# Guidelines on Preparing for the Mathematics Preliminary Exams

## Exam Workshops

The Preliminary Exams for the upcoming Fall Quarter are scheduled to take place on ** Wednesday, Sept 12, 2018, and Thursday, September 13, 2018**.

The exam workshops were initiated under the VIGRE program in 2002 and they continue today. We conduct review and problem-solving workshops aimed at helping students get ready for the graduate preliminary 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 exams.

**The following workshops are held in room 2112 of the Mathematical Sciences Building. Please note that days and times are subject to change.**

Workshop Topic | Instructor | Dates | Time |
---|---|---|---|

Workshop in Analysis | Nachtergaele | Aug 27-Aug 31 | 10am-12pm & 2pm-4pm |

Workshop in Algebra | Kuperberg | Sept 4-Sept 8 & Sept 10-11 | 10am-11:30am & 1pm-2:30pm |

There are separate guides, one for preliminary exams for analysis, and another for preliminary exams for algebra. These guides have been approved by the Graduate Program Committee as of Spring 2010.

- Guidelines for Math Prelim Exams in Analysis
- Guidelines for Math Prelim Exams in Algebra
- Sample Math Prelim Exams for M.A. and Ph.D.

### Guidelines on Preparing for the Mathematics Preliminary Exams in Analysis

General Background Required:

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

- Continuous functions: Convergence of functions; Spaces of continuous functions; Approximation by polynomials; Arzela-Ascoli theorem.
- Banach Spaces: Bounded linear operators; Different notions of their convergence; Compact operators; Dual spaces; Finite dimensional Banach spaces.
- Hilbert Spaces: Orthogonality; Orthonormal bases; Parseval's identity.
- Fourier Series: Convolution; Young's inequality; Fourier series of differentiable functions; Sobolev embedding theorem; Weak Derivatives.
- 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.
- The spectral theory for compact, self-adjoint operators: Spectrum; Compact operators; The spectral theorem; Hilbert-Schmidt operators; Functions of operators.
- Lp spaces: Inequalities of Jensen, Holder, Minkowski, Young; Completeness; Dual space of Lp.
- 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 .
- The Sobolev spaces; Weak derivatives; Negative-order spaces as distributions.

### Guidelines on Preparing for the Mathematics Preliminary Exams in Algebra

General Background Required:

- Undergraduate linear algebra, as covered in MAT 67.
- Undergraduate abstract algebra, as covered in MAT 150A, B, C.
- 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:

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

### Master's Sample Exams

- MA-Fall-2012-analysis
- MA-Fall-2012-algebra
- MA-Spring-2012-analysis
- MA-Spring-2012-algebra
- MA-Fall-2011-analysis
- MA-Fall-2011-algebra
- MA-Spring-2011-analysis
- MA-Spring-2011-algebra
- MA-Spring-2010-algebra
- MA-Fall-2008-analysis
- MA-Winter-2008
- MA-Fall-2007-analysis
- MA-Fall-2007-algebra
- MA-Winter-2007
- MA-Fall-2006
- MA-Winter-2006
- MA-Fall-2003
- MA-Fall-2001

### Ph.D. Sample Exams

- Prelim-Fall-2018-algebra
- Prelim-Fall-2018-analysis
- Prelim-Spring-2018-analysis
- Prelim-Spring-2018-algebra
- Prelim-Fall-2017-analysis
- Prelim-Fall-2017-algebra
- Prelim-Fall-2016-algebra
- Prelim-Fall-2016-analysis
- Prelim-Spring-2017-Analysis
- Prelim- Spring- 2017 Algebra
- Prelim-Spring-2016-algebra
- Prelim-Spring-2016-analysis
- Prelim-Fall-2015-algebra
- Prelim-Fall-2015-analysis
- Prelim-Spring-2015-algebra
- Prelim-Spring-2015-analysis
- Prelim-Fall-2014-analysis
- Prelim-Fall-2014-algebra
- Prelim-Spring-2014-algebra
- Prelim-Spring-2014-analysis
- Prelim-Fall-2013-algebra
- Prelim-Fall-2013-analysis
- Prelim-Spring-2013-analysis
- Prelim-Spring-2013-algebra
- Prelim-Fall-2012-analysis
- Prelim-Fall-2012-algebra
- Prelim-Spring-2012-analysis
- Prelim-Spring-2012-algebra
- Prelim-Fall-2011-analysis
- Prelim-Fall-2011-algebra
- Prelim-Spring-2011-analysis
- Prelim-Spring-2011-algebra
- Prelim-Fall-2010-analysis
- Prelim-Fall-2010-algebra
- Prelim-Spring-2010-analysis
- Prelim-Spring-2010-algebra
- Prelim-Fall-2009-analysis
- Prelim-Fall-2009-algebra
- Prelim-Winter-2009-analysis
- Prelim-Winter-2009-algebra
- Prelim-Fall-2008
- Prelim-Winter-2008
- Prelim-Fall-2007-analysis
- Prelim-Fall-2007-algebra
- Prelim-Winter-2007
- Prelim-Fall-2006
- Prelim-Winter-2006
- Prelim-Winter-2005
- Prelim-Fall-2004
- Prelim-Winter-2004
- Prelim-Fall-2003
- Prelim-Winter-2003
- Prelim-Fall-2002
- Prelim-Winter-2002
- SamplePrelim-11-01