25/26 Lecturer: Giulia Gugiatti (Edinburgh, Organiser) and Ivan Cheltsov (Edinburgh)

25/26 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.

25/26 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. 

25/26 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).

25/26 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.

25/26 Lecturer: Sayantani Bhattacharyya (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: D-modules play a central role in representation theory and an important role in many aspects of algebraic geometry. This course, which assumes only a basic knowledge of algebraic geometry, will develop from scratch the theory of D-modules.  The end goal of the course is to understand the statement of the Riemann-Hilbert correspondence, giving an equivalence between the category of regular holonomic D-modules and the category of perverse sheaves (with complex coefficients) on a smooth complex algebraic variety. Topics covered in the buildup to the equivalence include push-forward and pull-back of D-modules, good filtrations and characteristic varieties, holonomic D-modules, duality for holonomic D-modules and the classification of simple holonomic D-modules via minimal extensions.  

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

AGQ Interfaces in Geometry and Quantum Fields

Overview: Asymptotic, divergent power series are ubiquitous in mathematical physics and pure mathematics. In quantum mechanics, quantum field theory, and string theory, the quantization process typically introduces a small parameter, allowing relevant quantities to be
expressed as power series in that parameter. The coefficients of these series are often determined recursively, sometimes through Feynman diagrams, and tend to exhibit factorial growth, leading to the divergence of the series.
Resurgence is the systematic mathematical study of such divergent series, particularly those arising in quantization. The goal is to use Borel-Laplace summation to reconstruct an analytic function in some domain, whose asymptotic expansion matches the original formal power series. In general, there are multiple analytic functions with overlapping domains of validity, differing on their overlaps by Stokes factors.
This course will explore both the formal mathematical tools developed in the resurgence program and their applications in geometry and physics. A key geometric example arises in enumerative geometry: the Gromov-Witten invariants of Calabi-Yau manifolds are encoded in asymptotic series, while the Stokes factors capture another set of invariants the Donaldson-Thomas invariants. In physics, asymptotic series typically emerge from
perturbative expansions, while Stokes factors encode non-perturbative, exponentially suppressed effects. These effects play a crucial role not only in exact computations but also in the conceptual understanding of physical theories across their full parameter space and
dualities.

Prerequisites: Undergraduate analysis and linear algebra, very good knowledge of Complex Analysis. An exposure to ideas of quantum mechanics and quantum field theory is helpful but not
necessary.

25/26 Lecturer: Alessandro Sisto, Matthew Cordes (Heriot-Watt)

AGQ Interfaces in Algebra and Geometry

Overview: This course will be a general overview of the geometry and topology of low-dimensional manifolds, i.e., manifolds of dimensions 2 and 3. Our plan is as follows: We will first review constructions and classification of surfaces (2-manifolds) and discuss their mapping class group, i.e., their “topological symmetry group”. We will then turn our attention to geometry and construct hyperbolic metrics on surfaces, briefly touching upon Teichmüller space and moduli space, which are parameter spaces for these metrics. After this we jump up a dimension to 3-manifolds and show how algebraic structures of the mapping class group encode information about surfaces and 3-manifolds. . We will also discuss various constructions of 3-manifolds and describe canonical ways to split them into so-called geometric pieces, ending with the statement of Thurston’s Geometrisation Conjecture.

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 Topics in Scattering Amplitudes 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.

The course meets Monday 9-10:50 (lecture) and Thursday 9-10:50 (exercise session) in the Higgs Centre seminar room, JCMB 4305, starting 15 September. To register, email agq-cdt@ed.ac.uk with your name, the course number PGPH12001 and title, and “class with assessment” (homework assignments) or “class only.”

Further course information here.

Lecturer: Einan Gardi (Edinburgh)

Overview: The course provides an introduction to modern approaches to scattering amplitudes, exposing the students to universal properties of amplitudes and the profound connections between physics and mathematics concepts, which guide current research directions in this field. The course will build upon familiarity with the basics of QFT in general, and gauge theories in particular, as well as the properties of Feynman integrals taught in the first semester course, but it will not rely on any prior specialised knowledge on scattering amplitudes. To facilitate sufficient depth, the course will focus, in any given year, on just one or two areas of expertise, such as the IR structure of gauge theory amplitudes, on-shell methods, or applications of amplitudes to gravity.

To register, email agq-cdt@ed.ac.uk with your name, the course number PGPH12002 and title, and “class with assessment” (homework assignments) or “class only.”

Further course information here.

There are a wide array of courses offered as part of the Higgs Centre MSc programme in Theoretical Physics. These include

• Statistical Physics (S1) & Advanced Statistical Physics (S1)
• Quantum Field Theory (S1)
• Quantum Theory (S1)
• Computational Astrophysics (S1)
• Symmetries of Particles and Fields (S1)
• Cosmology (S2) & Advanced Cosmology (S2)
• Classical Electrodynamics (S2)
• General Relativity (S2)
• Hamiltonian Dynamics (S2)
• Gauge Theories in Particle Physics (S2)

For a complete list, visit the UoE Path page for Physics, and scroll down to Postgraduate.

More information on the SMSTC course website.

Lecturer: Alexandre Martin (Heriot-Watt)

Overview: Geometric group theory is an extremely active field, whose premise is to understand and exploit the actions of infinite groups on geometric spaces such as graphs, cube complexes, hyperbolic spaces and manifolds, in order to unveil key algebraic, algorithmic or topological properties of the groups. Since groups arise naturally in most parts of mathematics and physics, geometric group theory provides new tools to tackle problems from various research topics, and it has had striking applications to low-dimensional topology and birational geometry, among others.

This course will be an introduction to this dynamic field, and will be structured in three parts. In the first part we will give the necessary background on group presentations and Cayley graphs, group actions on graphs and spaces, and quasi-isometries, culminating with the Švarc-Milnor theorem. We will then illustrate its general philosophy through the study of two important classes of groups at the crossroads between several areas: in part two we will focus on hyperbolic groups and hyperbolic geometry, and in part three we will present reflection and Coxeter groups through their actions. In each case, students will see how the geometric tools allow them to obtain new algebraic information about the groups, such as solving the word problem in hyperbolic groups, or understanding the subgroups of Coxeter groups.

More information on the SMSTC course website.

Students may want to look at various activities offered by the Quantum Informatics CDT (in which Edinburgh and Heriot-Watt are partners). In particular, interested AGQ students have been invited to attend the weekly seminar Topics in Quantum Informatics, which gives an overview of the research topics in the remit of the QI CDT, of their interconnections, and of open challenges. Each session will take the form of two long-form seminars given by rotating speakers, followed by a discussion. For more information and meeting times and location, see the DRPS entry.