University at Albany
 

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

  • Algebra, by Hungerford.
  • Algebra, by Lang.
  • Advanced Modern Algebra, by Rotman.
  • Algebra, by Steinberger.
  • Modern Algebra, by van der Waerden.
This material in Steinberger is in chapters 2-5, 7, 8, 10 and 11.

 

Complex Analysis

Complex numbers, power series, analytic functions, Cauchy-Riemann equations, Poisson kernel, harmonic functions, Cauchy's theorem, Liouville's theorem, zeros of analytic functions, Laurent series and isolated singularities, the residue theorem, the argument principle, Rouche's theorem, Schwarz lemma, Montel's theorem

References

  • Complex Analysis, L.V. Ahlfors, McGraw-Hill, 1966.
  • Functions of one complex variable, J.B. Conway, Springer, 1973.

 

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

Axioms on real numbers, topology of real numbers, \sigma- algebra and Lesbesgue measure, Lesbegue measurable functions, Egoroff's theorem, Lusin's theorem, Lebesgue integration, functions of bounded variation, L^p space, Riesz representation theorem, basics of Hilbert space and Banach space (inner product, norm, linear functionals, dual space).

References

  • The Elements of Integration, R.G. Bartle, John Wiley & Sons 1966.
  • 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.