Symmetries and correspondences: intra-disciplinary developments and applications

Lead Research Organisation: University of Nottingham
Department Name: Sch of Mathematical Sciences

Abstract

Among the many sensibilities that humans have, two are very basic: the sensibility of the discrete, and the sensibility of the continuous. The sensibility of the discrete is at the basis of counting and hence of economics, that of the continuous at the basis of drawing, one of the arts. These two basic ways of apprehending the sensible have led to the development of arithmetic and of geometry respectively by means of finding formal languages to express them. Many fundamental changes in mathematics have arisen from insights into how one sensibility could be understood in terms of the other. From internet pages and enormously successful internet based startups to the pictorial presentation of quantum mechanical algorithms, the effectiveness of geometric cognition is seen all around us.

The natural numbers are the most basic object of mathematics. Yet, the most hard and unsolved problems in mathematics are about numbers. The simplicity of their definition hides an underlying immense complexity and profound depth. Clay Mathematical Institute's Millennium Problems include several problems on numbers.

Despite many previous great achievements, we are still missing a powerful geometric view of numbers that will reveal and apply their underlying continuous nature as opposed to their discrete appearance. Progress in solving difficult problems often involves methods and constructions from seemingly unrelated areas. It is a manifestation of deep harmony in mathematics when new structures are discovered which explain and solve very complex long-standing problems.

Using the features of EPSRC programme grants, our team will develop new fundamental insights and approaches to several key types of geometries, including very recent ones, and create many links between them. Using our united geometric vision, we will intra-disciplinary work on some of the most challenging problems in modern mathematics.

We will understand, develop and apply correspondences and symmetries. This includes the Langlands correspondences and generalisations to higher dimensions. Members of the team have already contributed to its areas. We will use several recent developments alternative and complimentary to the Langlands programme, such as adelic geometry, anabelian geometry, anabelian reciprocities, to extend it further. The Langlands programme is considered to be a Grand Unified Theory of mathematics. This programme is a sweeping network that interconnects many areas of mathematics and physics including electro-magnetic duality and conformal field theory. Our international visiting researchers will include the top leading researchers in the programme.

Uncovering hidden unifying fundamental structures in the mathematical universe will give us clearer vision and insight and lead to new amazing pathways. Our geometric work will be applied to other outstanding problems including two Millennium Problems and several important conjectures. Our proposal has 7 vertical threads of projects and 4 horizontal threads which interweave the vertical threads. The projects are interrelated through both vertical and horizontal threads, forming a multi-layered web of interactions. Vertical and horizontal threads consist of project of varying degrees of difficulty, ranging from projects where we can involve PhD students to projects which require a highly complex team work.

The expertise of the investigators ranges from differential and algebraic geometry, higher arithmetic geometry to model theory, anabelian geometry, geometric representation theory and infinite algebraic analysis. Our intra-disciplinary work will create new synergies among the leading contemporary research streams.

Planned Impact

Our proposal will be conducted in several strategic directions and areas of mathematics of the 21st century by world leaders in their areas, and by young researchers, selected for their highest quality, with input from the world's greatest experts.

Using recent advances and crucial developments in geometry and number theory our transformative synergetic proposal will fundamentally contribute to symmetries and correspondences and to solutions of exceptionally famous problems. The latter include key advances in the Birch and Swinnerton-Dyer conjecture and Generalized Riemann Hypothesis, two Millennium Problems, the Langlands and geometric Langlands correspondences and their higher developments. We will work on three of the five pure mathematics Challenges in the DARPA list of the fundamental Mathematics Challenges.

The immense difficulties of these research directions and problems are well known. Further progress requires a coordinated intra-disciplinary team work. It will reshape respective fields and influence their further development for many decades. Our team is unique in its strength to carry out the proposed research. The programme grant is ideal for our work in its flexibility, its longer term, its use for most strategic developments.

Mathematicians in the areas of our proposal and in related areas will greatly benefit from our research. Our work will enhance and create new relations between areas of mathematics. Our impact activities will be well integrated with our research activities. Our Pathways to Impact strategy will include 10 carefully designed pathways which will be regularly reviewed and updated.

We will run regular seminars, study groups, weekend meetings and workshops. Our team events will include Nottingham-Oxford seminars and video communication via internet. Overcoming the communication barriers between our fields, we will extend our expertise and its applications. Our proposal will educate young researchers, potential leaders of the future. Based on our previous history of training young researchers and their academic career, we will help them to gain an excellent expertise in both theory development and application to highly complex challenging problems. All this will be highly beneficial for this country and its economy via highly skilled and desirable workforce.

We will visit world-leading research centres and deliver impactful talks at major conferences to collaborate and disseminate our results. We will run a very strong programme of visiting researchers who will bring additional value to our proposal via encouraging international development. Since our proposal involves more than one institution, its implementation will have a much wider influence.

In addition to the academic influences such as enhancing the knowledge economy through new knowledge and scientific advancement and developing expertise in new disciplines, we will will reach out to other scientists through our research and meetings and via the internet. Our weekend meetings will be organized in such a way as to enable efficient participation of scientists who might benefit from this research via using parts of it and developing analogous studies in their areas.

We will publicise our activities to general audience through public engagement events, media reports and articles. Our networking events will be multiple, flexible and of great variety, to ensure the highest outcome. With the help of excellent writers, we will make ideas, theories and results of our research available to millions of people worldwide, using creative forms of its presentation and reaching the maximal levels of non-academic impact.

Publications

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Andreatta F (2015) A $p$ -adic nonabelian criterion for good reduction of curves in Duke Mathematical Journal

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Atiyah Michael Francis (2016) The Geometry and Dynamics of Magnetic Monopoles

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Balakrishnan J (2018) A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves in Mathematische Annalen

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Bambozzi F (2018) Stein domains in Banach algebraic geometry in Journal of Functional Analysis

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Beraldo D (2019) The topological chiral homology of the spherical category in Journal of Topology

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Beraldo D (2017) Loop group actions on categories and Whittaker invariants in Advances in Mathematics

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Bezrukavnikov R (2016) Microlocal sheaves and quiver varieties in Ann. Sci. Univ. Toulouse

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Bogomolov F (2015) Spitsbergen volume in European Journal of Mathematics

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Bogomolov F (2016) Division polynomials and intersection of projective torsion points in European Journal of Mathematics

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Bogomolov F (2015) Birationally isotrivial fiber spaces in European Journal of Mathematics

 
Title VIdeo Animation for Inter-Universal Teichmueller Theory of Shinichi Mochizuki 
Description a video illustration of various key features of the theory 
Type Of Art Film/Video/Animation 
Year Produced 2015 
Impact thousands of online viewers 
URL https://www.maths.nottingham.ac.uk/personal/ibf/files/iut.anima.html
 
Description Work on this grant is aimed at new fundamental progress in mathematics by building new links between several geometrical and other structures underlying key objects of study

Among many of its activities, the team of the grant (1) Internationallly leads the study of inter-universal Teichmueller theory of Shinichi Mochizuki and organised two major international workshops on it, as well as colllaborating on further extensions of the theory; (2) Developed and investigated various links between the IUT theory and other fundamental theories, such as two-dimensional adelic analysis and geometry, the Milnor-Bogomolov proof of the geometric version and Gromov bounded cohomology; (3) Extended measure and integration on abelian higher objects arising in arithmetic geometry to algebraic groups over such objects; (4) Identified the Gaiotto Lagrangian for the standard symplectic representation as a particular component of the nilpotent cone in the symplectic Higgs bundle moduli space and interpreted Lagrangian correspondences in products of moduli spaces and their mirror partners; (5) Developed a cohomological vision for the arithmetic linking numbers of primes, and established that they can be computed as a 'path integral' in the sense of quantum field theory with respect to a Chern-Simons counting measure; (6) Studied model theory of non-commutative geometry and quantum mechanics and developed a new proof of path integral formula for quadratic Hamiltonians.
Exploitation Route Our findings are affecting various areas of mathematics, and they are of cultural and societal value
Sectors Education,Culture, Heritage, Museums and Collections

URL https://www.maths.nottingham.ac.uk/personal/ibf/symcor.html
 
Description One of areas which this grant supports research activities in is the study of inter-universal Teichmueller theory of Shinichi Mochizuki. This theory is viewed as one of the top achievements in modern mathematics. It is attracting a huge interest and many mass media representatives have contacted members of the team for interviews. There are several mass-media publications, in Nature, Scientific American, New Scientist, etc., which are based on our interviews. Our article 'Fukugen' in Inference: International Review of Science has attracted more than 10,000 viewers during the first 4 months since its publication.
First Year Of Impact 2016
Sector Education,Culture, Heritage, Museums and Collections,Other
Impact Types Cultural,Societal

 
Description Collaboration with research group of Shinichi Mochizuki, RIMS, Kyoto University, Japan 
Organisation University of Kyoto
Country Japan 
Sector Academic/University 
PI Contribution Intensive collaboration between Symmetries and Correspondences and Center for Research in Next-Generation Geometry at Research Institute for Mathematical Sciences, Kyoto University, Japan Several international workshops coorganised. Collaboration on joint papers is ongoing.
Collaborator Contribution Intensive collaboration between Symmetries and Correspondences and Center for Research in Next-Generation Geometry at Research Institute for Mathematical Sciences, Kyoto University, Japan Several international workshops coorganised. Collaboration on joint papers is ongoing.
Impact to be added later
Start Year 2015
 
Description 10 hours of lectures at Kyoto University, July-August 2018 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact 10 hours of lectures, video recorded, at Kyoto University, July-August 2018
Year(s) Of Engagement Activity 2018
 
Description CMI Workshop 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Representatives of several mass media attended this workshop supported by EPSRC Programme grant Symmetries and Correspondences, interviewed many participants of our workshop or contacted its participants later, to publish numerous mass media reports about the workshop
Year(s) Of Engagement Activity 2015
URL https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html
 
Description Publication of an article of general interest about the work on the grant: 10,000 viewers in the first 4 months 
Form Of Engagement Activity A press release, press conference or response to a media enquiry/interview
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Public/other audiences
Results and Impact Article 'Fukugen' in Inference: International Review of Science
http://inference-review.com/article/fukugen
has attracted almost 10,000 viewers in the first 4 months after its online publication
Year(s) Of Engagement Activity 2016
URL http://inference-review.com/article/fukugen
 
Description interviews for a large article in Nature 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Media (as a channel to the public)
Results and Impact Interviews by members of the programme grant team for a large article published by Nature
Year(s) Of Engagement Activity 2015
URL http://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrabl...