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Details of Grant 

EPSRC Reference: EP/R006989/1
Title: 4D TQFT and categorified Hall algebras
Principal Investigator: Kremnizer, Dr Y
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematical Institute
Organisation: University of Oxford
Scheme: Standard Research
Starts: 01 October 2017 Ends: 30 September 2020 Value (£): 595,108
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Physics
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
07 Jun 2017 EPSRC Mathematical Sciences Prioritisation Panel June 2017 Announced
Summary on Grant Application Form
Quantum field theory is the tool used to describe and study fundamental particles. It is one of the most successful and precise scientific theories with a wide range of applications: from particle accelerators to superconductors and quantum computers. Quantum fields depend on the underlying geometry of space and time. In some cases the fields are independent of the measurment of length and duration and only depend on the underlying shape of space and time. In such cases the quantum field theory is called topological. Topological Quantum Field Theories (TQFTs) are now in the frontier of research in both physics and mathematics.

The mathematical construction of TQFTs is based on a collection of axioms, much like, for example, Euclidean geometry. This makes TQFTs a mathematical as well as physical theory. TQFTs were invented by physicists in the late 80's in an effort to make a mathematicallly rigorous construction of Quantum Field Theories. They were quickly understood to be of wide mathematical interest, being related to, among other things, Knot Theory and the theory of four-manifolds in Algebraic Topology, the theory of moduli spaces in algebraic geometry and Quantum Groups. For almost three decades TQFT has been one of the central research areas in mathematics and theoretical physics. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory. However substantial parts of the theory, in particular the generalization to the higher-dimensional case, still remain to be constructed. This is the ultimate goal of the proposed project.

The way we propose to achieve it is to create a coherent framework unifying several recent results in the approach to TQFT. One of the main tools currently at our disposal is Higher Category Theory which is a novel and very active area of pure mathematics. The fundamental idea behind the categorical approach is that formulating the problems on the most abstract level often leads to new insights and creation of radically new approaches to their solutions. For this reason the categorical language is becoming more and more widespread in Pure Mathematics and even in some areas of more applied disciplines, such as Chemistry and Computer Science. Higher Category Theory seeks to develop this language further in order to apply it to additional areas of mathematics, generally speaking of a more geometric flavor, including TQFT.

The second important object in our approach are Quantum Groups. Quantum groups were originally defined at around the same time as TQFTs. They were originally conceived as algebraic objects, that is their definition involved formulas and equations. Very soon they were found to have a number of surprising and deep connections to different areas of mathematics, in particular to TQFT. In several early influential works on the subject, Quantum Groups were shown to give rise to invariants of three-dimensional manifolds, i.e. objects of a purely geometric nature. In this way it became apparent that Quantum Groups contain a rich stock of geometric information.

To construct invariants of four-dimensional manifolds we must take this definition of Quantum Groups one level deeper, defining a similar object with an additional layer of information. Such a process is called categorification. We intend to do so using the wealth of research on Quantum Groups accumulated since their introduction and the new ideas coming from Higher Category Theory. This should then allow us to develop a systematic approach to constructing four-dimensional TQFTs as well as constructing new and concrete examples of these theories. The ultimate result would be to produce an algorithm assigning various numbers to a four-dimensional object, expressing it's deep intrinisic properties.
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