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Advice courses mathematics

Which courses to choose?

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Dear all,

I am currently taking some small step in my career, having studied financial analytics I decided to move forward in area of mathematics doing at first Graduate diploma in Mathematics with an expectation to move to further studies.

I am particularly interested in quant trading space; alpha modelling, portf construction. I would like to ask for bit of advice in the mater of choosing courses. There is several courses one may choose all in slightly different direction. Having some knowledge in pricing and financial modelling theory I would like to focus on mathematical concepts widely used in quantitative trading.
Here is the courses I am considering:

Advanced mathematical analysis (half course)
This is a course in real analysis, designed for those who already know some real analysis (such as that encountered in course 116 Abstract Mathematics). The emphasis is on functions, sequences and series in ndimensional real space. The general concept of a metric space will also be studied. After studying this course, students should be equipped with a knowledge of concepts (such as continuity and compactness) which are central not only to further mathematical courses, but to applications of mathematics in theoretical economics and other areas. More generally, a course of this nature, with the emphasis on abstract reasoning and proof, will help students to think in an analytical way, and be able to formulate mathematical arguments in a precise, logical manner. Specific topics covered are: series of real numbers; series and sequences in n-dimensional real space Rn ; limits and continuity of functions mapping between Rn and Rm ; differentiation the topology of Rn ; metric spaces uniform convergence of sequences of functions.

Optimisation theory (half course)

This course aims to bring together several parts of the wide area of mathematical optimisation, as encountered in many applied fields. The course concentrates on continuous optimisation, and in this sense extends the theory studied in standard calculus courses. In contrast to the Mathematics 1 and Mathematics 2 half courses, the emphasis in this Optimisation Theory course will be on the mathematical ideas and theory used in continuous optimisation. This course covers the following topics: Introduction and review of relevant parts from real analysis, with emphasis on higher dimensions. Weierstrass’ Theorem on continuous functions on compact set. Review with added rigour of unconstrained optimisation of differentiable functions. Lagrange’s Theorem on equality constrained optimisation. The Kuhn-Tucker Theorem on inequality constrained optimisation. Finite and infinite horizon dynamic programming.

Advanced statistics: distribution theory (half course)

Probability: Probability measure. Conditional probability. Bayes’ theorem. Distribution Theory: Distribution function. Mass and density. Expectation operator. Moments, moment generating functions, cumulant generating functions. Convergence concepts. Multivariate Distributions: Joint distributions. Conditional distributions, conditional moments. Functions of random variables.

Advanced statistics: statistical inference (half course)
Data reduction; Sufficiency, minimal sufficiency. Likelihood. Point estimation; Bias, consistency, mean square error. Central limit theorem. RaoBlackwell theorem. Minimum variance unbiased estimates, Cramer-Rao bound. Properties of maximum likelihood estimates. Interval estimation; Pivotal quantities. Size and coverage probability. Hypothesis testing; Likelihood ratio test. Most powerful tests. Neyman-Pearson lemma.

Discrete mathematics and algebra
This full course develops the mathematical methods of discrete mathematics and algebra and will emphasis their applications. Counting: selections, inclusion-exclusion, partitions and permutations, Stirling numbers, generating functions, recurrence relations. Graph Theory: basic concepts (graph, adjacency matrix, etc.), walks and cycles, trees and forests, colourings. Set Systems: matching, finite geometries, block designs. Abstract groups: revision of key concepts such as cyclic groups, subgroups, homomorphisms and Lagrange’s theorem. Conjugation and normal subgroups. Group actions. Applications of algebra to discrete mathematics I: permutations, orbits and stabilisers, the orbit-stabiliser theorem; applications to counting problems. Rings and polynomials: the Euclidean algorithm for polynomials, integral domains, ideals, factor rings, fields, field extensions. Finite fields: construction, the primitive element theorem, and finite linear algebra. Applications of algebra to discrete mathematics II: finite Geometry: designs, affine and projective planes. Error-correcting codes: linear codes, cyclic codes, perfect codes.

Please can someone working in the field give me advice that would provide me a bit of insight to what is well of use in business. One should choose 4 half or 1 full 2 half courses.

I thank you in advance.

Jakub M.H.