The geometry of the moduli spaces of morphisms and sheaves

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The PI will approach several questions in moduli theory. A first direction concerns the the moduli space of stable quotients, in particular the calculation of invariants for hypersurface and local geometries. Variations of stable quotient constructions will also be studied. Secondly, the PI will approach the Verlinde formula and the strange duality conjecture for moduli spaces of sheaves over surfaces. Strange duality concerns the spaces of generalized theta functions over moduli spaces of sheaves, and has applications to the study of the associated theta linear series. While the case of curves has been understood a few years ago, the case of surfaces is still largely open. Other projects concern topological invariants of the moduli spaces of genus 0 stable maps and stable quotients, and higher rank DT invariants for toric geometries.The PI works in algebraic geometry, which, roughly speaking, is the study of solutions of polynomial equations. The current proposal outlines several research directions all of which touching on aspects of moduli spaces e.g. spaces which classify algebro-geometric objects with similar properties. Moduli theory has traditionally been closely related to other fields such as combinatorics, symplectic geometry, number theory, representation theory, theoretical physics. The PI aims to better understand three kinds of moduli spaces all of which emerged in recent decades. In particular, the proposal will combine a quantitative side e.g. calculation of naturally constructed invariants, and qualitative aspects e.g. the geometric study of the moduli spaces.