EPSRC Reference: 
EP/S003657/2 
Title: 
Mirror symmetry, quantum curves and integrable systems 
Principal Investigator: 
Brini, Dr A 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

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 GromovWitten theory), inspired by the physics of quantum gauge and string theories, has revolutionised the field by providing the longawaited solutions to an immense range of enumerativegeometric 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 GromovWitten theory and its allied enumerative theories. This is typically encoded into rich numbertheoretic properties of the generating functions of GromovWitten counts; and sometimes it can be characterised in terms of a hidden infinitedimensional group of symmetries given by the flows of a classical integrable hierarchy: a very special nonlinear 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 crossdisciplinary 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 GromovWitten 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 longstanding conjecture relating the invariants of a class of real fourdimensional spaces related by surgerytype operations (the "quantum McKay correspondence") in full generality; and the study of the asymptotic properties of curve counts in a large class of fourdimensional spaces (complex surfaces).

Key Findings 
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Potential use in nonacademic contexts 
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Impacts 
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Summary 

Date Materialised 


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Project URL: 

Further Information: 

Organisation Website: 
http://www.shef.ac.uk 