Dualities in Enumerative Geometry and Representation Theory

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Enumerative geometry is a branch of mathematics which seeks to determine the number of geometric objects that satisfy certain conditions, for instance, the number of ways in which one space can be embedded into another. Representation theory is a part of linear algebra studying symmetries. This project is focused on deep interaction of these areas known as 3-dimensional mirror symmetry. This interaction leads to powerful identities between various objects in these fields and to discovery of new formulas which find applications in quantum field theory, algebraic geometry and combinatorics. This project includes opportunities for student research. More precisely, this is a project to investigate properties of the vertex functions and the stable envelope classes in equivariant K-theory and elliptic cohomology of symplectic varieties. Explicit formulas connecting these objects for pairs of varieties related by 3-dimensional mirror symmetry will be established. The main technical tools for this investigation include the abelianization of stable envelopes and the quantum difference equations. The resulting identities between the stable envelope classes lead naturally to new dualities in representation theory of quantum groups. The PI will also investigate important special cases, including Hilbert schemes of points on surfaces and instanton moduli spaces, in order to develop applications in combinatorics and theoretical physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.