General Background Required

1. Undergraduate linear algebra, as covered in MAT 67.
2. Undergraduate abstract algebra, as covered in MAT 150A, B, C.
3. Graduate algebra, as covered in MAT 250A, B.

The exam is based on the material covered in Serge Lang's textbook "Algebra" (Springer, 3rd Edition, ISBN-10 #038795385X, $63.00 on Amazon Books).

Specific Topics

1. Groups: Basic group theory; Group actions; Abelian groups; Group presentations.
2. Rings: Basic ring theory; Localization; Unique factorization; Polynomial rings.
3. Modules: Basic definitions; Operations; Free and projective modules.
4. Tensor products of modules and algebras.
5. Representation Theory: Matrix algebras; Semisimple rings; Representation theory of finite groups.
6. Field Extensions: Algebraic extensions; Galois Theory; Solvability.

General Background Required

1. Undergraduate real analysis, as covered in MAT 25 and MAT 125A, B.
2. Main concepts of complex analysis, as covered in MAT 185A. Minimum requirement: operations in the complex number system.
3. Main concepts of topology, as covered in MAT 147. Topological spaces, bases, product topology, limit points, continuity, metric spaces, complete, connected, compact spaces.
4. Graduate analysis, as covered in MAT 201A, B, and C.

The exam is based on the material covered in the textbook "Applied Analysis" by Hunter and Nachtergaele, and the textbook "Analysis" Chapters 1, 2, and 4-8 by Lieb and Loss. The Hunter and Nachtergaele text is available for free as a PDF File (Applied Analysis) or for purchase through Amazon Books (World Scientific, ISBN-10 #9810241917, $92.00). The Lieb and Loss text can also be purchased through Amazon Books (AMS, 2nd Edition, ISBN-10 #0821827839, $35.00).

Specific Topics

1. Continuous functions: Convergence of functions; Spaces of continuous functions; Approximation by polynomials; Arzela-Ascoli theorem.
2. Banach Spaces: Bounded linear operators; Different notions of their convergence; Compact operators; Dual spaces; Finite dimensional Banach spaces.
3. Hilbert Spaces: Orthogonality; Orthonormal bases; Parseval's identity.
4. Fourier Series: Convolution; Young's inequality; Fourier series of differentiable functions; Sobolev embedding theorem; Weak Derivatives.
5. Bounded Linear Operators on a Hilbert Space: Orthogonal projections; The dual of a Hilbert space (Riesz representation); The adjoint of an operator; Self-adjoint and unitary operators; Weak convergence.
6. The spectral theory for compact, self-adjoint operators: Spectrum; Compact operators; The spectral theorem; Hilbert-Schmidt operators; Functions of operators.
7. Lp spaces: Inequalities of Jensen, Holder, Minkowski, Young; Completeness; Dual space of Lp.
8. Fourier transform: L1(Rn) Fourier transform and its inversion; Schwartz functions of rapid decay; The Gaussian; Extension of Fourier transform to L2(Rn) and Plancherel's theorem; Distributions; Examples of the use of Fourier transform to Poisson, heat, and wave equations .
9. The Sobolev spaces; Weak derivatives; Negative-order spaces as distributions.

This information is currently being updated. In the mean time you may find Practice Problems: HERE