The conference is on complex analysis, in the broad sense, with connections to, among other topics, harmonic analysis, statistical physics, operator theory, and dynamical systems. It features a strong list of international researchers.

All talks are in room 332A, matematikcentrum.

Wednesday 3rd of June

9:30 - 10:00 Opening and coffee

10:00 - 10:50 Athanasios Kouroupis

On the cusp formation in the Hele-Shaw flow

Abstract:

The Hele–Shaw flow is a fluid dynamics model describing the evolution of a viscous fluid droplet confined between two parallel plates separated by a narrow gap, where the motion is driven by the injection or withdrawal of fluid through a point source. Under mild assumptions, the flow can be represented by a family of univalent functions (f_t) satisfying a Löwner–Kufareff equation. In this talk, we will discuss the formation and resolution of cusp singularities in the Hele–Shaw flow. This talk is based on joint work in progress with A. Nishry and A. Wennman.

11:00 -  11:50 Joakim Cronvall

A direct approach to soft and hard edge universality for random normal matrices

Abstract:

The random normal matrix model is a two-dimensional point process describing log-correlated particles in a confining field. The statistics are described by the reproducing kernel of a weighted polynomial Bergman space. Typically, one studies the kernel using asymptotics of orthogonal polynomials. In many interesting situations, such as when the eigenvalues are confined by a hard wall or when the eigenvalues concentrate on disconnected sets, this has turned out to be complicated. In this talk, I will discuss an alternative approach that altogether avoids orthogonal polynomials. Instead we work with Hilbert spaces of entire functions using Paley-Wiener type theorems and potential theory. This approach gives several new universality results and shows in particular how the microscopic behavior of the kernel depends on the local potential theory. 

Based on joint work with Aron Wennman.

12:00 - 14:00 Lunch

14:00 - 14:50 Olof Rubin

Chebyshev polynomials for Jordan arcs

Abstract:

Given a compact set K ⊆ C containing infinitely many points, there is a unique monic polynomial of degree n that minimizes the uniform norm on K. This polynomial is called the Chebyshev polynomial of degree n for K. A classical problem is to understand how the geometric properties of K are reflected in the asymptotic behavior of the associated Chebyshev polynomials as n grows. This picture is more or less complete when K consists of smooth Jordan curves that are mutually disjoint. Faber considered the case of a single analytic Jordan curve in 1920, and Widom considered several curves in 1969. Chebyshev polynomials for Jordan arcs have proven more elusive. Widom conjectured in 1969 that the interval should provide the model behavior in this setting. However, this conjecture was shown to be false by a counterexample involving arcs on the unit circle. Instead, the two sides of the arc turn out to play a crucial role. In this talk, I will explain recent progress toward a revised conjecture by Christiansen, Simon, and Zinchenko. This is based on joint work with Benedikt Buchecker, Benjamin Eichinger and Aron Wennman.

15:00 - 15:50 Mats Bylund

Talk: TBA

19:30 Conference dinner at Mat och destillat

Thursday 4th of June

9:30 - 10:00 Coffee

10:00 - 10:50 Ole Brevig

Revisiting the proofs of two Riesz results.

Abstract:

I will present simplifications of the following textbook proofs:

1. The proof of the F. & M. Riesz theorem on analytic measures based on the Poisson integral of the measure.

2. The proof of the M. Riesz theorem for conjugate functions that uses interpolation between even integer exponents and duality.

In each case the improvement is obtained by reworking the technical estimate at the heart of the argument.

11:00 - 11:50 Chris Felder

Shift invariant subspaces, metric projections, and an extremal problem in $H^p$

Abstract:

Given a shift-invariant subspace in the classical Hardy space of the unit disk, the orthogonal projection of the unit constant function onto that subspace recovers, up to a multiplicative constant, the inner function which generates the subspace. In H^p, when $p \neq 2$, what happens when the unit constant is projected onto a shift-invariant subspace? The projection here is no longer orthogonal, rather a \textit{non-linear} metric projection. This talk will present an answer to this question, along with several related results and observations. 

12:00 - 14:00 Lunch

14:00 - 14:50 Kristian Seip

Talk: TBA