Mathematical Physics projects

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

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 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.

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 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.