Training

24/25 Lecturer:  Clark Barwick (Edinburgh)

Overview: The goal of this course is to give students a zoomed-out perspective on algebraic geometry. The course starts with the introduction of affine varieties, schemes and sheaves, before introducing projective varieties, morphisms and fibre products. We will motivate the main definitions, discuss the basic results and provide plenty of examples.

The course ends with the introduction of vector bundles, (quasi-)coherent sheaves and sheaf cohomology, again with lots of examples from different areas.

24/25 Lecturer:  Murad Alim (Heriot-Watt)

Overview: This 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.

Finally, the material will be brought together via an introduction to Chern-Weil theory and characteristic classes of vector bundles on manifolds.

24/25 Lecturer:  Anatoly Konechny (Heriot-Watt)

Overview:  This course is an introduction to two-dimensional conformal field theory. This 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 aims at a thorough exposition of the main structures of conformal field theory, including conformal geometry on the (extended) complex plain, correlation functions, Ward identities, radial quantisation, operator product expansion, Virasoro algebra and conformal families. The emphasis in the course is on conceptual clarity rather than on the breadth of examples and connections. Another important aim is the development of basic computational skills needed to perform operator product expansions and to work with Virasoro algebra representations.

The course does not assume any prior knowledge of quantum field theory and can 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 will be provided along with suggestions for complementary reading. 

24/25 Lecturer:  Gwyn Bellamy (Glasgow)

Overview: Algebraic topology aims to associate 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, examples and techniques in algebraic topology.  Topics to be covered include homotopy theory, relative homotopy groups, CW-complexes, homology and cohomology theory. Roughly speaking, the course will cover chapters 0,1 and 4 of Hatcher’s excellent book Algebraic Topology, which is available freely online.

24/25 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.

24/25 Lecturer:  Sayantani Bhattacharya (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.

24/25 Lecturer:  Pavel Safronov (Edinburgh)

Overview: Topological quantum field theories (TQFTs) provide a pairing between manifolds and higher categories, thus giving rise to both invariants of manifolds (e.g. Reshetikhin—Turaev and Seiberg—Witten invariants) and invariants of algebraic structures (e.g. factorization homology). They also provide toy models of quantum field theories with trivial dynamics, describing for instance its ground states or possible topological orders. In this course we will study a functorial approach to TQFTs concentrating on algebraic and categorical structures (categories of cobordisms, cobordism hypothesis). We will also look at several examples including Dijkgraaf—Witten theory, ​Chern—Simons theory and topological A- and B-models relevant for mirror symmetry.

24/25 Lecturer:  Vaibhav Gadre (Glasgow)

Overview: The uniformisation theorem tells us that an orientable surface with finite genus and finitely many marked points admits a conformal structure, and thus is a Riemann surface. The space of conformal structures forms the moduli space of Riemann surfaces. These objects and spaces have been at the forefront of many modern developments in geometry, topology and dynamics. The course will survey the theory of Riemann surfaces and their associated moduli spaces (such as that of holomorphic quadratic differentials). The focus will be on geometry and topology initially, and on dynamics towards the end.

24/25 Lecturer: Matthew Walters (Heriot-Watt)

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. Similarly, the Conformal Field Theory and Vertex Algebras course offered in semester 1 is highly encouraged, but not required.

24/25 Lecturer:  Einan Gardi (Edinburgh)

Overview: This course provides a broad introduction to Feynman integrals in the context of perturbative scattering amplitudes in Quantum Field Theory (QFT), which draws out the connection between physical principles and mathematical concepts and methods. The course begins with an overview of different methods for evaluating Feynman Integrals, including direct evaluation and differential equation techniques. It proceeds with an introduction to the special functions Feynman Integrals evaluate to, such as multiple polylogarithms. Next, we discuss the analytic structure of Feynman Integrals, their cuts and discontinuities, and show how these, along with the differential equations, are encoded in a motivic coaction. The final chapters concern the singularity structure of integrals (Landau analysis), and asymptotic expansions (the method of regions).

While prior experience in QFT is useful for understanding how Feynman Integrals appear in physics, the course is designed to be self-contained and suits students with either maths or physics background.  The course will be followed by Scattering Amplitudes II in the second semester, which will focus on the infrared structure of scattering amplitudes in gauge theories, and will rely directly on knowledge of QFT.

24/25 Lecturer: Des Johnston (Heriot-Watt)

Overview: This course seeks to provide an introduction to the basics of the quantum circuit model of quantum computing and (at the risk of mixing metaphors…) to some of the “classic” quantum algorithms.