Paul Johnson Email: pdjohnso at umich dot edu Mailing Address: Department of Mathematics 2074 East Hall 530 Church Street Ann Arbor, MI 48109-1043 Office: 3084 East Hall

In Hilbert-Raum, Göttingen

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I am currently in the final year of my PhD under Yongbin Ruan; I'll graduate in the spring of 2009.
My main research interest is Gromov-Witten theory, particularly in that of stacks.
More generally, I'm also interested in other areas of enumerative geometry, particularly Hurwitz theory.
Most generally, I can get excited about a large swath of Algebraic Geometry / Combinatorics / Topology, especially when they interact.

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Application Documents:

• Research Statement
• CV

Preprints:

• Abelian Hurwitz-Hodge integrals, joint with Rahul Pandharipande and Hsian-Hua Tseng .
  The ELSV formula expresses certain integrals over the moduli space of curves in terms of Hurwitz numbers, and has been a vital tool in Gromov-Witten theory. This paper gives an analagous formula for moduli spaces of orbifold curves, and is the key tool in my thesis.
• Tropical Hurwitz Numbers, joint with Renzo Cavalieri and Hannah Markwig.
  We show that tropical double hurwitz numbers agree with classical double Hurwitz numbers. In the process, we discover a new method for calculating double Hurwitz numbers in terms of graphs. Work in progress exploits this method to prove that double Hurwitz numbers are piecewise polynomial, and to derive wall crossing formulas for these polynomials. This approach is both easier and stronger than existing results.

Other Writing:

• Equivariant Gromov-Witten theory of stacky P^1s
  A draft of my thesis, which studies the equivariant Gromov-Witten theory of one dimensional toric stacks. This is the starting point of a larger project to study the GW invariants of all stacky curves. The stack can be ineffective - that is, the generic point can have an isotropy group. For example, consider M_{1,1}: every genus 1 curve has an involution. The main result is an operator formula for the GW invariants in terms of a Fock space for wreath products. This formula leads to some nice results: a proof that the GW theory of ineffecitve orbifolds decomposes, and a proof that the GW invariants satisfy a form of the 2-Toda hierarchy.