Spectra, gaps, degenerations and cycles

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The PIs will continue their foundational work on Homological Mirror Symmetry (HMS), to develop the structures and theories involved in HMS, and to build on applications of these theories. One notable direction is the theory of higher symplectic structures, which brings out the duality between the ``stacky'' directions and the ``derived'' directions of most moduli problems in algebraic geometry. Exploiting the full depth of these structures will require a careful study of the moduli of various kinds of categorical and higher categorical entities. This is one of the main areas of expertise of all the PIs. Kontsevich introduced one of the main tools, derived schemes. Katzarkov came up with the idea that moduli of LG models and its monodromy can be interpreted as stability conditions and spectra.These activities fit into a more general and global philosophy designed to accompany Geometry in the 21st Century. The study of Geometry in the 20th Century was devoted, in large part and with astounding success, to the classification and parametrization of geometrical objects. However,these objects, of various kinds, were uniformly viewed somehow as ``sets of points''. Along the way, the relationship with categorical structures grew steadily, leading to the many inputs into our program as discussed above. The PIs themselves played a pivotal role in much of the progress that was made at the turn of the century. With PI Kontsevich's introduction of HMS, a subtle change was introduced, in that ``Geometry'' began to be seen within a categorical structure. And the concurrent development of the theory of higher stacks meant that geometric structures were no longer viewed just as ``sets of points'' but rather as objects enclosing a higher structure. This project is highly connected with theoretical physics. As we head into the second decade of the 21st Century, elementary particle physics is on the crux of a profound revolution to be brought about by the new experimental results coming out of the LHC at CERN. These will serve to identify which of the multitude of theoretical possibilities which are currently open, best address quantum field theory at the high energy scale. And for those theories, to tell which are the right parameters. So there will soon be a lot of work to do on the theoretical side, and this will surely require new tools and a new approach. With the relationship between HMS and supersymmetric theories, with the relationship between higher categories and TQFT, with the relationship between partition functions and nonabelian cohomology, the kinds of geometrical objects which we are going to investigate in this project are becoming crucial for understanding these new panoramas in theoretical physics. The project has an educational component - conferences and educating postdocs, This component has been hugely successful in the past and with more funding we plan to bring it to the next level.