26/27 Lecturer: Giulia Gugiatti (Edinburgh, Organiser) and Ivan Cheltsov (Edinburgh)

26/27 Lecturer: Matt Walters (Edinburgh)

Overview: Black hole geometries are curious solutions of the general theory of relativity. In this course we shall study such black hole geometries from both a classical and a quantum perspective. The course will effectively have two parts. In the first half we shall discuss different exact but `time-independent’ black hole solutions, their causal structures, and symmetries, finally leading to the thermodynamic nature of these black holes.

The second half of the course will be more fluid in nature and its structure will depend on how the first part goes. Here, we attempt to construct dynamical but approximate black hole solutions, we might study black holes in semi-classical gravity, leading to the topic of Hawking radiation, and we might contemplate the information paradox.

26/27 Lecturer:  Jim Belk (Glasgow)

Overview: The course will start with a refresher on smooth manifolds and their tangent bundles, before introducing more general vector bundles and fibre bundles. It will then discuss in detail various aspects of calculus on manifolds, introducing the Lie derivative, the exterior derivative, connections, holonomy, curvature, and Stokes’ Theorem. This will be followed by an exposition of de Rham cohomology and its relation to singular and Cech cohomology. We hope to end the course with an introduction to Chern-Weil theory and characteristic classes of vector bundles on manifolds.

Prerequisites: Multivariable calculus and point-set topology are essential. A first course in differential geometry, for example on curves and surfaces, and some familiarity with algebraic topology, Lie groups and Lie algebras is useful, but not essential.

26/27 Lecturer: Anatoly Konechny (Heriot-Watt) and Ioana Coman-Lohi (University of Edinburgh) 

Overview: This course is intended to provide both the introduction to two-dimensional conformal field theory (2D CFT) and to give an opportunity to learn more advanced topics linked to current research. 2D CFT is a well-developed branch of mathematical physics with many connections to other  topics in mathematics such as infinite-dimensional Lie algebras, vertex operator algebras, modular tensor categories and theory of modular forms. 

The course starts with lectures covering the basics  such as  conformal geometry on the (extended) complex plain, correlation functions, Ward identities, radial quantisation, operator product expansion, Virasoro algebra and conformal families. This part is planned to be covered in 5-6 lecturers to be followed by 2 lectures focussed on Vertex Operator Algebras. The students are then offered a number of guided reading projects on more advanced topics such as conformal blocks, WZW theories, AGT conjecture and more. Brief informal presentations on their reading topics are expected to be given by the students in the last 2 lectures.

The course does not assume any prior knowledge of quantum field theory and may serve as an introduction into this topic for mathematicians. The main prerequisite for the course is basic knowledge of quantum mechanics although the essential concepts will be reminded along the way. Some basic knowledge of groups, differential geometry, functional and complex analysis is also assumed. A detailed set of notes covering the introductory part along with 9 fully worked tutorial sheets for self practice will be provided.

26/27 Lecturer:  Livio Ferretti (Glasgow)

Overview: Algebraic topology aims to associate algebraic invariants (groups, vector spaces, algebras…) to topological spaces in order to be able to distinguish spaces and better understand their geometry. This course is intended to give an overview of basic concepts, focusing on the main results and examples. The first half of the course will cover homotopy theory, homotopy groups and fibre bundles. In the second half, we will study homology and cohomology theory. The topics of the course are covered, in much greater detail, in Hatcher’s excellent book Algebraic Topology, which is available freely online.

Prerequisites: Working knowledge of metric and topological spaces, linear algebra and basic group theory (groups and group actions).

26/27 Lecturer:  Christian Korff (Glasgow)

Overview: Broadly speaking, representation theory is the mathematical study of symmetries. The most successful physical theories which underpin our understanding of nature and new technologies rely on the algebraic description of symmetries. For example, the quantum fields in the standard model are governed by a representation of what physicists call a “gauge group”, an algebraic structure which allows one to predict and find quantum particles. The mathematical study of representation theory can be seen as the attempt of classifying all possible manifestations of a group or symmetry and making them concrete in terms of matrices. While as a mathematical subject representation theory sits within the larger topic of algebra, it heavily draws on and influences other areas of mathematics such as geometry, combinatorics and number theory. Its application in physical theories has led to spectacular successes in making predictions and developing technologies. As such it is a core subject within the CDT training.

26/27 Lecturer: Lotte Hollands (HWU) & Tudor Dimofte (Edinburgh)

Overview: Black hole geometries are curious solutions of the general theory of relativity. In this course we shall study such black hole geometries from both a classical and a quantum perspective. The course will effectively have two parts. In the first half we shall discuss different exact but `time-independent’ black hole solutions, their causal structures, and symmetries, finally leading to the thermodynamic nature of these black holes.

The second half of the course will be more fluid in nature and its structure will depend on how the first part goes. Here, we attempt to construct dynamical but approximate black hole solutions, we might study black holes in semi-classical gravity, leading to the topic of Hawking radiation, and we might contemplate the information paradox.

25/26 Lecturer:  Gwyn Bellamy (Glasgow)

AGQ Interfaces in Algebra and Quantum Fields

Overview: TBA  

25/26 Lecturer:  Murad Alim (Heriot-Watt)

AGQ Interfaces in Geometry and Quantum Fields

Overview: TBA

26/27 Lecturer: Alessandro Sisto, Matthew Cordes (Heriot-Watt)

AGQ Interfaces in Algebra and Geometry

Overview: TBA