MA480 Core Subjects

MA481 Measure Theory and Functional Analysis

• The Lebesgue integral: the deficiencies of the Riemann integral, Lebesgue measure, measurable functions, the Lebesgue integral
• Convergence theorems, functions of bounded variation and absolutely continuous functions, Vitali's Covering Theorem, integration and differentiation
• General measure and integration theory: outer measures, measures, measurable functions, modes of convergence
• Functional analysis: normed vector spaces and inner product spaces, bounded linear mappings, linear functionals and the dual space, the classical Banach spaces and their duals, Hilbert spaces, orthogonal decomposition, orthonormal bases, Fourier series

Texts

• H.L. Royden, "Real Analysis" (Collier Macmillan)
• E. Kreyszig, "Introductory Functional Analysis with Applications" (Wiley)

References

• R.G.Bartle, "The Elements of Integration" (Wiley)
• A. Brown & C. Pearcey, Introduction to Operator Theory, Vol I (Springer)
• E. Hewitt & K. Stromberg, "Real and Abstract Analysis" (Springer)
• M. Spiegel, "Real Variables" (Schaum Outline)
• W. Rudin, "Functional Analysis" (McGraw Hill)
• W. Rudin, "Real and Complex Analysis" (McGraw Hill)
• M. Schechter, "Principles of Functional Analysis" (Academic Press)
• N. Young, "An Introduction to Hilbert Spaces" (Cambridge)

MA416 Ring Theory

• Introductory examples of rings and fields
• Axioms. Subrings. Integral domains; theorems of Fermat and Euler
• Field of quotients of an integral domain
• Division rings. Quaternions.
• Rings of polynomials. Factorisation. Gauss's Lemma. Eisenstein's criterion
• Ideals, factor rings, ring homomorphisms. Homomorphism theorems
• Prime ideals, maximal ideals. Principal ideal rings
• Unique factorsation domains, Euclidean domains.
• Gaussian integers

MA491 Field Theory

• Field extensions - simple, algebraic, transcendental
• The degree of an extension
• Ruler and compass constructions
• Algebraically closed fields, splitting fields and finite fields
• Galois groups and the Galois correspondence
• Introduction to codes (ISBN, linear, cyclic)

Texts

• J.B. Fraleigh, "A First Course in Abstract Algebra" (Addison Wesley)

References

• I.T. Adamson, "Introduction to Field Theory" (Oliver & Boyd)
• I.N. Herstein, "Topics in Algebra" (Wiley)
• I.N. Stewart, "Galois Theory" (Chapman & Hall)

MA487 Statistics

• Introduction to statistical inference: methods of estimation, least squares and maximum likelihood, Bayes methods, properties of estimators
• Approaches to statistics, including data analytic, frequentist, Bayesian, robust, non-parametric, structural and fiducial. Ther role of probability in the frequentist approach to statistical inference.
• Brief review of random variables and their distributions. Some methods of point estimation needed in hypothesis tests, including maximum likelihood and method of moments.
• Hypothesis testing: likelihood ratio tests, Neymann Pearson theory. Exponential families of distributions. Derivation of uniformly most powerful tests when they exist.

Discussion of uniformly most powerful unbiased tests. Discussion of uniformly most accurate, and uniformly most accurate unbiased confidence sets. Applications.
• Principles of data reduction, with particular emphasis on the concept of sufficiency. The concept of completeness.
• Point estimation: criteria and derivations. Methods of estimation, especially minimum variance unbiased estimators. Basu's Theorem. Applications.

Texts

• R. Hogg & A. Craig, "Introduction to Mathematical Statistics" (Macmillan)

References

• L.J. Bain & M. Engelhardt, "Introduction to Probability and Mathematical Statistics" (van Nostrand Reinhold)
• E.J. Dudewicz & S.N. Mishra, "Modern Mathematical Statistics" (Wiley)
• P.G. Hoel, S.C. Port & C.J. Stone, "Introduction to Statistical Theory" (Houghton Mifflin)
• H.J. Larson, "Introduction to the Theory of Statistics" (Wiley)
• A.M. Mood, F.A. Graybill & D.C. Boes, "Introduction to the Theory of Statistics" (McGraw Hill)