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Details of Grant 

EPSRC Reference: EP/S003657/2
Title: Mirror symmetry, quantum curves and integrable systems
Principal Investigator: Brini, Dr A
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Wolfram Research Europe Ltd
Department: Mathematics and Statistics
Organisation: University of Sheffield
Scheme: EPSRC Fellowship
Starts: 01 September 2019 Ends: 30 September 2024 Value (£): 833,984
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Analysis
Mathematical Physics
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:  
Summary on Grant Application Form
The subject of the proposed research lies at the frontier between pure mathematics (geometry) and mathematics motivated by physics, especially high energy particle physics.

One basic motivation of this proposal is given by two fundamental classes of problems in mathematics: the enumeration and classification of geometric spaces. These type of questions hark back to Greek antiquity to form one of the most venerable branches of mathematics since classical times: enumerative geometry, that is, the count of the number of solutions of geometric problems. Whilst very classical, these questions have often eluded an answer through standard methods; however, since the early nineties, the introduction to the realm of algebraic geometry of a wide range of sophisticated curve counting theories (such as Gromov-Witten theory), inspired by the physics of quantum gauge and string theories, has revolutionised the field by providing the long-awaited solutions to an immense range of enumerative-geometric problems.

Besides its intrinsic interest for geometers, this close relation to modern theoretical high energy physics has sparked a paradigm shift across the subject, blending new physically motivated insights in geometry with an immense toolkit of powerful computational schemes; this has revealed an intricate web of highly sophisticated algebraic structures underlying Gromov-Witten theory and its allied enumerative theories. This is typically encoded into rich number-theoretic properties of the generating functions of Gromov-Witten counts; and sometimes it can be characterised in terms of a hidden infinite-dimensional group of symmetries given by the flows of a classical integrable hierarchy: a very special non-linear partial differential equation possessing infinitely many commuting conserved currents.

These discoveries have had a transformative impact in all fields concerned, in geometry, mathematical physics, and the theory of integrable systems, and they have provided a shared source of insights for very different communities of pure mathematicians and mathematical physicists. Furthermore, the last decade has witnessed a range of dramatic advances which have revolutionised methods and perspectives of the field, and have opened up important avenues of investigation.

These new directions are the subject of this project. The central tenet of the proposed research is the identification of novel, powerful, and truly cross-disciplinary methods stemming from recent work of the applicant to solve four central problems in a burgeoning area of theoretical science: these include the explanation of the interplay between higher genus Gromov-Witten invariants, the theory of modular forms, and the theory of integrable systems; a constructive proof of a recently proposed and surprising connection between analysis (spectral theory) and geometry (mirror symmetry); a proof of the strongest version of a long-standing conjecture relating the invariants of a class of real four-dimensional spaces related by surgery-type operations (the "quantum McKay correspondence") in full generality; and the study of the asymptotic properties of curve counts in a large class of four-dimensional spaces (complex surfaces).
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Organisation Website: http://www.shef.ac.uk