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.