Symmetries and correspondences: intra-disciplinary developments and applications

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.

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.