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.

 The Applied Math Prelim divides into these 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 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.

 

  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.