A workshop intended to bring together researchers into Thurston theory and those from transcendental dynamics, as well as PhD students in holomorphic dynamics. There will be a series of introductory talks to introduce and motivate the central conjectures of Transcendental Thurston Theory, as well as research talks describing the current state of the art. The workshop is kindly funded by the Dame Kathleen Olleranshaw Trust at the University of Manchester.

Prior knowledge of transcendental dynamics or of Thurston theory is not required.

Workshop description:

• Holomorphic dynamics concerns the study of systems whose state is described by a single complex variable, and that evolve over time according to a rule described by a holomorphic (complex differentiable) function. The field was founded by Fatou and Julia at the beginning of the 20th century, underwent a revival in the 1980s and continues to be the subject of an intensive world-wide research effort. Much work in the 1980s and 1990s centred on the case where the rule is described by a polynomial or rational function, with particular focus on the study of quadratic polynomials, giving rise to the famous Mandelbrot set. Among the foundational results of the 1980s is Thurston’s theorem, which characterises postcritically finite rational functions, i.e., those whose critical values only pass through finitely many different states under repeated application of the rule.

• Transcendental dynamics studies systems whose evolution rule is described by a function that is not rational or a polynomial, and has transcendental singularities at the boundary of its domain of definition. This field also has a long history, going back to a seminal article of Fatou in 1926. In the last two decades, the field has seen some spectacular advances, and a number of important open questions have been resolved. As a result, a detailed understanding of the dynamics of some very large classes of transcendental entire functions has come into reach, in particular those belonging to the important Speiser class. However, obtaining an analogue of Thurston’s theorem in this context remains a difficult unresolved challenge; this is becoming a bottleneck impeding further progress. 

• This area of study has become known as Transcendental Thurston Theory. A considerable amount of work on the problem was done by Prof. Dierk Schleicher (Univ. of Marseille) over the past two decades. He solved the problem for exponential maps in joint work with Hubbard and Shishikura. More recently, Schleicher and his doctoral students have made advances in a number of directions, but the problem remains wide open in general, and will benefit from new approaches. 

The goals of this workshop are: 

A) To bring together researchers with a background on Thurston theory, in the polynomial, rational and transcendental setting, with a focus on collecting as many different approaches as possible. There are now many ways to approach Thurston's theorem for polynomials, and we expect that bringing these together with experts in transcendental dynamics, we will provide a critical step towards the extension of Thurston's theorem to transcendental functions. 

B) To introduce researchers and doctoral students studying holomorphic dynamics to the techniques and problems of transcendental Thurston theory. 

Registration link.

Please register at the above link as soon as possible. If you would like to request financial support (see below), please make sure to do so before July 31.

Registration for online attendance.

If you cannot make the workshop in person, but may wish to attend remotely, please register your interest in this using the above link.

Schedule.

Workshop dinner:

There will be a workshop dinner in honour of Professor Schleicher's 60th birthday. The dinner will take place at 7pm on Thursday August 28 in the Laureate Restaurant at the Hyatt Regency Hotel Manchester, 55 Booth St W. (The hotel is a short walk from the Alan Turing Building.)

Peak district walk:

Monday August 25 is a public holiday in the UK, but for participants who arrive already on Sunday, we are planning to arrange a joint walk in the Peak District on Monday August 25, weather permitting. The walk will be a circular trail from Disley station to Lyme Park and back. We will leave at 14.30 from Disley station; participants travelling from Manchester will be taking the train at 13.51 from Manchester Piccadilly in the direction of Buxton and get off at Disley. You will need to book your ticket prior to taking the train.

Practical information:

The University of Manchester campus map shows the location of the Alan Turing Building (No 46), the Hyatt Regency Hotel for the workshop dinner (No 124) and Piccadilly and Oxford Road railway stations. Participants for whom we have booked accommodation will be staying in the Pendulum Hotel (No 11 on the map) and should already have received a booking confirmation. 

Organisers:

Konstiantyn Drach (Barcelona)

Daniel Meyer (Liverpool)

Malavika Mukundan (Boston)

Nikolai Prochorov (Marseille)

Bernhard Reinke (Marseille)

Lasse Rempe (Manchester)

Richard Webb (Manchester)

Financial support:

Limited financial support is available, thanks to support by the Dame Kathleen Ollerenshaw Trust. Participants who require financial support should indicate this on the registration form.

Introductory lectures:

There will be a series of approximately introductory talks, introducing different aspects of transcendental Thurston theory. If you are happy to give one of these talks, please indicate this on the registration form. A provisional list of topics is as follows:

• Post-critically finite polynomials: Dynamics and combinatorics

• The Speiser class (Walter Bergweiler)

• Quasiconformal maps (Malavika Mukundan)

• Teichmüller theory and quadratic differentials (John Hubbard) 

• Post-singularly finite (PSF) entire functions: Dynamics and combinatorics (Bernhard Reinke)

• Statement and consequences of Thurston's theorem for polynomials and rational maps (Bernhard Reinke)

• Thurston theory for PSF entire functions (Konstantin Bogdanov)

• Thurston iteration (Nikolai Prochorov)

• Finite-Type Maps (Adam Epstein)

• Proof(s) of Thurston's theorem for polynomials / rational maps (Mitsuhiro Shishikura)

• Proof of Thurston's theorem for exponential maps (Mitsuhiro Shishikura)

Research talks:

In addition to the introductory lectures, there will be a number of research talks (please see the schedule for more details).

Confirmed speakers:

Walter Bergweiler

Konstantin Bogdanov

Dzmitry Dudko (remote talk)

Adam Epstein

Darragh Glynn

John Hubbard

David Martí-Pete

Julia Münch

Nikolai Prochorov

Bernhard Reinke

Mitsuhiro Shishikura

One Day Function Theory Meeting

Particularly for international participants, it may be of interest that the the yearly One Day Function Theory Meeting will take place in London on the Monday following the workshop (September 1, 2025). This year's speakers will be Prof. Mario Bonk (UCLA), Dr Vasiliki Evdoridou (Open University), Prof. Caroline Series (University of Warwick) and Dr Nina Snigireva (University of Galway). There will also be two talks by PhD students that are announced later.

More information is available at

https://sites.google.com/view/onedayfunctiontheorymeeting/home