Objective The theme of this proposal is the study of random operators associated with some geometric structure, and the influence of the geometry on the spectral properties of the operator. Such operators appear in problems from theoretical physics, and lead to new and interesting mathematical structures.One circle of questions is related to random operators, which describe the motion of a quantum particle in a disordered medium, such as random band matrices. The behaviour of the particle is influenced by the underlying geometry, as quantified by the (non-rigorous) Thouless criterion for localisation in terms of the mixing time of the classical random walk; in the context of random band matrices, the predictions of the Thouless criterion are supported by additional (non-rigorous) arguments. These predictions have so far not been rigorously justified; an exception is my own result, validating it at the spectral edges. One of our goals is to develop new methods, which would be applicable in the bulk of the spectrum, for random band matrices and other operators with geometric structure.Another circle of questions is given by random processes taking values in large random matrices. The spectral properties of the random matrix at every point of the underlying space are described by the random matrix theory; but how does the spectrum evolve along the underlying space? The richness of this question is apparent from the one-dimensional case of Dyson Brownian motion. We intend to study the local eigenvalue statistics of general matrix-valued random processes with multi-dimensional underlying space; to give a complete description of the random processes which appear in the limit, first for the spectral edges and then for the bulk of the spectrum, and to explore the appearance of these processes in a variety of basic questions of mathematical physics. Fields of science natural sciencesmathematicsapplied mathematicsmathematical physicsnatural sciencesmathematicspure mathematicsmathematical analysisnatural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesphysical sciencestheoretical physics Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-StG-2014 - ERC Starting Grant Call for proposal ERC-2014-STG See other projects for this call Funding Scheme ERC-STG - Starting Grant Host institution QUEEN MARY UNIVERSITY OF LONDON Net EU contribution € 824 853,75 Address 327 MILE END ROAD E1 4NS London United Kingdom See on map Region London Inner London — East Tower Hamlets Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 824 853,75 Beneficiaries (2) Sort alphabetically Sort by Net EU contribution Expand all Collapse all QUEEN MARY UNIVERSITY OF LONDON United Kingdom Net EU contribution € 824 853,75 Address 327 MILE END ROAD E1 4NS London See on map Region London Inner London — East Tower Hamlets Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 824 853,75 TEL AVIV UNIVERSITY Participation ended Israel Net EU contribution € 166 896,25 Address RAMAT AVIV 69978 Tel Aviv See on map Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 166 896,25