For GAP 2013, we have decided to focus on the overarching theme of wall-crossing and integrable systems, which includes several related subthemes. What follows is a short overview of the chosen theme and some of its subthemes.

Wall-crossing and integrable systems

In the last five years, a discovery in quantum field theory about the counting of stable configurations of particles and black holes has shown itself to be an extremely powerful mathematical tool, with ramifications far beyond its initial applications in physics. The discovery concerns a "wall-crossing formula" which explains how certain numerical invariants, such as the number of stable configurations in a system, change when the parameters describing the system pass through a "wall", which is a real hypersurface in the parameter space. The change is discontinuous, but can be controlled by a remarkable formula of Maxim Kontsevich and Yann Soibelman, which involves a potentially infinite product in the group of symplectomorphisms of a torus, where the subtle issue of ordering is controlled by various stability conditions.

The discovery of the wall-crossing formula was inspired by several explicit studies of the jumping behaviour of certain well-known invariants such as the Gromov-Witten invariants and Donaldson-Thomas invariants. Building on earlier work of Cecotti-Vafa, Seiberg-Witten, and Douglas, the recent developments were triggered by work of many leading mathematicians and physicists, most notably Ooguri-Strominger-Vafa, Denef-Moore, Joyce-Song, Bridgeland, and Auroux. Through recent work of Gaiotto, Neitzke, and Moore, the physical meaning of the formula was explained in a remarkable way using the theory of supersymmetric sigma models: they interpret the jumping of invariants as a change in the behaviour of instanton contributions to a hyperKähler metric; the wall-crossing formula is then viewed as a sum of multi-instanton contributions to this metric. In this way, enumerative invariants in algebraic geometry become involved in the definition of a smooth, non-algebraic object, namely a Riemannian metric.

This discovery has brought tremendous excitement to several fields of mathematics: perhaps the greatest promise that it holds is the possibility that, using the wall-crossing ideas, it might actually be possible to explicitly determine the famous hyperKähler metrics on the Hitchin moduli space of Higgs bundles - an object of central study in differential geometry, algebraic geometry, and number theory through the geometric Langlands programme. These metrics, like the metrics proven to exist by Yau, were firmly believed to be "transcendental" and impervious to explicit determination. Now it appears to be possible to obtain explicit expressions for them using enumerative algebraic invariants such as the Donaldson-Thomas invariants.

Because of the fact that this theory touches upon foundational topics in both physics and mathematics, it has had a remarkably broad impact in a short amount of time. Below is a short list of the main areas which should be represented in an overview of the topic.

1. Donaldson-Thomas invariants.

The theory of Donaldson-Thomas invariants concerns the counting of stable objects in triangulated categories. Through the work of Bridgeland and Joyce--Song, it is a fundamental technique in the study of special Lagrangian subvarieties of Calabi-Yau 3-folds, which in turn are central in the Mirror Symmetry programme. The categorical approach, championed by Kontsevich-Soibelman, places the theory on a solid mathematical footing. Using the wall-crossing formula, the explicit determination of Donaldson-Thomas invariants in key examples has been made more tractable, leading to much current activity.

2. Cluster algebras.

Cluster algebras originate in the study of positivity in algebraic combinatorics. But in recent years, through work of Fomin, Zelevinsky, and others, they have had a major impact in differential and algebraic geometry, because of the fact that many interesting moduli spaces appear to have a cluster algebra structure. The most spectacular advance in this direction was made by Fock and Goncharov, who showed that certain character varieties and moduli spaces of connections have cluster coordinates. These cluster coordinates are, in simple cases, involved in the conjectural wall-crossing description of the metrics on these moduli spaces.

3. hyperKähler metrics.

Hitchin's hyperKähler metric on the moduli space of Higgs bundles is a key ingredient in the understanding by Hausel and Thaddeus of the Langlands mirror symmetry among these moduli spaces. This hyperKähler metric has been generalized to many other moduli spaces of singular connections by Boalch and Biquard. The immediate importance of these generalizations is that we now have several examples of hyperKähler moduli spaces in low dimension. Through work of Cherkis, we have an increasingly detailed understanding of these examples, and so they provide particularly good test cases for the conjectures of Gaiotto-Moore-Neitzke.

4. Quadratic differentials and spectral networks.

In work of Klemm-Lerche-Mayr-Vafa-Warner, a beautiful correspondence was established between counting special Lagrangian submanifolds in a particular class of Calabi-Yau three-folds and counting trajectories of quadratic differentials on associated Riemann surfaces. For this reason, Gaiotto-Moore-Neitzke have recently begun a potentially vast programme studying the impact of the wall-crossing formula on the classical theory of quadratic (and higher) differentials on curves. Their theory of spectral networks consists of a careful study of certain trajectories on the surface and how to enumerate them. In this way, they hope to realize the abelianization programme of Atiyah and Hitchin in their study of Chern-Simons theory. This represents a major opportunity for new ideas in a most classical topic: Teichmuller theory and the study of moduli of algebraic curves.

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