The PI's field of research is algebraic geometry. More specifically he investigates the intersection theory of geometrically defined classes on various moduli spaces related to curves. For this proposal there are roughly two major areas of research. A cluster of questions in open orbifold Gromov-Witten theory spring from the PI's previous work with Andrea Brini (University of Geneva) and his own graduate student Dusty Ross (Colorado State University). A second group of problems in Hurwitz theory continue along the line of the PI's previous work with Hannah Markwig (Saarbrucken), Paul Johnson (Imperial College) and Steffen Marcus (Brown University) and Johnatan Wise (Stanford University). Open orbifold GW theory for toric orbifolds was defined by Brini and the PI, with the scope of giving a mathematical definition to invariants predicted by mirror symmetry and to obtain combinatorial techniques to study (ordinary) GW invariants of orbifolds. The work with Ross gave some positive evidence that open invariants could be useful tool for questions such as the Crepant Resolution Conjectures of Ruan and others. We mention some direction for future investigation in the area: - setting up a convenient formalism for open GW invariants, that may help extend the applicability of these tools to the Crepant Resolution Conjecture question beyond the Hard Lefschetz cases. - studying the algebraic structure of the orbifold topological vertex, its relation with the analogous object in Donaldson Thomas theory, and its connection with representation theory of wreath products of groups. - applying techniques of OGW to a conjecture of Bouchard-Klemm-Marino-Pasquetti relating such invariants to quantities arising in mirror symmetry via topological recursions developed by Eynard-Orantin. In Hurwitz theory the PI has investigated with Johnson and Markwig the piecewise polynomiality of double Hurwitz numbers. The three authors gave a convincing and fairly exhaustive description of the combinatorial phenomenon, including some interesting wall crossing formulas. This opens up the quest of finding a geometric interpretation for such wall crossings. The author is seeking to develop a cohomological intersection formula (along the lines of the ELSV formula for simple Hurwitz numbers) on some appropriate moduli spaces that describes the double Hurwitz numbers and explains the piecewise polynomiality either in terms of birational modification of the moduli spaces involved, or of boundary corrections to the cohomology classes involved. An ingredient that should prove extremely helpful to this scope is the description of the pushforward to M_g-bar of the virtual fundamental class of moduli spaces of relative stable maps to the projective line in terms of standard generators of the tautological ring. This question has been investigated by the PI together with Marcus and Wise in genus 1. Richard Hain recently provides an answer for all genera but restricting to curves of compact type. The PI proposes to investigate this class on the full moduli space of curves, and exploit its polynomiality (or piecewise polynomiality) properties to prove an ELSV-type formula for double Hurwitz numbers. The PI also intends to work with Hannah Markwig to reinterpret this question tropically, with the twofold goal of obtaining better combinatorial tools to answer the question and to help the developments of the foundations of tropical moduli spaces of curves in arbitrary genus.
The PI's research explores interconnections among several areas of mathematics and mathematical physics. Creating "bridges" and "dictionaries" among several scientific disciplines is often a useful way to make science progress. The proposed research is inserted in a fertile and modern area of mathematics. The PI has been participating to several workshops and conferences. Throughout the period of the proposed project, the PI intends to support graduate students and help their mathematical development. He has taken some preliminary contacts about co-organizing research schools both in Colorado and internationally. He intends to give an opportunity to his own graduate students to travel to conferences and to interact with the broader mathematical community. Finally he is interested in developing partnership programs between Colorado State University and several international institutions, including University of Costa Rica, and University of Michoacan.