Overview

Slides

PhD Thesis

Collaborators

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 G. 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.

MSC2000: 32G15, 81T40, 17B69, 30F60, 46G20

• "A fiber structure of Teichmueller space and conformal field theory", To appear in a special issue of Journal of Geometry and Symmetry in Physics: Conference proceedings of the Xth International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, 2008.

• "A fiber structure on Teichmueller space" (with Eric Schippers),  in preparation.

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

• "A complex structure on the moduli space of rigged Riemann Surfaces" (in pdf), (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" (in pdf), (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" (zipped .dvi and figures), 1--163, PhD Thesis, Oct 2003,  Rutgers University. (Email me for the pdf file. It is too big for this server.)

• "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" (zipped .dvi and figures), pp 1--163. (Email me for the pdf file. It is too big for this server.)

• Introduction, pp 1--15

• Thesis defense slides

• 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.