As part of the VIGRE program at UC Davis, we are running some review and problem solving workshops aimed at helping students get ready for the graduate exams.

All students are welcome to participate, though it will be particularly useful for entering students in both the Mathematics and GGAM programs, and for first year students who have not yet passed their written graduate exams.

NOTE: The first workshop will be on Tuesday, September 2, 2003

Each week will feature two hours of review and sample problems in the mornings, as scheduled below. In the afternoons (2-4 PM) will have groups of postdocs and advanced graduate students available to help with problem solving in small groups.

While each week will have a theme for the morning meetings, the afternoon problem solving sessions will be more general. So all students should have appropriate problems and challenges on any given day.

More details and materials will appear later on this page.

September, 2003:
September 2-5	Sessions I, Prof. Thomas Strohmer (with Jared Tanner), Focus on Analysis, 10 AM-12 PM, 693 Kerr Hall.
September 8-12	Session II, Porf. Soshnikov, Focus on Analysis, 10 AM-12 PM, 693 Kerr Hall.
September 15-18	Session III, Focus on Linear Algebra and Algebra, Prof. Waldron, 10 AM-12 PM, 693 Kerr Hall.
September 18-19, 22-23	Session IV, Focus on Algebra, Prof. Anne Schilling (with Monica Vazirani), 10 AM-12 PM, 693 Kerr Hall.

Some sample problems in algebra (from Monica Vazirani) (pdf format)

Material covered

The exams also cover the prerequisites for these courses, namely standard undergraduate material on analysis and algebra, as reflected in the syllabus of 127ABC and 150ABC, as well as standard calculus and linear algebra material from lower division mathematics.

In the GGAM, the Ph.D. preliminary exam is a written examination covering MAT 119A and MAT 203ABC.

Following are the syllabi of these courses, which gives a view of the topics covered.

201AB and 203AB (Note: 201AB and 203AB are due to be merged into a single 2 quarter sequence in 2003.)

201ABC/203ABC Course description:
Topological, metric, normed spaces. Stone-Weierstrass theorem. Metric spaces. Contraction mapping theorem. Banach spaces. Bounded linear maps. Lebesgue measure. Fubini and Radon-Nikodym theorems. L^p and Sobolev spaces. Distributions, Fourier transform. Linear operators on Hilbert spaces. Spectral theorem. Variational methods. Applications.

Readings:
Main: "Applied Analysis" by Hunter and Nachtergaele.
Supplementary: "Real Analysis" by G.Folland, "Elements of the Theory of Functions and Functional Analysis" by A.N. Kolmogorov and S.V. Fomin

201A/203A Topical Outline:
Topological Spaces. Metric and normed spaces. Continuous maps. Compact Spaces. The Stone-Weierstrass theorem. Metric spaces. Contraction mapping theorem. Banach and Hilbert spaces. Bounded linear maps.

201B/203B Topical Outline:
Hilbert and L^p spaces. Fourier series. Distributions and Fourier transform. Linear operators on a Hilbert space. The spectral theorem for compact self-adjoint operators.

201C Topical Outline:
Abstract measure space. Lebesgue measure and integrals. Product measures and Fubini's theorem. The Radon-Nikodym theorem.

203C Topical Outline:
Linear differential operators and Green's function. Distributions and Fourier transform. Measure theory. L^p spaces. Sobolev spaces. Differential calculus and variational methods. Calculus of Variations.

250AB Topical outline:
Text: Algebra, by Dummit and Foote.
1. Review of chapter 1, chapter 2, and sections 3.1 and 3.2 First examples of groups, group actions, subgroups, and cosets
2. Sections 5.1 and 5.2 Direct products, classification of finite abelian groups
3. Chapter 4, sections 1-4 Group actions, permutation representations, class equation
4. Sections 4.4, 4.5, and 5.5 Sylow's theorems and semidirect products
5. Sections 3.3, 3.4, and 6.1 Jordan-Holder theorem, subgroup series
6. Section 6.3 Free groups and group presentations
7. Review of Chapter 7, sections 1-4 Ring structure
8. Chapter 8 Unique factorization
9. Chapter 9, sections 1-3 Polynomial rings.

250B Topical Outline
1. Sections 10.1, 11.1, 10.2, 10.3, and 11.2 Modules and vector spaces and maps between them
2. Sections 11.3 and 11.4 Dual vectors, multilinear functions, and determinants
3. Supplementary topic: bilinear forms
4. Sections 10.4 and 11.5 Tensor products and tensor algebras.
5. Chapter 12 Modules over PIDs: .
6. Chapter 13, sections 1-4 Field theory.
7. Chapter 14, sections 1-3 Galois theory and finite fields