Syllabus

MA390 Science (honours)

 MA341 Metric Spaces

• Metric spaces: examples of metric spaces; convergence in metric spaces; pointwise, uniform and mean convergence; continuity; open and closed sets; closure, interior and boundary; compactness in metric spaces; the Bolzano-Weierstrass theorem
• Completeness: Rⁿ and C[a,b]; contractions; the fixed point theorem; applications to differential equations etc
• Fractal geometry: the space of fractals; the Hausdorff metric; iterated function systems; algorithms for generating fractals; fractal dimension

Texts

• A.N. Kolmogorov & S.V. Fomin, "Real Analysis" (Dover)

MA342 Topology

• Topological spaces: examples; continuity and convergence; subspaces, quotients and product spaces
• Connectedness and path connectedness: components; totally disconnected spaces
• Compactedness and its applications: the Heine-Borel theorem; compactness of subspaces and product spaces; compactness and sequential compactness
• Convergence: the Hausdorff and other separation properties; inadequacy of sequences; nets; filters and ultrafilters

Texts

• S. Lipschutz, "General Topology" (Schaum Outline)

References

• A. Armstrong, "Basic Topology" (Springer)
• S. Barr, "Experiments in Topology" (Murray)
• M. Barnsley, "Fractals Everywhere" (Academic Press)
• N. Bourbaki, "General Topology" (Hermann)
• V. Bryant, "Metric Spaces, Iteration and Applications" (Cambridge)
• E. Copson, "Metric Spaces" (Cambridge)
• K. Falconer, "Fractal Geometry" (Wiley)
• G. Francis, "A Topological Picturebook" (Springer)
• V. Mendelson, "Introduction to Topology" (Alyn & Bacon)
• J. Munkres, "Topology, A First Course" (Prentice Hall)
• I.M. Singer & J.A. Thorpe, "Lecture Notes on Elementary Topology and Geometry" (Springer)
• W.A. Sutherland, "Introduction to Metric and Topological Spaces" (Oxford)
• S. Willard, "General Topology" (Addison Wesley)

MA343 Group Theory (part one)

• Group axioms, cyclic groups, permutation groups.
• Normal subghroups, homomorphisms, isomorphism theorems.
• Permutation groups, isomorphism theorems, linear groups
• Direct and semidirect products
• Finitely generated Abelian groups
• Automorphisms, groups of automorphisms

MA344 Group Theory (part two)

• Group actions, automorphism groups of graphs, application to enumeration.
• Sylow's Theorem, groups of small order
• Simple groups, the Jordan-Hölder Theorem
• Soluble and nilpotent groups
• Semigroups, machines, flip-flop and simple memory machines, flip-flop with reset

References

• J.B. Fraleigh, "A First Course in Abstract Algebra" (Addison Wesley)
• W. Ledermann, "Introduction to the Theory of Finite Groups" (Oliver & Boyd)
• I.D. Macdonald, "The Theory of Groups" (Oxford)
• J.S. Rose, "A Course on Group Theory" (Cambridge)

MA387 Statistics

• Introduction to probability theory: probability spaces, properties of probabilities, conditional probability, independence
• Combinatorial analysis and counting
• Random variables: discrete and continuous variables; expectation, variance, covariance, moments
• The Markov, Cauchy-Schwartz-Bunjakowski, and Chebysev's Inequality; correletion coefficients
• Jointly distributed random variables, marginal and conditional densities, iterated expectation
• Distributions of sums, products, and quotients of random variables
• Order statistics, sampling statistics
• Moment generating functions, and characteristic functions, the Uniqueness and Continuity Theorems
• The Weak Law of Large Numbers, the Central Limit Theorem, the normal and Poisson approximations to the binomial density
• Relations between the normal, Chi Squared, Tau, and F densities, and applications to sampling

Texts

• H.J. Larson, "Introduction to Probability" (Addison Wesley)
• Hogg & Craig . "Introduction to Mathematical Statistics" (Pentice Hall)

References

• P. Brémaud, "An Introduction to Probabilistic Modelling" (Springer)
• K.L. Chung, "Elementary Probability Theory with Stochastic Processes" (Springer)
• W. Feller, "An Introduction to Probability Theory and its Applications, Vol I" (Wiley)
• P.G. Hoel, S.C. Port & C.J. Stone, "Introduction to Probability Theory" (Houghton Mifflin)
• R.V. Hogg & A.T. Craig "Introduction to Mathematical Statistics" (Collier Macmillan)

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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)

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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

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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)

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