Mirror symmetry, quantum curves and integrable systems

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).

While the proposed research is rooted in its majority in blue-skies pure mathematics, this project takes concrete measures to make a tangible impact on a plurality of fronts.

One factor of impact of the proposal is its prominent intra-disciplinary character, which aims at bridging problems, ideas, and methods from areas of Mathematics that are seldom communicating and have precious few joint practitioners. A particularly fitting, although certainly not the only example is given by the study of the GHM correspondence (Milestones 2.1-2), which is of simultaneous and complementary interest for functional analysts working on spectral theory and for algebraic geometers working on GW theory and its allied curve-counting theories (such as Donaldson-Thomas theory).

On a different front, this projects puts forward a detailed array of training measures for the hired RAs and PhD student with a focus on inter-sectoriality, including a private sector secondment of the PhD student at the Technical Services Unit of Wolfram Research Europe. This will give a significant and genuine exposure to an exceptional forerunner of mathematical R&D in Industry, with an interaction that is not only fully coherent with the academic research endeavour, but is also explicitly conceived as a two-way street. In particular this partnership could also lead to objectively measurable deliverables in the form of extensions of the current functionality of Wolfram's flagship software (Mathematica), as well as an expansion of its knowledge-base; and what is more important, it will expose the group of the PI to hands-on training and supervision by leading-edge software developers, providing them with a rarely available well-rounded array of skills. This will leave them in an exceptionally strong position to achieve their career aspirations both vis-a-vis a progression through the ranks of Academia, as well as in making a smooth and successful transition towards industry, where they would make a highly valuable and innovative addition to the British workforce.

On this note, we remark that the research fields at the heart of this proposal have been the subject of enormous interest worldwide, and new centres of excellence are being created or further expanded overseas (in the US, Brazil, China and India). For these institutions, the study of the interactions between Geometry and Physics at the heart of our proposal represents a central research activity, when not an explicitly declared mission, and they are attracting a growing number of scientists from abroad (and in particular from Britain) in the relevant fields. It is absolutely imperative that Britain consolidate a leading position and remain a global pole of attraction for the best scientists in these internationally competitive, and continuously expanding, research fields at the frontier of Geometry and Physics; this proposal and its synergetic position in the British eco-system would be a means to help maintaining the attractiveness of the UK as a whole to the best young mathematicians in the field.