Research  

I'm interested in various areas of geometry, algebra, and mathematical physics.  Much of my research has been on the relationships between the moduli spaces of Riemann surfaces and various algebraic structures, particularly cyclic A-infinity algebras.  Recently I've been trying to understand renormalisation of quantum field theories.  

Work in progress 

A Wilsonian approach to renormalization of quantum field theories

This is a very preliminary draft of a book/paper I'm writing about quantum field theory. The proofs are not complete, and the final version will include a lot more. However, it should be readable in its present state.

This work develops a general theory of renormalization based on Wilson's ideas and the Batalin-Vilkovisky formalism, and applies this to prove perturbative renormalizability of Yang-Mills theory in 4 dimensions.  

Papers and preprints 

  1. arXiv:0706.1533 Renormalisation and the Batalin-Vilkovisky formalism.
  2. math.QA/0605647 Topological conformal field theories and gauge theories. Geometry & Topology, 11 (2007) 1539-1579.
  3. math.GT/0601130 A dual point of view on the ribbon graph decomposition of moduli spaces. Geometry & Topology, 11 (2007) 1637-1652.
  4. math.QA/0509264 The Gromov-Witten potential associated to a TCFT. Submitted to Journal of Topology.
  5. math.QA/0412149 Topological conformal field theories and Calabi-Yau categories. Advances in Mathematics, Volume 210, Issue 1, March 2007.
  6. math.AG/0402015 The A-infinity operad and the moduli space of curves.  Unpublished preprint, superseded by "TCFTs and Calabi-Yau categories" and "A dual point of view on the ribbon graph decomposition of moduli space".
  7. math.AG/0310189 Hilbert schemes, Hecke algebras and the Calogero-Sutherland system. Joint with I. Grojnowski.
  8. math.AG/0303387 Higher-genus Gromov-Witten invariants as genus 0 invariants of symmetric products.  This is my Ph.D. thesis from Cambridge University. Annals of Mathematics, Volume 164, Number 2, September 2006.