Dynamical Systems

Bridgeland Stability Spaces and Complex Function Theory

Univesity of Liverpool, 19-21st September 2023

In 2007, Bridgeland introduced the notion of "stability conditions" on triangulated categories. The space of all such stability conditions, called the stability space of the category, comes with a natural complex structure and has been studied extensively in the intervening years.

In certain cases, stability spaces (or certain slices thereof) are two-dimensional complex manifolds with a natural one-dimensional complex action. Taking a quotient, one obtains a Riemann surface. There are indications that the complex-analytic properties of this surface (such as the conformal type of the surface: is it hyperbolic or parabolic) reflects intrinsic properties of the associated category.

The question of how to determine the type of a simply-connected Riemann surface (the "type problem") and similar problems are long-standing subjects of classical complex function theory. It appears that applying the methods of function theory may lead to fruitful new insights into stability spaces. In turn, stability spaces may present new examples of Riemann surfaces and associated functions with interesting function-theoretic properties.

The goal of this informal workshop is to bring together researchers from both sides - classical function theory / complex analysis, on the one hand, and the study of stability conditions, on the other. The goal is to give an introduction to (2-dimensional) stability spaces, accessible to complex analysts, and to the type problem and related topics, accessible to algebraic geometers and category theorists. The goal is to develop a common language and identify promising avenues of research for investigating these intriguing spaces.

We are very happy that Prof. Alexandre Eremenko (Purdue University), a world expert on classical function theory and Leverhulme Visiting Professor at Liverpool, will give a two-part minicourse as an introduction to the type problem from the point of view of complex analysis. We are also delighted that Prof. Tom Bridgeland (Sheffield) will be participating in part of the workshop.

Interested researchers are welcome to attend either in person or remotely (we plan to hold the workshop in a hybrid format). This workshop is organised jointly by the Algebraic Geometry and Dynamical Systems research groups at the University of Liverpool; the organisers are Prof. Lasse Rempe and Dr Jon Woolf.


All talks in Room 106, Mathematics Building, University of Liverpool, Peach Street (L69 7ZL).

Tuesday 19/09

10.00 - 11.00 Woolf: Bridgeland stability conditions: an introduction
11.00 - 11.30 Questions and discussion
11.30 - 12.30 Rempe: Conformal modulus and quasiconformal mappings

12.30 - 14.00: Lunch and discussion

Wednesday 20/09

10.00 - 11.00 Woolf: Bridgeland stability conditions: phases and masses
11.00 - 11.30 Questions and discussion
11.30 - 12.30 Eremenko: An introduction to the type problem, I

12.30 - 14.00 Lunch and discussion

14.00 - 15.00 Woolf: Auto-equivalences and mass-growth
15.00 - 15.30 Questions and discussion
15.30 - 16.30 Eremenko: An introduction to the type problem, II

Thursday 21/09

10.00 - 12.00 Discussion, questions, and further steps
12.00 - 14.00 Lunch (and final discussions)

GLEN Workshop

Following the workshop, on Thursday and Friday there will be a meeting of the GLEN UK Algebraic Geometry workshop, with Prof. Bridgeland giving the opening talk at 2pm on Thursday afternoon. See https://pcwww.liv.ac.uk/~arizzard/GLEN/GLEN2023.html.


Jon Woolf (University of Liverpool)
Bridgeland stability conditions: an introduction

Abstract: The first goal of this talk is to introduce the notion of Bridgeland stability condition, motivated by the problem of classifying modules over a finite-dimensional algebra. One of the key features of the theory is that the space of all stability conditions (on a given triangulated category) has a natural complex manifold structure. This arises from the charge projection, which is a local homeomorphism from the stability space to a complex vector space. The second goal is to give some simple 2-dimensional examples of these stability spaces, and their charge projections.

Lasse Rempe (University of Liverpool)
Conformal modulus and quasiconformal mappings

Abstract: In this talk, we will introduce two concepts of complex analysis that are essential in the study of the type problem (and for Prof. Eremenko's minicourse). The first is the notion of the conformal modulus of an annulus (or, more generally, a familiy of curves), which measures the "thickness" of the annulus, in a manner that is invariant under conforrmal mappings. In particular, a punctured disc has infinite modulus, while a bounded round annulus has finite modulus. Hence the notion of modulus allows one to distinguish conformally between a puncture and a non-degenerate boundary component of our surface, and thus between the disc and the plane. The second topic of the lecture is quasiconformal mappings, which are homeomorphisms that are not necessarrily conformal, but distorrt angles (and moduli) by only a bounded amount. These mappings are more flexible than conformal mappings, but still share many properties with the latter. In particular, a quasiconformal map does not change the conformal type of a surface, and hence by building a quasiconformal mapping of a given surface we may be able to determine its type. I will illustrate both ideas with a simple example.

Jon Woolf (University of Liverpool)
Bridgeland stability conditions: phases and masses

Abstract: A Bridgeland stability condition on a triangulated category assigns masses and phases to the objects of the category. The distribution of phases is closely related on the one hand to the algebraic complexity of the heart of the stability condition and on the other to the geometry of the charge projection. The goal of this talk is to explain this link and illustrate it via 2-dimensional examples.

Alexandre Eremenko (Purdue University)
An introduction to the type problem, I

Abstract: TBA

Jon Woolf (University of Liverpool)
Symmetries of stability spaces and mass growth

Abstract: The symmetries of a triangulated category naturally act as symmetries of its stability space. One way to study this action is by considering the dynamical behaviour of masses and phases. In particular, Ikeda introduced the notion of mass growth which can be seen as an analogue of the growth of the length of a curve on a surface under the repeated application of a diffeomorphism. This is part of a wider analogy between stability spaces and Teichmuller spaces.

Alexandre Eremenko (Purdue University)
An introduction to the type problem, II

Abstract: TBA