Moduli spaces in algebraic geometry

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The investigator works in algebraic geometry, although his interests connect to other areas of mathematics, including topology, combinatorics, physics (string theory), number theory, and symplectic and differential geometry. This proposal, continuing various strands of the investigator's work, deals with moduli spaces and related notions in a variety of settings. In particular, the proposal deals with a number of fundamental questions regarding the moduli space of curves, deformation theory, the Schubert calculus, Geometric Invariant Theory, Gromov-Witten theory, and fundamental groups.Algebraic geometry provides a powerful theory with which to define moduli spaces of interesting objects. Once they are defined, natural compelling questions are: what do they look like? Are they irreducible? Connected? Of what dimension? Are they smooth? If not, what singularities arise? What structure is exhibited by their cohomology rings, and why should it be geometrically expected? What are their equations? The investigator intends to address many of these fundamental problems in a number of cases. The investigator has a track record of sustained and serious effort both in outreach to students at all levels (high school, undergraduate, and graduate), and in building institutions in which algebraic geometry can grow. The investigator will continue to attract graduate students into algebraic geometry, both at Stanford and elsewhere, and to continue to nurture the careers of graduate students, post-docs, and young researchers. The investigator will continue to work with large numbers of students at the secondary and undergraduate levels, attracting students into the mathematical sciences.