production

20 credits

Level 5

First Sub Session

This SMSTC course covers the following topics:

Measure and integration: concrete examples (Riemann and Lebesgue);abstract theory - convergence theorems; signed, product and Radon

measures; fractal sets and Hausdorff dimension; L^p spaces,differentiation and Fourier series.

Functional analysis: Banach and Hilbert spaces; Baire category, Open Mapping and Uniform Boundedness Principle; Weak and weak* topologies; Compact operators; Spectral theory and C*-algebras.

http://www.smstc.ac.uk/streams/pure_analysis20 credits

Level 5

First Sub Session

This course is dispensed by the Scottish Mathematical Sciences Training Centre, with different teachers taking turns broadcasting their lectures from and to universities around Scotland.

This course revises and extend standard undergraduate algebra in various directions.

The first part is on group theory, and includes topics on group actions, simple groups, soluble groups, Sylow’s Theorems, and computational aspects.

The second part is on rings and modules, presenting basic definitions and examples (in particular the ring of algebraic integers), chain conditions (Artinianity and Noetherianity), Hilbert’s Basis Theorem and Nullstellensatz.

20 credits

Level 5

First Sub Session

The aim of this course is to broaden students' knowledge of algebraic topology and algebraic geometry. The weekly lectures cover interesting topics quickly, and guide students through their own reading on the subject. Topics covered include point-set topology, homology, the fundamental group and covering spaces, the classification of surfaces, and elementary algebraic geometry. The course is based on a series of SMSTC lectures which are broadcast between several Scottish universities. See smstc.ac.uk for more information.

20 credits

Level 5

First Sub Session

Measure theory provides a systematic framework to the intuitive concepts of the length of a curve, the area of a surface or the volume of a solid body. It is foundational to modern analysis and other branches of mathematics and physics.

20 credits

Level 5

First Sub Session

20 credits

Level 5

First Sub Session

20 credits

Level 5

First Sub Session

This course covers the fundamental mathematical concepts required for the description of dynamical systems, i.e., systems that change in time. It discusses nonlinear systems, for which typically no analytical solutions can be found; these systems are pivotal for the description of natural systems in physics, engineering, biology etc. Emphasis will be on the study of phase spaces.

Next to the theory of relativity and quantum mechanics, chaos and dynamical systems theory is been considered as one of three major advances in the natural sciences. This course offers the mathematics behind this paradigm changing theory.

20 credits

Level 5

Second Sub Session

Measure and integration: concrete examples (Riemann and Lebesgue); abstract theory - onvergence theorems; signed, product and Radon measures; fractal sets and Hausdorff dimension; L^p spaces, differentiation and Fourier series.

Functional analysis: Banach and Hilbert spaces; Baire category, Open Mapping and Uniform Boundedness Principle; Weak and weak* topologies; Compact operators; Spectral theory and C*-algebras.

http://www.smstc.ac.uk/streams/pure_analysis_220 credits

Level 5

Second Sub Session

This course is dispensed by the Scottish Mathematical Sciences Training Centre, with different teachers taking turns broadcasting their lectures from and to universities around Scotland.

It builds on MX5002 Algebra 1 to further extend the algebraic knowledge of students.

The first part continues the study of rings and modules begun in MX5002. Starting with free and projective modules, it culminates with the Artin-Wedderburn Theorem on the structure of modules.

The second part is an introduction to representation theory of groups, starting with finite groups, representations and characters, Maschke’s Theorem and the Orthogonality Relations, and ending with compact groups.

20 credits

Level 5

Second Sub Session

The aim of this course is to broaden students' knowledge of differential geometry. The weekly lectures cover interesting topics quickly, and guide students through their own reading on the subject. Topics covered include manifolds in Euclidean space, curves and surfaces, geodesics and Riemannian curvature, the Gauss-Bonnet theorem, differentiable manifolds, and de Rham cohomology. The course is based on a series of SMSTC lectures which are broadcast between several Scottish universities. See smstc.ac.uk for more information.

20 credits

Level 5

Second Sub Session

20 credits

Level 5

Second Sub Session

20 credits

Level 5

Second Sub Session

20 credits

Level 5

Second Sub Session

20 credits

Level 5

Second Sub Session

20 credits

Level 5

Second Sub Session

One of the aims of the course is to understand the mathematical concept of curvature. We will do this by first studying the geometry of polygonal surfaces, and then by looking at smooth surfaces in Euclidean space.

Polygonal surfaces provide a set of very easy examples with which we can explore the new ideas and quantities. They also allow us to develop the intuition needed in the later part of the course.

60 credits

Level 5

Second Sub Session

Studying mathematical topics, researching literature, solving relevant problems, and reporting clearly and rigorously on findings is the heart and soul of mathematics as an occupation. While these abilities are essential for a mathematician, they are also precious as general skills for a graduate who does not necessarily intends a mathematical career. A specialist topics, compatible with the student’s particular mathematical interests, will be chosen with the assistance of an assigned supervisor. The topic will be studied in depth and a dissertation will be written, which will be required to be mathematically rigorous, clearly presented and of a high standard throughout.

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