This project studies connections between different areas of geometry that are inspired by ideas from theoretical physics. An important idea in theoretical physics is the notion of a duality between physical theories. A consequence of such a duality is a deep connection between the mathematical models that describe those theories. These connections allow us to look at a mathematical question in one model from another perspective, and thus derive new results. For this project, the mathematical models come from algebraic geometry (the geometry of sets defined by polynomial equations) and representation theory (the study of symmetry) on the one hand, and symplectic geometry (the geometry of the phase spaces of classical mechanics) on the other. The duality relating them is known as homological mirror symmetry. The focus of this project is to mine this connection for new insights into structures arising on each side of the duality. One part of the project studies how symmetries arise in symplectic geometry, and how this new perspective can give insights into representation theory. Another part studies the relationship between dynamics in symplectic geometry and one of the most basic objects in algebraic geometry, namely functions. The project also supports the training of graduate and early-career mathematicians in this rapidly-developing area of research. More broadly, this project fits into the ongoing interaction between mathematics and physics that has, over the centuries, led to theoretical advances that have enabled the transformative technologies of our time.
The organizing principle for this project is homological mirror symmetry for log Calabi-Yau varieties (varieties arising as the complement of an anticanonical divisor in a compactification). We consider such varieties both for the algebraic and the symplectic sides of the correspondence. On the symplectic side, the main structure we consider is symplectic cohomology, an algebraic structure that is built out of periodic orbits of certain Hamiltonian flows on a symplectic manifold (hence the connection to dynamics). The heart of the project is to relate this structure to functions and vector fields on the mirror algebraic variety. The connection to representation theory appears by considering the flag variety of a semisimple algebraic group G on the algebraic side. These are not log Calabi-Yau, but they contain open subsets which are, and part of the project is to understand better how to pass between the two situations (this involves considering a potential function on the symplectic side). The symmetries of the flag variety, namely the group G and its Lie algebra, should appear in the symplectic side as well. The natural home for the Lie algebra is symplectic cohomology, and the group action itself is manifested in the action of this Lie algebra on the Floer cohomology of equivariant Lagrangian submanifolds, which are the counterpart of equivariant vector bundles in algebraic geometry. Ultimately, one expects to obtain representations of G in the Lagrangian Floer cohomology groups. The pay-off for this effort is that these Floer cohomology groups come with a distinguished basis, and this project seeks to understand how that basis is related to the various known canonical bases in representation theory of Lusztig, Mirkovic-Vilonen, and others. In approaching these problems, the project uses ideas from the Strominger-Yau-Zaslow approach to mirror symmetry, as developed by Gross-Siebert and Gross-Hacking-Keel, as well as techniques developed by the PI in previous work on the case of log Calabi-Yau surfaces (complex dimension two).
