Research objectives and content
The new and active subject of Mirror Symmetry (MS) and Quantum Cohomology (QC) can be viewed as an international effort to formulate a mathematically rigorous theory of certain aspects of string theory. At the present level of understanding of this theory it is of great importance to study specific objects emerging from the general setup in detail. There has been a great deal of progress at the so-called tree level theory which in mathematical terms corresponds to a positive integer parameter, the genus, being zero. My research activities so far have also focused on this part of the picture.
The two main objectives of the proposed project are to investigate the higher genus case (genus > 0), about which much less is currently known, and to include the so-called gravitational descendants, another structure naturally furnished by theoretical physics, into the theory taking the recently discovered action of Virasoro differential operators into account. Another concrete goal of the project is the extension of the genus zero results of a joint work with Manin and Zagier on higher analogues of Weil-Petersson volumes and their generating functions to higher genus. Training content (objective, benefit and expected impact)
A collaboration with Maxim Kontsevich who is one of the world leading scientists in the field of MS and QC would provide a unique opportunity to get an inside view of the theory of MS in its most advanced form. Furthermore, a possibility to work at the IHES will enable me to broaden my mathematical horizon. One of the concrete objectives in this direction is a more intensive study of symplectic geometry and the theory of pseudo-holomorphic curves of M. Gromov.
A stay at the world renowned IHES would certainly yield a wealth of new inspirations and result in a long lasting stimulation for my further research.
Links with industry / industrial relevance (22)