Overview

I obtained my undergraduate degree from the University of Queensland (Australia), and my PhD from Rutgers University (USA) in October 2003 under the direction of Yi-Zhi Huang. From 2003 to 2006 I was a Hildebrandt Assistant Professor at the University of Michigan. I spent the 2004/5 academic year at the Max Planck Institute for Mathematics in Bonn, Germany.

My research interests lie at the interface of Conformal Field Theory and the complex analytic theory of  the Teichmüller spaces of Riemann surfaces. This work fits within the program of using Vertex Operator Algebras to rigorously construct conformal field theory in the sense of Graeme Segal as reviewed here.

Key Terms: Teichmüller and moduli space theory. Mathematical foundation and construction of conformal field theory. Rigged Riemann surfaces, quasiconformal mappings, sewing, conformal welding, lambda-lemma, Schiffer variation, geometric function theory, determinant line bundles, modular functors, and vertex operator algebras.

MSC2010: 32G15, 81T40, 17B69, 30F60, 46G20

Publications

• "The semigroup of rigged annuli and the Teichmueller space of the annulus", (with Eric Schippers), submitted. arXiv:1001.5211

• "Fiber structure and local coordinates for the Teichmüller space of a bordered Riemann surface" (with Eric Schippers), Conformal Geometry and Dynamics, 14 (2010), 14-34. arXiv:0906.3279

• "A complex structure on the set of quasiconformally extendible non-overlapping mappings into a Riemann surface" (with Eric Schippers), Journal d'Analyse Mathématique, 108 (2009), 277–291. arXiv:0803.3211

• "A complex structure on the moduli space of rigged Riemann Surfaces", (with Eric Schippers),  Journal of Geometry and Symmetry in Physics, Vol 5 (2006), 82--94 . arXiv:math-ph/0512080. Note: This is an expository article related to the paper below.

• "Quasisymmetric sewing in rigged Teichmueller space", (with Eric Schippers), Communications in Contemporary Mathematics, Vol. 8, No. 4 (2006) 481-534. arXiv:math-ph/0507031.

• "Schiffer variation in Teichmueller space, determinant line bundles and modular functors", 1--163, PhD Thesis, Oct 2003,  Rutgers University, NJ, USA.

• "From infinite-dimensional Teichmüller to conformal field theory and back". Geometry and Quantum Field Theory, Max Planck Institute for Mathematics, Bonn, 2010.

• "A fiber structure of Teichmueller space and conformal field theory". Xth International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, 2008.

• "Quasisymmetric sewing in rigged Teichmueller space". Related talks given at: (1) CMFT 2005, Joensuu, Finland, (2) XXIVth WGMP, Bialowieza, Poland, (3) XIVth OMGTP, Oporto, Portugal.

• Thesis defense slides

• "Schiffer variation in Teichmueller space, determinant line bundles and modular functors" (in pdf), pp 1--163.

• Abstract: The pure mathematical incarnation of conformal field theory was introduced by Segal and Kontsevich around 1987. Recently, Hu and Kriz further rigorized Segal's definition. Conformal field theory is intimately connected to vertex operator algebras and the complex geometry of Riemann surfaces with analytically parametrized boundaries. This thesis is centered on the analytic and geometric aspects of this theory.
  
  An explicit description of the complex structure of the infinite-dimensional moduli space of Riemann surfaces with analytically parametrized boundary components is given and the holomorphicity of the sewing operation is proved. The determinant line bundle is shown to be a holomorphic bundle over this moduli space and the sewing operation is proved to be holomorphic on these bundles. Applications to modular functors, which are high-rank generalizations of the determinant line bundle, are discussed.
  
  All these results are needed in order to have rigorous definitions of a holomorphic (or chiral) conformal field theory and a holomorphic modular functor. So certainly this is a necessary step in the on-going project to construct higher-genus conformal field theories from vertex operator algebras.
  
  The formulation and proofs of these results rely on deep aspects of analytic Teichmüller theory and quasiconformal mappings, the uniformization of higher-genus Riemann surfaces, and Schiffer variation. To my knowledge these techniques have not previously been applied in this context and will have continued applicability.

Collaborators

• Yi-Zhi Huang, Professor of Mathematics, Rutgers University, USA

• Eric Schippers, Associate Professor, University of Manitoba, Canada.

• Wolfgang Staubach, Heriot-Watt University, Scotland.