noetherian rings, Hilbert basis theorem.
     
    References:     Goldhaber  Ehrlich, Ch. II, III  1,2,4, IV, VII, VIII;  Hungerford, Ch. III except 4,6, IV 1,2,3,5,6, VIII 1,4,6.

    3.  Fields:     Algebraic and transcendental extensions, separable extensions, Galois theory of finite extensions, finite fields, cyclotomic fields, solvability by radicals.
     
    References:     Goldhaber  Ehrlich, Ch. V except 6;  Hungerford, Ch. V, VI;  Kaplansky, Part I.

    References:
            Goldhaber  Ehrlich, Algebra, reprint with corrections, Krieger, 1980.
            Hungerford, Algebra, reprint with corrections, Springer, 1989.
            Isaacs, Algebra, a Graduate Course, Wadsworth, 1994.
            Kaplansky, Fields and Rings, 2nd Edition, University of Chicago Press, 1972.
            Rotman, An Introduction to the Theory of Groups, 4th Edition, W.C. Brown, 1995.
     
     

    Syllabus for the Preliminary Examination in Analysis

    It is assumed that students are familiar with the subject matter of the undergraduate analysis  course M365C, a syllabus for which follows this exam syllabus.

    Real Analysis
     
    1.  Measure theory:     Lebesgue measure on  Rn and its elementary properties, regularity of Lebesgue measure; measurable functions, Egorov's Theorem, Lusin's Theorem; measure spaces and measurable functions in an abstract setting.
     
    2.  Integration:     Lebesgue integration in Rn and in an abstract setting; Fatou's Lemma, the Monotone and Dominated Convergence Theorems; product measures and the Fubini-Tonelli Theorems with applications such as convolution.
     
    3.  Differentiation:     Properties of monotone functions, absolutely continuous functions and functions of bounded variation; the Fundamental Theorem of Lebesgue Integral Calculus; singular functions, the Cantor function.
     
    4.   Lp-spaces:     Convergence almost everywhere, in norm and in measure; approximation in Lp-norm; the Jensen, Holder and Minkowski inequalities; Lp-spaces,  Lp- Lq duality.
     
    Complex Analysis
     
    1. Analytic functions:     Definition and elementary examples; power series and their properties; the Cauchy-Riemann equations; Mobius transformations.
     
    2. Complex integration:     Paths and contour integrals; homotopy; Cauchy's Theorem and Integral Formulae; power series representation; the Maximum Principle, Schwarz's Lemma.
     
    3.  Singularities:     Laurent series and classification of isolated singularities; the Residue Theorem and its applications; counting zeros and poles --- the Argument Principle; Rouche's Theorem, the Open Mapping Theorem.
     
    4.  Convergence:     Uniform convergence on compact sets and its implications for analytic functions.

     
    References:
            Ahlfors, Complex Analysis, 3rd Edition, McGraw-Hill, 1979, Ch. 1-6.
            Conway, Functions of One Complex Variable, Springer, 1978, Ch. 1-7.
            Palka, An Introduction to Complex Function Theory, corrected 2nd printing, Springer, 1995, Ch. 1-10.
            Royden, Real Analysis, 3rd Edition, Prentice-Hall, 1988, Ch. 1-7, 9-11 (omit 11.4-11.6), 12.1-12.4.
            Rudin, Real and Complex Analysis, 3rd Edition, McGraw-Hill, 1987, Ch. 1-4, 6-8, 10, 11.1-11.4, 12, 13, 14.5-14.6.



    Syllabus for the Preliminary Examination in Analysis

    Effective August 2000


    The objective of this syllabus is to aid students in attaining a broad understanding of analysis techniques that are the basic stepping stones to contemporary research. The prelim exam normally consists of eight to ten problems, and the topics listed below should provide useful guidelines and strategy for their solution. It is assumed that students are familiar with the subject matter of the undergraduate analysis courses M365C and M361. analysis courses M365C and M361.

    1. Measure Theory and the Lebesgue Integral
    Basic properties of Lebesgue measure and the Lebesgue integral on Rn (see [5], Ch. 1-4) and general measure and integration theory in an abstract measure space (see [5], Ch. 11-12; and especially [6], Ch. 1-2). Lp spaces (see [6], Ch. 3); convergence almost everywhere, in norm and measure; approximation in Lp-norm and Lp-Lq duality; integration in product spaces (see [6], Ch. 8) and convolution on Rn; and the concept of a Banach space, Hilbert space, dual space and the Riesz representation theorem.

    2. Holomorphic Functions and Contour Integration
    Basic properties of analytic functions of one complex variable (see [1], Ch. 4-5; [2], Ch. 4-7; [4], Ch. 4-8; or [6], Ch. 10-12 and 15). Integration over paths, the local and global forms of Cauchy's Theorem, winding number and residue theorem, harmonic functions, Schwarz's Lemma and the Maximum Modulus theorem, isolated singularites, entire and meromorphic functions, Laurent series, infinite products, Weierstrass factorization, conformal mapping, Riemann mapping theorem, analytic continuation, "little" Picard theorem.

    3. Differentiation
    The relationship between differentiation and the Lebesgue integral on a real interval (see [5], Ch. 5), derivatives of measures (see [6], Ch. 5), absolutely continuous functions and absolute continuity between measures, functions of bounded variation.

    4. Specific Important Theorems
    Students should be familiar with Monotone and Dominated Convergence theorems, Fatou's lemma, Egorov's theorem, Lusin's theorem, Radon-Nikodym theorem, Fubini-Tonelli theorems about product measures and integration on product spaces, Cauchy's theorem and integral formulas, Maximum Modulus theorem, Rouche's theorem, Residue theorem, and Fundamental Theorem of Calculus for Lebesgue Integrals. Students should be familiar with Minkowski's Inequality, Holder's Inequality, Jensen's Inequality, and Bessel's Inequality.

    References:
    1. L. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1979.
    2. J.B. Conway, Functions of One complex Variable, second edition, Springer-Verlag, New York, 1978.
    3. G.B. Folland, Real Analysis, second edition, John Wiley, New York, 1999.
    4. B. Palka, An Introduction to Complex Function Theory, second printing, Springer-Verlag, New York, 1995.
    5. H.L. Royden, Real Analysis, Macmillan, New York, 1988.
    6. W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill, New York, 1987.
    7. R. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, New York, 1977.
     
     

    Syllabus for M365C -- Introduction To Analysis

    The real number system and euclidean spaces:     The axiomatic description of the real number system as the unique complete ordered field; the complex numbers; euclidean space R.

    Metric spaces:     Elementary metric space topology, with special emphasis on euclidean spaces; sequences in metric spaces --- limits, accumulation points, subsequences, etc.; Cauchy sequences and completeness; compactness in metric spaces; compact sets in R; connectedness in metric spaces; countable and
    uncountable sets.

    Continuity:     Limits and continuity of mappings between metric spaces, with particular attention to real-valued functions defined on subsets of R; preservation
    of compactness and connectedness under continuous mapping; uniform continuity.

    Differentiation on the line:     The definition and geometric significance of the derivative of a real-valued function  of a real variable; the Mean Value Theorem and its consequences; Taylor's theorem; L'Hospital's rules.

    Riemann integration on the line:     The definition and elementary properties of the Riemann integral; existence theorems for Riemann integrals; the Fundamental Theorems of Calculus.

    Sequences and series of functions:     Uniform convergence, uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation.

    (An appropriate text might be Rudin's Principles of Mathematical  Analysis, and the course should cover roughly its first seven chapters.)
     
     

    Syllabus for the Preliminary Examination in Topology

    It is assumed that students have a working knowledge of the equivalent of a one semester course in general topology (for example, see the appended syllabus for the undergraduate course M367K). For the semester in differential topology, it will also be assumed that students know the basic material from an undergraduate linear algebra course.
     
    Algebraic Topology

    1. Manifolds:     Identification (quotient) spaces and identification (quotient) maps; topological n-manifolds, including surfaces, Sn, RPn, CPn, and lens spaces.

    2. Triangulated manifolds:     Representation of triangulated, closed 2-manifolds as connected sums of tori or projective planes.

    3. Fundamental group and covering spaces:     Fundamental group, functoriality, retract, deformation retract; Van Kampen's Theorem, classification of surfaces by abelianizing Pi1; covering spaces, path lifting, homotopy lifting, uniqueness of lifts, general lifting theorem for maps, covering transformations, regular covers,
    correspondence between subgroups of Pi1(x) and covering spaces of X;  computing  Pi1 of the circle, RPn, lens spaces via covering spaces.

    4. Simplicial homology:     Homology groups, functoriality, topological invariance, Mayer-Vietoris sequence; applications, including Euler characteristic, classification of closed triangulated surfaces via homology and via Euler characteristic and orientability; degree of a map between oriented manifolds, Lefschetz number, Brouwer Fixed Point Theorem.
     
     References:
            Armstrong, Basic Topology, Springer, 1983  (principal text).
            Greenberg, Lectures on Algebraic Topology, W.A. Benjamin, 1967.
            Massey, Algebraic Topology, an Introduction, 4th corrected printing, Springer, 1977.
            Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984.
     
    Differential Topology

    1.  Smooth mappings:     Inverse Function Theorem, Local Submersion Theorem (Implicit Function Theorem).

    2. Differentiable manifolds:     Differentiable manifolds and submanifolds; examples, including  surfaces, Sn, RPn, CPn  and lens spaces; tangent bundles; Sard's Theorem and its applications; differentiable transversality; orientation.

    3. Vector fields and differential forms:     Integrating vector fields; degree of a map, Brouwer Fixed Point Theorem, No Retraction Theorem, Poincare-Hopf Theorem; differential forms, Stokes Theorem.
     
    References:
            Guillemin  Pollack, Differential Topology, Prentice-Hall, 1974 (basic reference).
            Hirsch, Differential Topology, Springer, 1976.
            Milnor, Topology from the Differentiable Viewpoint, University of Virginia Press, 1965.
            Spivak, Calculus on Manifolds, Benjamin, 1965 (differentiation, Inverse Function  Theorem, Stokes Theorem).

    For the examples indicated we refer to the books of Greenberg, Hirsch and Munkres.
     

    Syllabus for M367K -- Topology I

    Cardinality:     1-1 correspondence, countability, and uncountability.

    Definitions of topological space:     Basis, sub-basis, metric space.

    Countability properties:     Dense sets, countable basis, local basis.

    Separation properties:     Hausdorff, regular, normal.

    Covering properties:     Compact, countably compact, Lindelof.

    Continuity and homeomorphisms:     Properties preserved by continuous functions, Urysohn's Lemma, Tietze Extension Theorem.

    Connectedness:     Definition, examples, invariance under continuous functions.
     
    Reference:     Munkres, Topology: a First Course, Prentice-Hall, 1975.
     
     

    Syllabus for the Preliminary Examination in Applied Mathematics

    It is assumed that students are familiar with the subject matter of the undergraduate analysis course M365C (see the Analysis section for a syllabus of that course) and an undergraduate course in linear algebra.

     

    1. Banach spaces: Normed linear spaces and convexity; convergence, completeness, and Banach spaces; continuity, open sets, and closed sets; continuous linear transformations; Hahn-Banach Extension Theorem; linear functionals, dual and reflexive spaces, and weak convergence; the Baire Theorem and uniform boundedness; Open Mapping and Closed Graph Theorems; Closed Range Theorem; compact sets and Ascoli-Arzelà Theorem; compact operators and the Fredholm alternative.

    2. Hilbert spaces: Basic geometry, orthogonality, bases, projections, and examples; Bessel’s inequality and the Parseval Theorem; the Riesz Representation Theorem; compact and Hilbert-Schmidt operators; spectral theory for compact, self-adjoint and normal operators; Sturm-Liouville Theory.

    3. Distributions: Seminorms and locally convex spaces; test functions and distributions; calculus with distributions.

    4. The Fourier Transform and Sobolev Spaces: The Schwartz space and tempered distributions; the Fourier transform; the Plancherel Theorem; convolutions; fundamental solutions of PDE’s; Sobolev spaces; Imbedding Theorems; the Trace Theorems for Hs.

    5. Variational Boundary Value Problems (BVP): Weak solutions to elliptic BVP’s; variational forms; Lax-Milgram Theorem; Green’s functions.

    6. Differential Calculus in Banach Spaces and Calculus of Variations: The Fréchet derivative; the Chain Rule and Mean Value Theorems; Banach’s Contraction Mapping Theorem and Newton’s Method; Inverse and Implicit Function Theorems, and applications to nonlinear functional equations; extremum problems, Lagrange multipliers, and problems with constraints; the Euler-Lagrange equation.

    7. Asymptotic Analysis: Definitions and fundamental properties; examples of transcendental equations and initial-value problems.

    References:

    The first four references cover most of the syllabus for the exam. The other references also cover some topics in the syllabus.

    1. C. Carath'eodory, Calculus of Variations and Partial Differential Equations of the First Order, 2nd English Edition, Chelsea, 1982.

    2. F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974.

    3. M. Reed and B. Simon, Methods of Modern Physics, Vol. 1, Functional analysis.

    4. R.E. Showalter, Hilbert Space Methods for Partial Differential Equations, available at World Wide Web address http://ejde.math.swt.edu//mono-toc.html .

    5. A. Avez, Introduction to Functional Analysis, Banach Spaces, and Differential Calculus, Wiley, 1986.

    6. L. Debnath and P. Mikusi'nski, Introduction to Hilbert Spaces with Applications, Academic Press, 1990.

    7. I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice-Hall, 1963.

    8. E. Kreyszig, Introductory Functional Analysis with Applications, 1978.

    9. J.T. Oden and L.F. Demkowicz, Applied Functional Analysis, CRC Press, 1996.

    10. W. Rudin, Functional Analysis, McGraw-Hill, 1991.

    11. W. Rudin, Real and Complex Analysis, 3rd Edition, McGraw-Hill, 1987.

    12. K. Yosida, Functional Analysis, Springer-Verlag, 1980.