TITLES AND ABSTRACTS

Transcription

1 TITLES AND ABSTRACTS Florian Beck (Hamburg) Title: Hitchin integrable systems and Calabi-Yau threefolds Abstract: Integrable systems are often constructed from geometric and/or Lietheoretic data. Two important example classes are Hitchin systems and Calabi- Yau integrable systems. A Hitchin system is constructed from a compact Riemann surface together with a complex Lie group (with mild extra conditions). In contrast, Calabi-Yau integrable systems are constructed from (a priori) purely geometric data, namely certain families of Calabi-Yau threefolds. In this talk, I will describe a non-trivial relation between these two classes of integrable systems which is based on work by Diaconescu-Donagi-Pantev and my own. Fran Burstall (Bath) Title: Isothermic surfaces and integrable systems Abstract: The study of isothermic surfaces was initiated more than 100 years ago but there are still interesting things to say about them. In this talk, I shall describe both classical and modern aspects of the theory and discuss the contributions of Darboux, Bianchi and Pedit, among others. Josef Dorfmeister (TU Munich) Title: Minimal Surfaces in the Three-Dimensional Heisenberg Group Abstract: The goal of this talk is a construction scheme for all minimal surfaces in the Heisenberg group Nil 3. We will start by following the twistor approach, arriving at the non-linear Dirac equation and the Lax pair of Berdinskii and Taimanov. From here we will reproduce the characterization of never-vertical minimal surfaces by the harmonicity of their normal Gauss map as a map into the two-dimensional hyperbolic space. This will be used to explain a construction principle via loop groups which applies equally well to spacelike surfaces in Minkowski 3-space as to minimal surfaces in Nil 3. Using the close connection between these two surface classes we will present a simple proof for the solution to the Bernstein problem (reproducing a result of Fernandez and Mira). 1

2 2 TITLES AND ABSTRACTS Karsten Große-Brauckmann (TU Darmstadt) Title: New minimal surfaces in the 3-sphere Abstract: In 1970 Lawson constructed complete embedded compact minimal surfaces in the 3-sphere by solving a Plateau problem for a geodesic polygon and extending it by Schwarz reflection. Due to the lack of methods, since then explicit examples have only been constructed in 1988 by Karcher/Pinkall/Sterling and, more recently, by Choe/Soret. In current work extending a PhD thesis by T. Alex we construct further examples, using the method of Lawson or Choe/Soret. We make use of the structure of the 3-sphere as a Riemannian fibration with nonzero bundle curvature, given by the Hopf fibration. Thus our method applies to other homogeneous 3-manifolds as well. Kohei Iwaki (Nagoya) Title: Exact WKB analysis and Voros symbols Abstract: Exact WKB analysis, developed by Voros et.al., is an effective method for global study of ODEs with a small parameter defined on a Riemann surface. It is known that, for 2nd order meromorphic ODEs with a small parameter, the monodromy matrices of the (Borel resumed) WKB solutions are described by infinite sum of integrals over the spectral curve (Aoki-Kawai-Sato-Takei). These integrals are called Voros symbols. In this talk I?ll show several properties of WKB solutions and how the Voros symbols describes their monodromy. If time allows, I?ll also show a cluster algebraic structure in the exact WKB analysis (joint work with Tomoki Nakanishi). Shimpei Kobayashi (Hokkaido) Title: Minimal surfaces with symmetries in the three-dimensional Heisenberg group Abstract: This is a continuation of the talk given by J. Dorfmeister. Since the normal Gauss map of a minimal surface in the three-dimensional Heisenberg groupcan be considered as a harmonic map into the hyperbolic two space, which is a symmetric space, one can apply the loop group method, that is, the so-called generalized Weierstrass type representation.i will discuss symmetries of minimal surfaces by using the loop group method. In particular, we will classify all equivariant minimal surfaces. This is a joint work with J. Dorfmeister (TU Munich) and J. Inoguchi (Tsukuba).

3 TITLES AND ABSTRACTS 3 Panagiotis Konstantis (Marburg) Titel: Singular oscillatory integrals and symplectic reductions Abstract: We study the topology of symplectic reductions for a hamiltonian action on a symplectic manifold from an analytical viewpoint. In this talk I will show how to use singular oscillatory integrals to connect the cohomology of a symplectic reduction to the equivariant cohomology on the symplectic manifold. This is a generalization of a theorem of Jeffrey and Kirwan, where in our case the symplectic manifold has not to be compact and where the symplectic reduction is not smooth. In the end we will see, that this talk has nothing to do with Higgs bundles, harmonic maps or integrable systems. Rob Kusner (UMASS, Amherst) Title: Morse Index and Willmore Stability of Minimal Surfaces in Spheres Abstract: Minimal surfaces in the round n-sphere are prominent examples of surfaces critical for the Willmore bending energy W; those of low area provide candidates for W-minimizers. To understand when such surfaces are W-stable, we study the interplay between their Laplace-Beltrami, area-jacobi and W-Jacobi operators. We use this, e.g. to prove: 1) the square Clifford torus in the 3-sphere is the only W-minimizer among tori in the n-sphere; 2) the hexagonal Itoh-Montiel- Ros torus in the 5-sphere is the only other W-stable minimal torus in the n-sphere, for all n. We also show: 3) the Itoh-Montiel-Ros torus is a local minimum for the conformally-constrained Willmore problem, evidence for a recent conjecture of Lynn Heller and Franz Pedit. [This is joint work with Peng Wang from Tongji University in Shanghai, China, who visited UMass Amherst this past year.] Ernst Kuwert (Freiburg) Title: Willmore minimizers with prescribed isoperimetric ratio Abstract: We discuss the existence of surfaces of type S 2 minimizing the Willmore functional with prescribed isoperimetric ratio, and some asymptotics as the ratio goes to zero. Katrin Leschke (Leicester) Title: Transformations of minimal surfaces

4 4 TITLES AND ABSTRACTS Abstract: A minimal surface gives rise to three different families of flat connections: one can consider the families defined by the harmonic Gauss map and the harmonic conformal Gauss map, as well as the family which arises when viewing the minimal surface as an isothermic surface. In this talk I will compare various transformations which are given by parallel sections of each of these families, e.g., the Darboux transforms and the simple factor dressings. Hui Ma (Tsinghua, bejing) Title: Uniqueness of closed self-similar solutions to σk α -curvature flow Abstract: In this talk, we show that any closed strictly convex self-similar solutions to general σk α flow for α >= 1/k must be a round sphere. In fact, we prove a uniqueness result for any strictly convex closed hypersurface in R n+1 satisfying F + C = X, ν, where F is a positive homogeneous smooth symmetric function of the principal curvatures and C is constant. The talk is based on the joint work with Shanze Gao and Haizhong Li. Cheikh Birahim Ndiaye (Basel, Howard U) Title: Recent progress on the constrained Willmore problem Abstract: In this talk, we plan to describe two joint works, one with Reiner Michael Schaetzle an other one with Lynn Heller, both on the conformally constrained Willmore problem.we will explain our solution of the constrained Willmore problem for rectangular conformal classes close to the one of Clifford torusnamely the square one-for which the solution was known since the 1982 work of LI and Yau. Furthermore, we will explain our solution for some non-rectangular classes close to rectangular ones which are enough close to the square class with an emphasis on the difference of the techniques used with respect to the rectangular case. Saskia Roos (MPI Bonn) Title: Dirac eigenvalues under codimension one collaps Abstract: After giving a characterization of a collaps of codimension one we study the behavior of Dirac eigenvalues in that situation. We show that there are converging eigenvalues if and only if there is an induced spin structure on the limit space. In addition, we determine the limit operator which corresponds to the limit spectrum.

5 TITLES AND ABSTRACTS 5 Markus Röser (Hannover) Abstract: Hypersymplectic geometry is the natural geometry of the moduli space of solutions to the gauge-theoretic harmonic maps equations a Riemann surface. These equations differ from Hitchin s self-duality equations by a sign. We shall investigate moduli spaces of solutions to the Nahm-Schmid equations, a system of non-linear ODEs arising from the harmonic map equations by dimensional reduction. We obtain hypersymplectic structures on certain open subsets of cotangent bundles and regular semi-simple adjoint orbits of complex semi-simple Lie groups by identifying these spaces with appropriate Nahm-Schmid moduli spaces. This is joint work with Roger Bielawski and Nuno Romao. Laura Schaposnik (UI, Chicago) Title: Higgs bundles, branes and applications. Abstract: We shall begin the talk by first introducing Higgs bundles for complex Lie groups and the associated Hitchin fibration, and recalling how to realize Langlands duality through spectral data. We will then look at a natural construction of families of subspaces which give different types of branes, and explain how the topology of some of these branes can be described by considering the Hitchin fibration. Finally, we shall give some applications of the above approaches in relation to Langlands duality, and other correspondences between integrable systems. Some of the work presented during the talk is in collaboration with David Baraglia, Steve Bradlow and Sebastian Heller. Title: Flows of CMC surfaces Nick Schmitt (Tübingen) Abstract: In this talk I discuss the generalized Whitham flow for CMC surfaces from an experimental point of view. In general, a CMC surface in the 3-sphere can be described by a family of flat connections (its DPW potential). I use a master DPW potential on the 4-punctured sphere which allows us to flow from the Clifford torus to all sorts of more complicated CMC surfaces including the Lawson surfaces, the Karcher-Pinkall-Sterling surfaces and generalizations thereof, and CMC tori with Delaunay ends. Jan Swoboda (LMU Munich) Title: The large scale geometry of the Higgs bundle moduli space

6 6 TITLES AND ABSTRACTS Abstract: In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiß and Frederik Witt on the asymptotics of the natural L 2 -metric G L 2 on the moduli space M of rank-2 Higgs bundles over a Riemann surface Σ as given by the set of solutions to the so-called self-duality equations { 0 = A Φ 0 = F A + [Φ Φ ] for a unitary connection A and a Higgs field Φ on Σ. I will show that on the regular part of the Hitchin fibration (A, Φ) det Φ this metric is well-approximated by the semiflat metric G sf coming from the completely integrable system on M. This also reveals the asymptotically conic structure of G L 2, with (generic) fibres of the above fibration being asymptotically flat tori. This result confirms some aspects of a more general conjectural picture made by Gaiotto, Moore and Neitzke. Its proof is based on a detailed understanding of the ends structure of M. The analytic methods used there in addition yield a complete asymptotic expansion of the difference G L 2 G sf between the two metrics, with leading order term having polynomial decay and a rather explicit description.