Conformally invariant Gaussian analytic functions, holomorphic correlations, and operator symbols of contractions
Time: Thursday, 6 March 2025 at 16:15. Note the exceptional weekday!
Place: Over Zoom and in colloquium watch events at mathematics departments (see bottom of page)
Speaker: Håkan Hedenmalm, KTH Stockholm
Chaired by: Risto Korhonen, University of Eastern Finland
Abstract:
The classical Dirichlet space of holomorphic functions on the unit disk is invariant under Möbius transformations, except that it is equipped with a marked point where the functions vanish. Associated with such a Dirichlet space with a marked point, we get a Gaussian analytic function in a canonical fashion. Then, if we take two such Gaussian analytic functions, say with the same marked point at the origin, we consider the holomorphic correlation function of the two. It turns out to be given in terms a contraction on the area-L2 space on the disk. More precisely, we obtain the operator symbol of the contraction. Some contractions on L2 are perhaps more natural than others. For instance, we can consider the multiplication operator associated with a Beltrami coefficient μ. But we can also consider Grunsky operators, which are prominent in the theory of conformal mapping. We obtain a characterization of the operator symbols of Grunsky operators as solutions to a nonlinear wave equation. We also study the average growth of the L2 means of the operator of a general contraction.
Håkan Hedenmalm is Professor of Mathematics at KTH Stockholm. He has developed a habit to look at classical topics, with the aim to find something new and interesting.
Where to watch?
The colloquium will be broadcast over Zoom on Thursday, March 6, at 16:15.
The Zoom link will appear here on the day of the event.
There will be colloquium watch events at most Finnish universities' mathematics departments:
- Aalto University: U6 (Otakaari 1)
- LUT University: 2411 (Lappeenranta)
- Tampere University: Tietotalo TC210 (Hervanta)
- University of Eastern Finland: M107 (Joensuu)
- University of Helsinki: Exactum B121 (Kumpula)
- University of Jyväskylä: MaD302
- University of Oulu
- University of Turku: Quantum QM3
- University of Vaasa
- (Åbo Akademi: No watch event this time)
The talk, excluding audience questions, will be recorded. Recordings of previous colloquium talks are available on FMS' YouTube channel.