Short Courses in Mathematical Physics

Tanja Hinderer: Analytical methods in general relativity

24th January, 2024

A linear code is a vector subspace of 𝔽qn, where 𝔽q is a finite field with q elements. The family of linear error-correcting codes are specially important when one is attempting to transmit messages across a noisy communication channel. Data can be corrupted in transmission or storage by a variety of undesirable phenomenon, such as radio interference, electrical noise, scratch, etc.. It is useful to have a way to detect and correct such data corruption. An error-correcting code can correct more errors larger is its minimum distance. This course aims to introduce a family of error-correcting codes, the Algebraic Geometry Codes, and show how to use the theory of semigroups to improve the minimum distance of the code. This construction of codes make use of a function field in one variable over a finite field. We will show how the local information in one or two rational places, the knowledge of the semigroup in these places, can be used to improve the minimum distance of the code.

Anton Ilderton: Strong-field and non-perturbative amplitudes

22nd January, 2024

A linear code is a vector subspace of 𝔽qn, where 𝔽q is a finite field with q elements. The family of linear error-correcting codes are specially important when one is attempting to transmit messages across a noisy communication channel. Data can be corrupted in transmission or storage by a variety of undesirable phenomenon, such as radio interference, electrical noise, scratch, etc.. It is useful to have a way to detect and correct such data corruption. An error-correcting code can correct more errors larger is its minimum distance. This course aims to introduce a family of error-correcting codes, the Algebraic Geometry Codes, and show how to use the theory of semigroups to improve the minimum distance of the code. This construction of codes make use of a function field in one variable over a finite field. We will show how the local information in one or two rational places, the knowledge of the semigroup in these places, can be used to improve the minimum distance of the code.

Pierre Raphaël: Non-linear waves

1st October, 2021

A series of talks about non-linear waves, energy supercritical problems, solitons, defocusing Non-Linear Schrödinger equation and shock waves.

Fiona Torzewska: Mapping class groupoids and motion groupoids

4th November, 2020

An LMS online course in mapping class groupoids.

A topological phase of matter is a physical system whose behaviour may be effectively described via a topological quantum field theory i.e. functor from cob to vect. The study of topological quantum field theories has applications in quantum computing but also involves a lot of beautiful mathematics which is interesting in its own right. A central role in the description of topological phases of matter in 2 (spatial) dimensions is played by the representations of braid groups. A natural generalisation to study the statistics of higher (spatial) dimensional phases of matter is then to look for generalisations of the braid group. Braid groups can be equivalently defined as the mapping class groups or as the motions groups of points in a disk, as well as in several other equivalent ways. In these lectures we will introduce generalisations of these two definitions. In each case we will show first that these give us groupoids and then that we can get back to the classical definitions by considering the endomorphisms of a single object. The mapping class groupoid is a simpler construction but is not in general the right notion to take when considering particles moving through space. We will construct a functor from the motion groupoid to the mapping class groupoid and hence see which cases we can study only the mapping class groupoid. We will use lots of examples to aid intuition and intend this talk to be accessible to those with minimal knowledge of topology.

Various Speakers: An Introduction to Quantum Field Theory in Curved Spacetime

12th October, 2020

An LMS online lecture course in quantum field theory.

In this mini-course we will give an introduction to quantum field theory in curved spacetime, one of the dominant research areas in modern algebraic quantum field theory. It is aimed at postgraduate students unfamiliar with the topic, although some preliminary knowledge of elementary differential geometry (notions of manifold, (co)tangent bundle, covariant derivative), as well as some familiarity with quantum mechanics and standard quantum field theory would be helpful.

The course will cover some of the underlying concepts of the algebraic approach to quantum field theory in curved spacetime and discuss several physically relevant examples. The focus will be on the linear (i.e., non-interacting) scalar field, favouring depth over breadth.

Topics for the lectures are:

   •  Lorentzian geometry, causality and the Klein–Gordon equation;
   •  Algebraic quantum field theory and the quantised scalar field;
   •  Aspects of QFT in curved spacetimes;
   •  Introduction to spacetimes;
   •  An introduction to Hadamard states.