What is Taught Course Centre

Starting from this year, @SwanseaMaths participates in the collaboration between the Mathematics Departments at the Universities of BathBristolImperialOxford and Warwick. The centre is supported by a grant from the EPSRC. The Centre offers graduate-level courses over the academic year. The web-page of the centre is https://www.maths.ox.ac.uk/groups/tcc 

Which Courses are run

The courses for this term are available by the link:
https://www.maths.ox.ac.uk/groups/tcc/tcc-current-courses

Swansea will run the following courses:

  • Criticality theory for Schrödinger operators by Prof Vitaly Moroz.

    Abstract: This course is an introduction to Agmon's Criticality Theory of Schrödinger operators. It will focus on two core but not widely known ideas, namely Allegretto-Piepenbrink positivity principle and Phragmen-Lindelöf comparison principle. We will see how these fundamental principles enable to prove a range of Hardy type inequalities, and at the same time provide a powerful tool in the analysis of the structure of positive solutions for large classes of nonlinear elliptic equations. 

    For further details, see the course web-page.

  • Distribution-dependent stochastic differential equations by Prof Feng-Yu Wang.

    Abstract: In 1966 H.P. McKean proposed a distribution-dependent stochastic differential equation (DDSDE for short) to characterize the nonlinear partial differential equation for Kac's caricature of a Maxwellian gas. Since that time, this type of SDEs has attracted increasing attention. In this course we will discuss some important results of the theory of DDSDEs. 

    For further details, see the course web-page.

  • Algebraic Structures by Prof Tomasz Brzeziński
    Abstract: An algebraic structure is a collection of sets with operations. Typical and most widespread across mathematics are systems such as a semigroup, monoid, group, ring, field, associative algebra, vector space or module. In this lecture course we will revisit these well-known systems and look at them from a universal algebra perspective, as well as study some simple algebraic systems which have recently gained prominent position in algebra and topology such as braces, racks or quandles (sets with two operations interacting with each other in prescribed ways). In particular we will explore a little known fact (first described nearly 100 years ago by Pruefer and Baer) that one can give a definition of a group without requesting existence of the neutral element and inverses by using a ternary rather than a binary operation (i.e. an operation with three rather than the usual two inputs). A set with such a suitable ternary operation is known as a heap. By picking an element in a heap, the ternary operation is reduced to the binary group operation, for which the chosen element is the neutral element (the resulting group is known as a retract). We will study properties and examples of heaps and relate them to the properties of corresponding groups (retracts). Next we will look at heaps with an additional binary operation that distributes over the ternary heap operation, known as trusses, relate them to both rings and braces, and study their properties.

    For further details, see the course web-page.