Training
Our training consists of scientific core and interface courses, advanced training in collaborative research computing, studentled activities such as cohort working groups, communication skills, ethics & RRI workshops, industryfocused initiatives, and an external placement with one of our academic, industry or third sector partners.
News: AGQ Core and Interface coursework is now available to students UKwide. Click here for details.

24/25 Course offerings now available for registration!
Our core and interface courses are now available to mathematics students UKwide. Click here for more information.
A timetable of all Semester 1 courses is available here.
Core Courses
Our core courses are designed to establish a common language and foundational approach to mathematics.
24/25 Lecturer: Clark Barwick (Edinburgh)
Overview: The goal of this course is to give students a zoomedout 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 (HeriotWatt)
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 ChernWeil theory and characteristic classes of vector bundles on manifolds.
24/25 Lecturer: Anatoly Konechny (HeriotWatt)
Overview: This course is an introduction to twodimensional conformal field theory. This is a welldeveloped branch of mathematical physics with many connections to other topics in mathematics such as infinitedimensional 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, CWcomplexes, 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 `timeindependent’ 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 semiclassical gravity, leading to the topic of Hawking radiation, and we might contemplate the information paradox.
Interface Courses
Our interface courses get our students thinking about the frontiers of mathematical research, about topics that bridge traditional mathematical disciplinary boundaries.
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 Bmodels 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 (HeriotWatt)
Overview: This course will provide an introduction to the holographic correspondence connecting theories of quantum gravity with nongravitational 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 stronglyinteracting quantum field theories. We will take a “bottomup” 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.
Aligned courses
We are pleased to advertise the following two courses, offered by Higgs Centre and by SMSTC, which are likely to be of interest to AGQ students, and which can be taken as part of the Degree Programme.
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 selfcontained 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 (HeriotWatt)
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.
 Quantum mechanics: Bras, kets, qubits and all that – the state vector formalism for quantum mechanics (in strange physicists’ notation). I won’t cover the more general approach using density matrices.
 Quantum circuit model of quantum computing: A pictorial way of representing the unitary operations and measurements carried out in a quantum computer.
 Party tricks: Dense coding and teleportation – some (small) quantum circuits doing nonclassical things.
 Quantum advantage: Deutsch’s (and DeutchJosza’s) algorithm – the first evidence that a quantum computer could do something a classical one couldn’t.
 Finding a needle in a haystack: Grover’s algorithm – a quantum algorithm for unstruc tured database search that offers a square root speedup over a naive exhaustive classical search.
 Breaking the code: Shor’s algorithm. What really got folk interested in the field – the prospect of breaking public key cryptography using the efficiency of quantum period finding algorithms.
 Solving linear systems and PDEs: The HHL algorithm – something that would be a very bad idea for solving a linear system classically turns out to be very effective in the quantum case.
 Quantum Simulation: Feynman was one of the first to point out that quantum comput ers would be good simulators of quantum systems – some simple examples are discussed.
Communication Skills
Through the Examples Showcase and the Group Project, our students learn important scientific communication skills. Students further hone these skills by presenting at annual conferences and workshops throughout their studentship. Finally, they learn to present themselves and their work professionally to the wider world during Landing the Transition sessions in their final years.
Cohort Working Groups
Students are invited to organise studentled cohort working groups to learn a particular topic together. The CDT will support these students with a working group webpage, hybrid room booking, and advertisement of the working group to the cohort.
Advanced Research Computing
The research computing courses will explore a variety of connections between Algebra, Geometry, and Quantum Fields and the world of modern computing and data analysis. One consistent goal of the course will be to increase students’ fluency and familiarity with a modern programming environment, including coding in Python and other languages, and collaborative programming with Git.
The course running in Semester 2 in 24/25 will focus on machine learning, surveying applications to quantum collider physics and to topological structures in biology, and on topological data analysis. Future topics may include mathematical computing (for problems in algebra, geometry, number theory, etc), quantum field theory techniques in machine learning, as well as interactive theorem proving with Lean.
Responsible Research & Innovation
During Semester 1 in 24/25 we will have a series of workshops led by Dr. Michael Barany, exploring the role of individual identity in mathematical research and the impacts of mathematics on society and vice versa. We will discuss what lessons we can draw about doing research in a way that is ethical and meaningful both in an academic context, and with a view towards its wider impact on society.