Preliminary Exams

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.

The first part of the Prelim examination will cover Real Analysis. The second part of the Prelim examination will cover Complex Analysis.

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). L^p spaces (see [6], Ch. 3); convergence almost everywhere, in norm and measure; approximation in L^p-norm and L^p-L^q 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

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.)

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.

The Applied Math Prelim divides into six areas.

The first three are discussed in M383C and will be covered in the first part of the Prelim examination:

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.

These three areas are discussed in M383D and will be covered in the second part of the Prelim examination:

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 H^s.

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.

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.

The Prelim sequence is M387C and M387D.

The first part of the Prelim examination will cover algebra and approximation and the second part of the Prelim examination will cover diferential equations.

Principles of discretization of differential equations

• ODEs: Stability and convergence theory, Stiff problems,Symplectic integrators
• FEM (finite element method) and FDM (finite difference method) for boundary value problems
• FEM for PDEs (main focus on elliptic problems): Basic theory, weak formulations, Lax-Milgram theorem, finite element spaces, approximation theory, a priori and a posteriori error estimates, practical algorithms, extensions, mixed methods etc.
• FDM for PDEs (main focus on hyperbolic and parabolic problems): Lax equivalence theorem, Von Neumann and other stability analysis, nonlinear conservation laws, shocks, entropy, practical algorithms

Brief survey of other methods for PDEs

• FVM, DG, Spectral and particle methods
• Applications: Elasticity (FEM), Fluids (FVM), and Waves (FDM)
• Solution of linear and nonlinear equations
• Solution of integral equations
• Eigenvalues
• Optimization
• Monte Carlo methods
• Fast Fourier, wavelet transforms, approximation theory
• Basic undergraduate numerical methods
  • Interpolation, fixed point iterations, Newton's method for root finding
  • Direct and iterative methods for solving linear equations
  • Quadratures

Recommended texts

• Dahlquist and Bjorck, Numerical methods. Dover
• Lambert, Numerical methods for ordinary differential systems. Wiley
• Gustafsson, Kreiss, and Oliger, Time dependent problems and difference methods
• Iserles, A first course in the numerical analysis of differential equations, Cambridge
• Claes Johnson, Numerical solution of partial differential equations by the finite element method. Cambridge University Press

The first part of the Prelim examination will deal with the material covered in M385C and the second part of the Prelim examination will deal with the material covered in M385D.

1. Theory of Probability I - M385C

Prerequisites:

• Real Analysis (M365C or equivalent),
• Linear Algebra (M341 or equivalent),
• Probability (M362K or equivalent).

Literature:

• R. Durrett, Probability: theory and examples, third ed., Duxbury Press, Belmont, CA, 1996. (required)
• D. Williams, Probability with martingales, Cambridge University Press, Cambridge, 1991. (recommended)

Syllabus:
(Note: all references are to Durrett's book)

Foundations of Probability:

• Random variables (Sections 1.1, 1.2): probability spaces, σ-algebras, measurability, continuity of probabilities, product spaces, random variables, distribution functions, Lebesgue-Stieltjes measures (without proof), random vectors, generation, a.s.-convergence
• Expected value (Section 1.3): abstract Lebesgue integration (without proofs), inequalities (Jensen, Cauchy-Schwarz, Chebyshev, Markov, Hölder, Minkowski), limit theorems (Fatou's lemma, monotone convergence and dominated convergence theorems), change-of-variables formula,
• Dependence (Section 1.4): independence, pairwise independence, Dynkin's - theorem, convolution of measures, Fubini's theorem, Kolmogorov's extension theorem (without proof)

Classical Theorems:

• Weak laws of large numbers (Sections 1.5, 1.6): the L² -weak law of large numbers, triangular arrays, Borel-Cantelli lemmas, modes of convergence, inequalities (Markov, Chebyshev, Jensen, Hölder), the weak law of large numbers
• Central limit theorems (Sections 2.2, 2.3a, 2.3b, 2.3c, 2.4a, 2.9part ): weak convergence of distributions, the continuous mapping theorem, Helly's selection theorem, tightness, characteristic functions, the inversion theorem, continuity theorem, the central limit theorem, multivariate normal distributions

Discrete-Time Martingale Theory:

• Conditional expectation (Sections 4.1a, 4.1b): Radon-Nikodym theorem (without proof), conditional expectation, filtrations, predictability and adaptivity
• Martingales (Sections 4.2, 4.4, 4.5, 4.6part , 4.7): martingale transforms, the optional sampling the- orem, the upcrossing inequality, Doob's decomposition, Doob's inequality, Lp -convergence, maxi- mum inequalities, L² -theory, uniform integrability, backwards martingales and the strong law of large numbers.

2. Theory of Probability II - M385D

Prerequisites:

• Graduate-level probability (M385C or equivalent).

Literature:

• I. Karatzas and S. Shreve, Brownian motion and stochastic processes, second ed., Springer, 1991 (required)
• D. Revuz and M. Yor, Continuous martingales and stochastic processes, third ed., Springer, 1999 (recommended)

Syllabus:
(Note: all references are to the book of Karatzas and Shreve)

Continuous-Time Martingale Theory:

• General theory of processes (Sections 1.1, 1.2) : Continuous-time processes and filtrations, types of measurability (optional, predictable, progressive), continuous stopping/optional times
• Path regularity of martingales (Section 1.3 A): existence of RCLL modifications, usual conditions for filtrations
• Convergence and optional sampling (Section 1.3 A-C): martingale inequalities, convergence theorems, optional sampling, uniform integrability and martingale with a last element
• Quadratic variation (Section 1.5 or Section IV.1 in Revuz-Yor): quadratic variation for continuous martingales, local martingales and localization, spaces of martingales
• Doob-Meyer decomposition (Section 1.4): no proof

Brownian Motion:

• Definition, construction and basic properties (Sections 2.1, 2.2): construction via Kolomogorov extension theorem, Hölder regularity of paths (Kolmogorov-Centsov), Gaussian processes
• The canonical space (Section 2.4): weak convergence on C[0, infinity), invariance principle, Wiener measure
• Markov and strong Markov property of Brownian motion (Sections 2.5-2.8, selected topics): reflexion principle, density of hitting times, Brownian filtrations, Blumenthal zero-one law

Stochastic Integration:

• Construction of the Stochastic Integral (Sections 3.1, 3.2): stochastic integration with respect to continuous local martingales, quadratic variation and Itô isometry
• Itô formula (Section 3.3): Itô formula, exponential martingales, linear stochastic differential equations

Applications (and extensions) of Itô's formula:

• Paul Léavy's characterization of Brownian motion (Section 3.3 B):
• Changes of measure (Section 3.5): Girsanov theorem, Brownian motion with drift Representations of martingales (Section 3.4): predictable representation property and Kunita-Watanabe decomposition, time-changed Brownian motions (Dambis-Dubins-Schwarz), Knight's theorem on orthogonal martingales
• Local time (Sections 3.6, 3.7): local time for Brownian motion and continuous semimartingales, Tanaka's formula, generalized Itô's formula for convex functions.

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.

The first part of the Prelim examination will deal with Algebraic Topology and the second part will deal with Differential Topology.

 
Algebraic Topology

A brief and fairly accurate description of this syllabus is
"Hatcher chapters 0-2".

1. MANIFOLDS AND CELL COMPLEXES
Identification (quotient) spaces and maps;
Topological n-manifolds including surfaces, Sⁿ, RPⁿ, CPⁿ;
CW decompositions, including these examples.

2. FUNDAMENTAL GROUP AND COVERING SPACES
Fundamental group, examples S¹, Sⁿ, RPⁿ;
Functoriality and homotopy-type invariance;
Retraction and deformation retraction;
Van Kampen's Theorem;
Covering spaces and lifting properties;
Covering transformations;
The covering space versus subgroups of π₁ correspondence;
Regular covers;
Further examples including RPⁿ and lens spaces.
Standard presentation of π₁ of a closed surface.

3. SINGULAR HOMOLOGY
Definitions, functoriality, homotopy-type invariance;
Relative homology, excision, the Eilenberg-Steenrod axioms;
Mayer-Vietoris and examples, including Sⁿ, CPⁿ;
Cellular homology as a consequence of singular homology;
Further examples, including RPⁿ;
 H₁ is the abelianization of π₁;
Local homology and orientations of manifolds, degree of a map between
closed oriented manifolds;
Brouwer fixed point theorem, Jordan separation theorem;
Euler characteristic.
Statement & applications of the Lefschetz fixed point theorem.

PRINCIPAL TEXT
Hatcher, Algebraic Topology (available for free download)

OTHER REFERENCES
Armstrong, Basic Topology, Springer
Greenberg, Lectures on Algebraic Topology
Massey, Algebraic Topology, an Introduction
May, A Concise Course in Algebraic Topology
Munkres, Elements of Algebraic Topology

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.