CAREER: Knot invariants, moduli spaces of sheaves and representation theory

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The subject of this project is the geometry of configuration spaces of collections of points inside varieties of small dimension, and more generally, the moduli spaces of sheaves on these varieties. The main objective is to reveal new and further explore previously known links between the moduli spaces and objects in other fields of mathematics, in particular Representation Theory and Lower Dimensional Topology. The PI will work toward a proof of the mathematical conjecture relating the topological invariants of the Hilbert scheme of points on plane singular curves and the HOMFLY knot homology of the links of the singularities of the curve (Hilb/HOMFLY formula). The conjecture also reveals unexpected symmetries of the homology of torus knots: conjecturally, they form an irreducible representation of the rational Cherednik algebra of type A. The PI will explore the generalized Hilb/HOMFLY conjecture that relates the representation theory of the symplectic reflection algebras and the rational Cherednik algebras of types other than A. Finally, the PI describes the cohomology ring of the compactified Jacobians of quasi-homogeneous singularities. The PI (jointly with Zhiwei Yum) conjectures a relation between the cohomology ring of the compactified Jacobian of the curve and the structure ring of the moduli space of the rational maps to the curve: a local variation of the Gromov-Witten/Donaldson-Thomas relation. The educational component of the project offers a new model for the UMass REU program. Knot invariants and topological invariants allow us to analyze the global structure of complicated shapes by collecting local information about the shape. Complicated shapes occur naturally in biology (e.g. proteins, DNA), theoretical physics (strings), and other areas of natural science. Thus developing new invariants and computational methods for understanding of the global structure of complex shapes is an important mathematical problem with many potential applications. The PI strives to understand the hidden symmetries of already discovered invariants, develop new invariants, and find unexpected applications of these invariants to other areas of mathematics. The PI will also involve undergraduate students in cutting edge research through a summer research program integrating mentorship by faculty and graduate students. The PI aims to attract more students from underrepresented groups to mathematical research by reserving specific spaces in the summer research program for students from two local women's colleges. The PI will prepare graduate student mentors during the year by teaching related graduate classes and a reading seminar. This new summer research program structure will increase diversity and strengthen vertical integration in academia and improve the communication and flow of ideas between different generations of present and future researchers.