TOPOLOGICAL MIRROR SYMMETRY

Dualities in theoretical physics are important tools to gain information on a physical model from its proposed duality with another typically more accessible physical theory. One such duality is mirror symmetry, which is a duality theory stemming from string theory. The mathematical implications of this duality are manifold. In the proposed project we are interested in relating these mathematical implications with ideas coming from number theory and representation theory. Namely, we propose to find patterns in the character tables of some finite matrix groups which explain this mirror symmetry from the perspective of Langlands duality. This latter is a vast program in modern number theory which in a special case implies Fermat's Last Theorem by the work of Andrew Wiles. In this proposal we are connecting via the study of the character tables of finite matrix groups, these two seemingly far dualities: mirror symmetry in string theory, and Langlands duality in number theory.

Researchers working in algebraic and differential geometry, representation theory, number theory and string theory will benefit from this research. They will benefit from the research by being in possession of new techniques to evaluate Hodge numbers of open varieties,to study the character table of SL_n(F_q). The mirror symmetry phenomena being unearthed will open up the possibility of understanding deeper properties of the representation theory of SL_n(F_q) and its connection with the Langlands program. This will bring together ideas originating in string theory with those arising in number theory. The beneficiaries will be made aware of this research by both visits to seminars, workshops and conferences at UK institutions and abroad, posting of preprints on the arXiv and via publications in reputable journals.