Aspects of the moduli theory of sheaves and varieties

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The PI proposes to study various spaces of sheaves on curves and surfaces, often in relation to the geometry of the moduli space of curves and of some moduli spaces of surfaces. In particular, the PI will investigate how the geometry of the Verlinde sheaves over the moduli space of smooth curves of genus g is reflected in the structure of the kappa ring. The PI also plans to further the current understanding of the geometry of determinant line bundles over moduli spaces of sheaves on a surface. For abelian or K3 surfaces, when the surface is allowed to vary, interesting questions arise about the cohomology of the moduli space of polarized abelian or K3 surfaces. In a different direction, the PI will interpret geometrically a q-defomation of two-dimensional Yang-Mills theory. In the mathematical study of important geometric objects, there is considerable interest in describing geometrically the set itself of all objects of a certain type. Such a set is called a moduli space. In particular, moduli spaces of sheaves on varieties are important in both mathematics and theoretical physics. Spaces of sheaves over a variety often reflect geometric properties of the underlying variety, while in physics, moduli spaces of vector bundles arise naturally in the fundamental study of gauge theories. The PI plans to analyze moduli spaces of sheaves in various settings, with the potential to shed light on a range of interesting geometries.