Integrability in Geometry and Mathematical Physics

Hausdorff Trimester Program

January 2 - April 30, 2012

Organizers: Franz Pedit, Ulrich Pinkall, Iskander A. Taimanov, Alexander Veselov, Katrin Wendland

The past two decades have seen substantial developments in the theory of integrability, to the extent that integrable systems nowadays comprise a vital ingredient in algebraic, di fferential and enumerative geometry as well as in mathematical physics. Contributions range from Dubrovin's pioneering work on Frobenius manifolds over Hitchin's discovery of integrability of moduli spaces of stable vector or Higgs bundles up to results by Kontsevich and Okounkov-Pandharipande on the Witten conjecture and Gromov-Witten theory, and Krichever's proof of the Welter's trisecant conjecture. Furthermore, integrable systems play a pivotal role in the resolution of longstanding conjectures about the global structure of minimal, constant mean curvature and Willmore surfaces.

The most striking aspects of the theory of integrable systems, which however are becoming apparent only very slowly, feature integrability as a binding element between seemingly unrelated areas in mathematics. For example, the role of integrability in singularity theory is not well studied to the very day, although for quantum cohomology, the importance of both singularity theory and integrability is undisputed, such that direct links between the two must exist. On another account, the tt* geometry of topological quantum field theory also provides aspects of the Pedit-Dorfmeister-Wu construction in the context of Willmore surfaces.

This Hausdorff Trimester Program brought together some of the world leading scientists in the study of integrable systems in geometry and quantum field theory. One main aspect of the program was the aim to bridge between the research topics, to spark interactions, and to shed light on the binding role that integrability can play for all the themes involved.