# Algebra

Gauss elimination, dimension theory, determinants, characteristic polynomials, the Cayley-Hamilton theorem, invariant subspaces and invariant direct sums, minimal polynomials, diagonalizability, rational canonical form, Jordan canonical form.

Noether isomorphism theorems, symmetric and alternating groups, solvable groups, Jordan-Hoelder theorem, Sylow theorems, modules over a PID, algebraic and transcendental field extensions, cyclotomic and Kummer extensions, classification of finite fields, Galois group of a polynomial, fundamental theorem of Galois theory, solvability by radicals.

## References

The material can be found in any graduate level algebra book, for example in *Algebra* by Hungerford, *Algebra* by Lang, *Advanced Modern Algebra* by Rotman, *Algebra* by Steinberger, or *Modern Algebra* by Van der Waerden. This material in Steinberger is in chapters 2-5, 7, 8, 10 and 11.

# Complex Analysis

Algebraic, geometric and topological properties of complex numbers, analytic functions, power series, conformal mapping, linear transformations, exponential, trigonometric and logarithmic functions, Cauchy's theorem and formula, isolated singularities, maximum modulus theorem, Schwartz's Lemma, general form of Cauchy's theorem, Residue theorem, argument principle, Rouche's theorem, harmonic functions, reflection principle, Taylor and Laurent series, Mittag-Leffler theorem, meromorphic functions, Weierstrass factorization theorem, Gamma function, normal and compact families of analytic functions, Riemann mapping theorem, analytic continuation, monodromy theorem.

## References

The main reference is the book *Complex Analysis*, L.V. Ahlfors, McGraw-Hill, 1966. This book represents the level and scope of coverage of the topics listed above. The main material in Ahlfor's book is Chapters 1 through 6 (up to p. 227) plus section 1 of Chapter 8.

Most of the syllabus is covered by the material in the book *Functions of one complex variable*, J.B. Conway, Springer, 1973, in Chapters 1 through 7 plus the first few sections of Chapters 9 and 10.

# Probability

Random variables; distribution functions; expectations; independence; conditional expectation; zero-one laws; convergence concepts; martingales; sum of independent variables; classical limit theorems, e.g.: central limit theorem, law of large numbers, law of iterated logarithm; characteristic functions.

## References

Chung, *A Course in Probability Theory* (any edition). In terms of 1st edition: Chapters 2, 3, 4, 5, 6, 7, section 1 of Chapter 8, and the sections in Chapter 9 dealing with martingales. Also: Breiman, *Probability*. Chapters 2, 3, 4, 5, 8, 9, and 12.

# Real Analysis

Measurable spaces, measurable and non-measurable sets, Borel sets, outer measure, measurable functions, Monotone Convergence Theorem, Fatou's Lemma, Lebesgue Dominated Convergence Theorem, Lp spaces, convergence in mean: convergence in measure, convergence almost everywhere, almost uniform convergence, Egoroff's Theorem, Vitali Convering Theorem, Radon-Nikodým Theorem, Riesz Representation Theorem, Caratheodory Extension Theorem, product measures, Lebesgue and Lebesgue-Stieltjes measures, the Tonelli and Fubini Theorems, functions of bounded variation and differentiation of such functions, absolute continuity and differentiation of the integral.

## References

The main references are *The Elements of Integration*, R.G. Bartle, John Wiley & Sons 1966, and Chapters 2, 3, 4, 5, 6, 11 and 12 of *Real Analysis*, H.C. Royden, MacMillan, third ed., 1988.

# Topology

Metric spaces, topological spaces, subspaces, products, quotients, continuous functions. Connectedness, path connectedness, local connectedness, compactness, local compactness, Tychonoff Theorem.

Countability axioms, Urysohn's Lemma, Urysohn's metrization theorem, Tietze's extension theorem.

Homotopies, fundamental groups, covering spaces, and applications.

#### References

The standard text used in recent years is Munkres: *Topology*, 2nd edition (Prentice-Hall). The topics are covered in chapters 1-4, 5.1, and chapters 7,9.

# Mathematical Statistics

Probability spaces and axioms, conditional probability and independence, random variables and vectors, expectation, generating functions, convergence, limit theorems, distribution theory (e.g., relationships among binomial, normal, bivariate normal, chi-square, gamma, beta, t, F, etc.), order statistics, sufficient statistics, consistency, expontial families, method of moments, maximum likelihood estimation, minimum variance unbiased estimation, information inequality, confidence intervals, hypothesis testing, Neyman-Pearson lemma, optimality properties of tests, likelihood ratio tests, Cochran's theorem, basic Bayesian methods.

## References

- Bickel, P.J. and Doksum, K.A.:
*Mathematical Statistics*(1977): Chapters 1-6 and Appendix. - Rao, C.R.:
*Linear Statistical Inference and Its Applications*(1973), 2nd ed.: Chapters 2, 3, 5, and sections 6a, 6b, 6e-g, 7a, 7b, 8a. - Dudewicz, E.J. and Mishra, S.N.:
*Modern Mathematical Statistics*(1988), Chapters 2-10.

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