26/27 Lecturer: Clark Barwick (Edinburgh)

26/27 Lecturer: Matt Walters (HWU)

Overview: This course will provide an introduction to the holographic correspondence connecting theories of quantum gravity with non-gravitational theories in one fewer spacetime dimension. Over the past 30 years, holography has developed into a powerful tool for studying both fundamental questions about black holes and making concrete predictions in strongly-interacting quantum field theories. We will take a “bottom-up” approach to this correspondence, discussing very general features of gravitational theories and how they can be reinterpreted as statements in the language of conformal field theory. The main prerequisite for this course is comfort with the concepts and calculations in quantum mechanics, as well as some very basic knowledge of group theory and differential geometry. Experience with quantum field theory will be a significant help but is not required.

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:  Gwyn Bellamy (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:  TBA

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: TBA

26/27 Lecturer: Clark Barwick (University of Edinburgh) 

Overview: A geometric object (variety, scheme, manifold, analytic space, etc.) X almost always has an attached category of sheaves A(X) (coherent sheaves, D-modules, étale sheaves, crystals. etc.). This is typically a symmetric monoidal category, and the assignment of A(X) to X often has two kinds of functoriality that fit together in an interesting way. Taken together, such assignments are called six functor formalisms​. The main part of the course will be concerned with how one might try to identify inverses​ to these assignments. This will lead us to contemplate stratified spaces​, tannakian duality​, and versions of the Balmer spectrum​. We will also think about how these structures help us understand some of the remarkable structures that appear in modern stable homotopy theory.

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: Anton Izosimov (Heriot-Watt) 

Overview: TBA