By Teresa Crompton
David Radnell is an Associate Professor in the Department of Mathematics and Statistics. He was awarded a Faculty Research Grant (FRG3) Travel Grant in Academic Year 2012-2013
How did the FRG3 award facilitate your research?
It allowed me to participate in a three-person ‘research in teams’ meeting at the Banff International Research Station (BIRS) in Canada. My visit was co-sponsored by BIRS. The meeting had the title ‘Moduli spaces in conformal field theory and Teichmuller theory,’ and, in addition to myself, the organizers were Yi-Zhi Huang (Rutgers University, New Jersey, UK), Eric Schippers (University of Manitoba, Winnipeg, Canada), and Wolfgang Staubach (Uppsala University, Sweden). Meeting with my long-term collaborators was essential to the rapid development of my research goals.
What is the Banff International Research Station?
Its full title is Banff International Research Station for Mathematical Innovation and Discovery. It opened in 2003 and is an independent research institute for the mathematical sciences, which hosts researchers attending workshops, research groups, summer schools and training groups in pure and applied mathematics. It is housed at the Banff Center, which is one of the world’s largest centers for arts and creativity. The station is funded by four governments: the federal government of Canada, the provincial government of Alberta, the US National Science Foundation (NSF), and the National Science and Technology Council of Mexico.
In 2012/13 alone the station hosted over 2,000 visits by researchers from hundreds of institutions in more than 60 countries who participated in over 70 different programs spanning almost every aspect of pure, applied, computational and industrial mathematics, statistics, computer science, physics, biology, engineering, economics, finance, psychology and scientific writing. The contribution to research arising from these activities is enormous.
What was the subject of your meeting?
Our long term research projects are at the interface of analysis, geometry and algebra in pure mathematics and quantum field theories in theoretical physics. We are working on establishing the mathematical foundations of conformal field theory, which plays an important role in both string theory and condensed matter physics. The focus of the team meeting was the following problems, which are central to the deﬁnition and construction of conformal ﬁeld theory from vertex operator algebras and furthering the connections with infinite-dimensional Teichmuller theory.
1. Deﬁne the determinant line of a Riemann surface whose boundary is parameterized by a
2. Construct local trivializations of the determinant line bundle over the rigged moduli space
using ‘Faber polynomials,’ a canonical basis for the set of holomorphic functions on a domain.
3. Construct projectively ﬂat connections on the determinant line bundle.
4. Give a rigorous and complete deﬁnition of conformal ﬁeld theory and a modular functor.
5. Establish a conjectured ‘sewing property’ for suitable ‘meromorphic functions’ on the rigged moduli space.
6. Using the ‘sewing property’ for these ‘meromorphic functions’ on the rigged moduli space to show that traces of products or iterates of intertwining operators for suitable vertex operator algebras satisfy differential equations with regular singular points and thus are absolutely convergent.
What did the meeting achieve?
Some of the problems mentioned above have been on our agenda for more than 15 years. Due to all our previous research we are finally at the stage to solve these problems. Getting our whole team in one place and formulating these solutions was very exciting. We solved problems 1, 2 and 5, and have some new insight as to how to proceed on problems 3, 4 and 6. A full technical summary can be found in the team’s official ‘final report’ of the meeting on the BIRS website. http://www.birs.ca/events/2012/research-in-teams/12rit139
Do you expect any publications to result from the meeting?
At least five papers related to our meeting will ultimately be published over the next several years. So far we have submitted one paper to a top ranked journal: Radnell, D., Schippers, E. and Staubach, W., ‘Dirichlet space of multiply connected domains with Weil-Petersson class boundaries.’ We have been waiting over 15 months for an answer from a journal for our previous publication; one has to be very patient in mathematics.
What did the BIRS environment contribute to the research?
BIRS has excellent facilities, and the peaceful location in the mountains puts you in a very calm and focused state of mind. The housing, dining, and work rooms are all within a few minutes’ walk, and so you spend all waking hours with your colleagues. The result is that you end up discussing research problems over breakfast. This level of intensity with colleagues from different areas of mathematics is what is needed to make real progress. There are only a handful of centers existing worldwide that offer this kind of environment.
Did your research findings surprise you?
There are always surprises in research - sometimes bad and sometimes good. The surprises at BIRS were mostly good, because we discovered that some technical results we had proved earlier could be used to solve what at first seemed like an unrelated problem. The beauty in mathematics often lies in the unexpected connection between disparate ideas.
What are your future research plans?
My long-term research goals were partly formulated already in my PhD thesis (2003) and their full completion is still several years away. As this work is completed we have been finding deep connections to other problems in mathematics, so new research directions are constantly arising.
Teresa Compton is a Grants Writer at the Office of Research and Graduate Studies at American University of Sharjah.