Postgraduate Mathematical Sciences 2014-2015

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_analysis

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.

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.

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.

Galois theory is based around a simple but ingenious idea: that we can study field extensions by instead studying the structure of certain groups associated to them. This idea can be employed to solve some problems which confounded mathematicians for centuries, including the impossibility of trisecting an angle with ruler and compass alone, and the insolubility of the general quintic equation.

The ability to study and understand a complex mathematical text or topic independently is an essential skill to a mathematics graduate. This course serves as a preparation.  The topic or topics studied in this course are chosen by the student in coordination with an assigned supervisor. The student will read assigned text, and study the subject of their choice in depth, while reporting to the supervisor weekly on progress. At the same time the student will have an opportunity to clarify points of difficulty with the supervisor.

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.

This SMSTC course covers the following topics:

 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_2

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.

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.

A knot is a closed curve in three dimensions.  How can we tell if two knots are the same?  How can we tell if they are different?  This course answers these questions by developing many different "invariants" of knots.  It is a pure mathematics course, drawing on simple techniques from a variety of places, but with an emphasis on examples, computations and visual reasoning.

Algebraic topology is a tool for solving topological or geometric problems with the use of algebra. Typically, a difficult geometric or topological problem is translated into a problem in commutative algebra or group theory. Solutions to the algebraic problem then provide us with a partial solution to the original topological one.

This course concerns the integers, and more generally the ring of algebraic integers in an algebraic number field.  The course begins with statements concerning the rational integers, for example we discuss the Legendre symbol and quadratic reciprocity. We also prove a result concerning the distribution of prime numbers. In the latter part of the course we study the ring of algebraic integers in an algebraic number field. One crucial result is the unique factorisation of a nonzero ideal as a product of primes, generalising classical prime factorisation in the integers.

The ability to study and understand a complex mathematical text or topic independently is an essential skill to a mathematics graduate. This course serves as a preparation.  The topic or topics studied in this course are chosen by the student in coordination with an assigned supervisor. The student will read assigned text, and study the subject of their choice in depth, while reporting to the supervisor weekly on progress. At the same time the student will have an opportunity to clarify points of difficulty with the supervisor.

This course was designed to show you what you can do with everything you learnt in your degree. We will use mathematical techniques to describe a fast variety of “real-world” systems: spreading of infectious diseases, onset of war, opinion formation, social systems, reliability of a space craft, patterns on the fur of animals (morphogenesis), formation of galaxies, traffic jams and others. This course will boost your employability and it will be exciting to see how everything you learnt comes together.

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.

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.