# Research

**Collaborators**

Yi-Zhi Huang, Rutgers University, USA.

Kalle Kytölä, Aalto University, Finland

Eric Schippers, University of Manitoba, Canada

Mohammad Shirazi, University of Manitoba, Canada

Wolfgang Staubach, Uppsala University, Sweden

**Students**** (future, current, and former)**

**Motivation:**

Conformal field theory is a mathematically rich two-dimensional quantum field theory (CFT) that appears in condensed matter physics, statistical mechanics and string theory. Understanding the mathematical structures has led to developments in algebra, representation theory, geometry, topology, operator algebras, and SLE. Even though CFT is now over 30 years old the story is very far from finished. In many cases a rigorous understanding and connection between the different approaches to CFT is only just beginning.

One approach to axiomatizing CFT and rigorously encoding its mathematical structures is the so called functorial approach originally due to G. Segal and M. Kontsevich. In a program led by Yi-Zhi Huang (read his blog, research interests and research description for an up to date overview of the status of this program) , vertex operator algebra theory and its corresponding representation theory has been successfully used to mathematically construct a large class of conformal field theories in genus-zero and genus-one. In higher-genus, even rigorizing the definition of CFT requires significant work in geometry and analysis. The construction program in higher-genus also needs some new results in moduli space theory before it can be completed.

In particular, one needs to carefully define and study the infinite-dimensional moduli space of Riemann surfaces with parametrized boundaries components. The appropriate mathematical setting is quasiconformal Teichmüller theory (see our recent article arXiv:1605.00449). As well as developing new mathematical structures and results needed for CFT, my motivation lies in obtaining results of interest purely in Teichmüller theory by using ideas from CFT. A lot of progress has been made over the past 15 years, starting with my PhD thesis and continuing in joint work with Eric Schippers and Wolfgang Staubach. Significant interesting work remains to be done as continually deeper connections are made between CFT and geometric function theory.