PhD projects

Positive geometries (Tomasz Lukowski)

Positive geometries provide us with a new, geometric understanding of many observables in quantum field theories. They have emerged in the study of scattering amplitudes for maximally supersymmetric Yang-Mills theory in four dimensions, in the form of the Amplituhedron. Subsequently, they have been applied to variety of models, ranging from scalar quantum field theories to string theory. Nevertheless, not many is known about positive geometries in general and many statements in physics are still conjectural. The project will focus on uncovering new structure for positive geometries and applying them to physics.

Relevant keywords for this project are:

Amplituhedron
Kinematic Associahedron
Cosmological Polytope
Positive geometries in Conformal Field Theories

From Yangians to Shifted Yangians and W-algebras (Vidas Regelskis)

Yangians are some of the most elegant examples of infinite-dimensional quantum groups related to rational solutions of the Yang-Baxter equation. Originally introduced in the context of quantum integrable systems they have since expanded their authority to other areas of mathematics such as symplectic geometry and geometric representation theory. The elegance of Yangians is due to their relatively simple algebraic structure and highest weight representation theory. Yangians have some very interesting subalgebras and quotients that have a much more intrinsic structure and thus have not been studied extensively so far, especially beyond type A. This project will focus on finding new links between Yangians, Shifted Yangians and W-algebras, and advancing their representation theory with a particular emphasis on the cases when the underlying Lie algebra is of type B, C, or D.

Generalised geometry (Charles Strickland-Constable)

Generalised geometry is a relatively new mathematical structure which provides elegant geometrical descriptions of the supergravity theories underlying string theory and M-theory, with enhanced symmetry (the continuous versions of the string duality groups) playing an essential role. This description is especially suited to studying supersymmetric solutions with non-trivial internal fluxes and consistent truncations of supergravity, areas of study which are important for string phenomenology and AdS/CFT. Much of the existing work in this area has concerned the construction of the formalism and applications of the technology are relatively unexplored in comparison. The project will explore such applications and could involve aspects such as moduli spaces, quantisation of associated functionals, classification problems and connections with other areas such as higher gauge theory.

Categories in quantum field theory and gravity (Leron Borsten)

The early 1900s saw group theory derided by some leaders of theoretical physics as inscrutable and, worse, unnecessary pure mathematics. The gruppenpest, as Erwin Schrödinger put it. Despite this initial scepticism, group theory is now a cornerstone of fundamental physics. At the turn of the XX-century category theory occupied a similar position (although with somewhat less strident opposition; we did learn some lessons). Today, category theory is increasingly prevalent in theoretical physics and this project will develop such applications in quantum field theory (QFT) and gravity. The core theme will be the homotopy algebraic formulation of QFT. Homotopy algebras have emerged as the appropriate language for the BV formalism in QFT and this picture will be developed throughout the project to address various fundamental aspects of QFT and gravity: renormalisation, categorical symmetries, twists and (topological) phases, integrability and holography.

Relevant keywords for this project are:

Homotopy algebras, especially L infinity (homotopy Lie) algebras
BV formalism
Operads and Koszul duality
Twisting
Categorical symmetries

Gravity as the square of Yang—Mills: colour-kinematics duality and the double copy (Leron Borsten)

Recent years have witnessed dramatic progress in the idea that, at least in certain regimes, gravity can be reimagined as the “square” or “double copy” of Yang—Mills theory. More precisely, gluon scattering amplitudes can be “squared” to give gravitational amplitudes. The secret ingredient underlying this remarkable observation is colour-kinematics duality, which asserts that the kinematic data of gluon amplitudes obeys Jacobi identities that mirror the familiar Jacobi identities of the colour gauge group. Colour-kinematics duality and the double-copy have now been generalised to varied gauge and gravity theories, and has found applications ranging from gravity wave astronomy to testing the UV properties of supergravity. Yet, we are still lacking a complete understanding. In this project you will develop the underlying mathematical principles governing colour-kinematics duality and the double-copy, while exploring generalisations (e.g. to string field theory, curved spacetimes, condensed matter, lattice gauge theory…), physical implications, and applications. This will involve significant overlap with the “categories in quantum field theory and gravity” project, but with a focus on scattering amplitudes.

Relevant keywords for this project are:

Scattering amplitudes
Double copy
Colour-kinematics duality
Kinematic Lie algebras
C infinity and BV infinity algebras

Dualities in quantum field theory and string/M-theory (Leron Borsten)

Dualities connect seemingly distinct theories, phases or solutions. Often, highly non-trivial insights can be gleamed by considering questions through the dual system. Paradigmatic examples include S-duality in super Yang-Mills theory, particle-vortex duality in condensed matter, and U-duality in M-theory. The possible research directions offered include: (1) strong/weak electromagnetic/gravitational dualities (on manifolds with boundary), their anomalies and physical implications; (2) the conjectured D=6 (4,0) theory appearing M-theory and its implications for dualities in D=4; (3) black holes in string/M-theory, U-duality, and the higher composition laws of Manjul Bhargava.

Relevant keywords for this project are:

S-duality, T-duality, U-duality
Partition functions and the Ray—Singer Torsion
Extremal black holes
Supergravity, string/M-theory
Higher composition laws

Quantum integrals of motion for the Virasoro and W-algebras (Charles Young)

The Virasoro algebra plays a central role in conformal field theory, both in stringy and condensed matter settings. Its envelope contains a remarkable commutative subalgebra called the algebra of Quantum Integrals of Motion. The first open problem is very simple to state: find an explicit construction of these Quantum Integrals of Motion. (At the moment there is an existence proof, but closed formulas are known only for the first few.) The Virasoro algebra is (also) the quantization of the classical KdV system, which is the prototypical example of an integrable field theory (solitons, hierarchy of Hamiltonians, etc). So from another perspective, the task is to find the Hamiltonians of quantized KdV theory. This question turns out to be the tip of an iceberg of highly topical mathematical physics. At the level of keywords, here are some approaches that should be important and which would make good projects:

Quantum Toroidal algebras and the AGT correspondence;
(Affine) Quantum Gaudin models and the geometric Langlands correspondence;
The ODE/IM correspondence and (affine) opers; W algebras and coset constructions;
Vertex algebras and chiral algebras.

There turn out to be deep links between this project and the topic of higher- and homotopy- algebras.

Applied Category Theory (Luigi Alfonsi, Severin Bunk, Charles Young)

Applied category theory is an emerging interdisciplinary field that uses concepts and techniques from category theory beyond pure mathematics, to solve real-world problems and gain insights in various areas of science, engineering, and industry.

Categories provide a unifying framework for understanding relationships between different structures. They put on firm conceptual ground the key idea of compositionality, which is the property of certain systems to compose to give rise to new systems of the same species.

Examples of areas studied by applied category theory include artificial intelligence, data science, linguistics, quantum computing, and information theory.

Parallel transport in higher geometry and field theories (Severin Bunk)

In classical differential geometry, parallel transport is an operation which moves geometric data along paths in a differentiable space (a manifold) without changing it. Although seemingly a simple process, it lies at the heart of gauge theory, general relativity, and much of differential geometry: parallel transport along loops detects the field strength of gauge fields and the curvature of Riemannian manifolds.

One of the challenges of modern differential geometry is to lift the idea of parallel transport from one-dimensional paths to higher-dimensional geometric shapes, such as surfaces inside the background manifold. This requires the use of higher category theory in geometry, which is another ongoing modern development.

The project’s goal is to develop a two-dimensional parallel transport for a geometric object called a gerbe. (A gerbe is a categorified line bundle which models, for instance, the B-field in string theory and higher Berry phases in condensed-matter physics). Some models for this already exist, but exhibit various shortcomings. This project will resolve these shortcomings through a particular, novel approach, and if time permits will produce important applications to Stolz-Teichner’s functorial field theories and anomalies.

Computable Deformation Theory (Severin Bunk)

Deformation theory analyses the deep question how and whether given algebraic or geometric structures can be altered, without changing the type of structure. For instance, in how many ways can we change the multiplication operation on a commutative algebra so that we still get a commutative algebra?

It turns out that this can be answered by studying a completely different type of algebraic object: by a famous theorem due to Lurie and Pridham, deformations of commutative algebras are encoded entirely in homotopy-coherent versions of Lie algebras. The Lurie-Pridham Theorem was a major advance in pure mathematics and has found applications in areas including mathematical physics (such as in studying perturbations of solutions to field equations and perturbative quantisation).

However, the theorem and its proof provide only few tools for actually computing which homotopy-coherent Lie algebra controls a given deformation problem (only the other, arguably less important, direction is known). That is a obstacle in many applications of deformation theory. This project aims to close that gap by taking a new route in encoding the algebraic data, coming from (higher) category theory and homotopy theory. If time permits, the newly developed tools will then be applied to problems in higher differential geometry.

Algebraic methods for QCSPs (Catarina Carvalho)

The aim in a Constraint Satisfaction Problem (CSP) is to find an assignment of values to a given set of variables which are subject to a number of constraints. In the logical setting, the CSP can be formulated as a model-checking problem, in which we restrict ourselves to syntactic fragments of first-order logic with sentences built from conjunction, ∧, and existential quantifiers, ∃ (and perhaps =). In this setting, the Quantified Constraint Satisfaction Problem, QCSP, is the natural extension of the CSP where the universal quantifier is also allowed. The question in a QCSP is to decide, given a structure, whether or not a quantified conjunctive sentence, i.e. a first-order sentence built from atoms, conjunction, and both quantifiers ∀, ∃ (and perhaps =), evaluates to true on the structure (model). The QCSP can be used to model and solve complex problems containing contingency or uncertainty, allowing us to model problems that are difficult to express and solve in the context of the classical CSP. Applications are found in areas of game playing, uncertainty handling, conformant planning, robust scheduling, model checking, and testing. We are interested in developing and using algebraic methods in order to answer questions on the computational complexity of QCSPs, as well studying its connections with graph theory.

Wall's D(2)-problem (John Evans)

Wall’s D(n)-problem arose from an attempt during the 1960s to classify compact manifolds by means of ‘surgery’. It asked what conditions were necessary and sufficient for a complex of geometrical dimension n+1 to be homotopy equivalent to a complex of dimension n. The simply connected case was resolved by Milnor, and the non-simply connected case resolved in all dimensions other than n=2 by Wall, although the D(1)-problem was only cleared up completely once the Stallings-Swan proof showed that groups of cohomological dimension one are free. This left just the D(2)-problem. To date, the D(2)-problem remains unsolved in general, although it has been solved in several cases. A short but non-exhaustive list of fundamental groups for which the D(2)-problem has been solved is as follows:

Cyclic groups

Dihedral groups of order 4n+2

Certain quasi-dihedral groups

The quaternion groups Q(8), Q(12), Q(16), Q(20)

Certain metacyclic groups, e.g. the metacyclic group of order 21

Since the proof for the dihedral groups of order 4n+2, a recurring theme has been the construction of so-called diagonalised resolutions. These have been instrumental, for example, in solving the D(2)-problem for many metacyclic groups. However, it is also clear that it is too optimistic to expect these resolutions to exist for all groups. As such, it would be of great theoretical interest to start looking at these (metacyclic) groups which in some way misbehave. One such way a metacyclic group of order pq (p odd prime and q a positive integral divisor of p-1) can ‘misbehave’ is when the projective class group of 𝑍[𝐶𝑞] is trivial. A different direction may be to look for a counterexample to the D(2)-problem. A good family to look at would be the quaternion groups which admit non-trivial stably free modules (we therefore require their order to be >20). The consequence here is that there are now at least two minimal algebraic homotopy 2-types which we need to be realisable. The case Q(32) would be of particular significance here.

Topological Algebra (Yann Peresse)

Many important groups, semigroups and rings etc. can be given a natural topology that is compatible with their algebraic operations. Studying this combination of algebraic and topological structure can lead to a nice interplay of Topology and Algebra and a deeper understanding of these objects. For example, a topological group is a group equipped with a topology under which the group operation and inversion are continuous; an easy example is the group of real numbers under the usual addition and topology. Questions and problems in this area include:

Given a specific group (semigroup, ring) specify what sort of topologies it admits. For example, the group S∞ of all permutations of a countably infinite set admits a Polish (completely metrizable and separable) topology called the pointwise topology. The pointwise topology is known to be the unique second countable Hausdorff topology on S∞ . Many other interesting groups and semigroups also admit unique Polish topologies.
Given a group (semigroup, ring) with a fixed topology we can study subgroups with certain topological properties (open, closed, dense, compact. ...). For example, the closed subgroups of S∞ under the pointwise topology are precisely automorphism groups of relational structures.
Study the class of all groups (semigroups, rings) that have some topological property. For example, there is a rich body of research into groups admitting compact Hausdorff topologies.

Key words:

Topological groups;
Semigroup Theory;
Polish spaces, Polish groups, Polish semigroups.

Combinatorial algebra (Catarina Carvalho)

The field of combinatorial semigroup theory deals with infinite semigroups, which are natural generalisations of groups, defined by means of a presentation.

If an infinite semigroup can be defined by a finite presentation, which includes a finite generating set and a finite number of defining relations, then it can be encoded in a way that can be easily dealt with using a computer, making working with these semigroups a lot easier. These are naturally relates with computational algebra, automata theory, and the study of formal languages. Some problems in this area are:

When is a given semigroup construction finitely generated and/or finitely presented?
How does the finite presentability/ generation of a semigroup related with the finite presentability/generation of its group of units?

Higher Structures in Quantum Field Theory (Hyungrok Kim, Leron Borsten, Charles Young)

Physical systems are described by various symmetries. Physical systems also come with a notion of homotopy: in the presence of gauge symmetry, the BRST operator Q renders states and operators into chain complexes, which are homotopical objects. Consequently, the various symmetries hold only up to homotopy, and the homotopy-coherent closure of the symmetries is captured by the presence of a tower of higher operations. Such structures have begun to receive intense interest in recent years, in high-energy physics, condensed matter and beyond.

Here are some of the avenues that we can pursue depending on the candidate's background and interest:

homotopy-coherent notions of colour-kinematics duality and double copy, using homotopy versions of Batalin-Vilkovisky algebras;

higher gauge theory, higher symmetries, and higher Noether theorems in the presence of boundaries and defects;

applications of homotopy algebras to holography and gravitational physics, especially in lower-dimensional gravity (Chern-Simons, Jackiw-Teitelboim, ...);

higher vertex algebras, factorisation algebras, ravioli, and topological-holomorphic twists of supersymmetric field theories.