Leicester mini workshop, July 2018
Browse the talks and speakers who participated in our mini-workshop at the University of Leicester.
Discrete omega surfaces - a rough introduction
Omega-surfaces were discovered by Demoulin, and one of the motivations of studying Omega-surfaces arise from the fact that well-known classes of surfaces such as constant mean curvature surfaces, constant Gaussian curvature surfaces, and linear Weingarten surfaces are all examples of Omega-surfaces. Demoulin showed that Omega-surfaces are the class of surfaces that envelop a pair of isothermic sphere congruences, allowing one to apply the rich theory of isothermic surfaces to examine the properties of Omega-surfaces. Recent interest in integrable systems has led to renewed interest in Omega-surfaces, exemplified in works by Burstall, Cecil, Ferapontov, Hertrich-Jeromin, Musso, Nicolodi, etc..
In this talk, we look at discrete Omega-surfaces. First, we give different characterizations of discrete Omega-surfaces: via Konigs dual lifts, via Moutard lifts, and via gauge-theoretic description. Then, using the transformation theory developed for discrete isothermic surfaces, we discuss Darboux transformations of discrete Omega-surfaces. This talk is based on the joint work with Fran Burstall (University of Bath), Udo Hertrich-Jeromin (Technische Universität Wien), Mason Pember (Technische Universität Wien), and Wayne Rossman (Kobe University).
Symmetric calorons and the rotation map
For quite some time, there has been much interest in Yang-Mills instantons. These are manifested as finite-action anti-self-dual connections over some four manifold. They possess a very rich geometry, providing examples of hyperkahler metrics, and various applications to physics. A great deal of the progress in understanding the most basic examples of these objects has been through studying fixed points under isometries. In this talk, I will discuss calorons, which are instantons on S^1\times R^3. In particular, I shall present a classification result of cyclically symmetric calorons, where the cyclic groups considered are coupled to a non-spatial isometry known as the rotation map.
Dispersionless integrable systems in 3D and Einstein-weyl geometry
For several classes of second-order dispersionless PDEs, we show that their characteristic varieties define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of dispersionless PDEs can be seen from the geometry of their characteristic varieties. (Based on joint work with Boris Kruglikov).
Conformal volume, eigenvalue problems, and related topics
I will give a short survey about the classical inequalities for the first Laplace eigenvalue on Riemannian manifolds, such as the Reilly inequality, the Li-Yau inequality, and tell about related history and questions. I will then discuss results concerning their rsions for the higher Laplace eigenvalues and concerning estimates for the number of negative eigenvalues of Schrodinger operators.
Constant mean curvature surfaces and positon-like solutions
The classical Bianchi-Baecklund transformation for constant mean curvature surfaces in Euclidean 3-space has been studied by many researchers. In this talk, we introduce the method to construct positon-like solution of elliptic sinh-Gordon equation via successive Bianchi-Baecklund transformations with a single spectral parameter. We also show the recipe of the corresponding constant mean curvature surfaces of positon-like solutions.
Combescure transformations – a helpful tool for the study of Guichard nets
Two surfaces are Combescure transformations of each other if they have parallel tangent planes along corresponding curvature lines. The existence of particular Combescure transformations can be used to characterize various classes of integrable surfaces, e.g. the Christoﬀel transformation for isothermic surfaces. In this talk we investigate this concept for triply orthogonal systems and discuss the subclass of Guichard nets, which arise as special coordinate systems of 3-dimensional conformally ﬂat hypersurfaces.