Geometry of Moduli Spaces

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In geometry, one often tries to classify all possible shapes of objects of a given type: for example, all triangles are classified by the lengths of their sides - which must then be positive, and satisfy the triangle inequalities. Such parameter spaces often themselves have rich geometric structures, and are called moduli spaces. In complex geometry, one studies objects that have complex coordinates - that is, from close up look like complex numbers. In complex algebraic geometry, one further restricts to studying shapes defined by polynomial equations in complex numbers - and the basic classification problem is to study all such shapes of a given type, find the parameters for such shapes, and what conditions these parameters must satisfy. Moduli spaces are ubiquitous in algebraic geometry, and in recent times have provided some of the most powerful tools for understanding individual geometric objects, by deformation and degeneration. In particular, algebraic curves (Riemann surfaces) permeate many constructions in algebraic geometry; abelian varieties appear naturally in varied contexts ranging from number theory to integrable systems to physics. The proposed research aims to obtain new information about the geometry of moduli spaces and relations among them. The investigator will seek new deep relations and properties of various moduli problems with the aim of providing further tools that could be used by researchers in complex and algebraic geometry, Teichmuller theory, string perturbation theory, and integrable systems.The investigator will work to further understand the geometry of moduli spaces over complex numbers, especially focusing on the moduli spaces of abelian varieties, and of curves. The investigator will build on the techniques and results he developed with Hulek, Tommasi, and Zakharov to define and study an extended tautological ring for suitable compactifications of the moduli space of abelian varieties, trying to determine whether it may be Gorenstein, and whether the intersection numbers may satisfy an interesting recursion relation. The investigator will apply the real-analytic techniques he developed with Krichever, and inspired by integrable systems, to study the classical problem of bounding the number of cusps of plane curves, and to study complete subvarieties of the moduli space of curves. The investigator will use this work to characterize geometrically the locus where the Prym map fails to be injective. With Salvati Manni, the investigator will study the slope of the effective cone of the moduli space of curves.