Certain differential equations are studied from the point of view of geometry and the theory of integrable systems. The solution of such an equation can be simulated, and visualized, as the motion of a lattice of point masses with nonlinear interactions between neighbouring masses. An important example from quantum field theory is the tt*-Toda lattice, first studied by Cecotti and Vafa. Guest and Lin have obtained theoretical results on the existence of solutions, consistent with computer simulations. Other examples such as the Kuramoto lattice are being studied. Future applications, e. g. to the mathematical interpretation of synchronization, are anticipated.

We have made progress with some key examples, which demonstrate interesting and nontrivial phenomena. In "Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa" (M. Guest and C.-S. Lin, J. reine angew. Math., 2012, in press) the existence of a family of smooth solutions of the tt*-Toda equation was established. This was a technical breakthrough: p.d.e. methods are well suited to the noncompact case, where standard loop group methods fail. In "Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data" (M. Guest, A. Its, and C.-S. Lin, arXiv:1209.2045), a second technical breakthrough was made, by relating the global smoothness of the solutions to the monodromy data (Stokes data) of an associated linear equation. This Stokes data was computed explicitly for all globally smooth solutions of the tt*-Toda equation. We expect that these techniques will be applicable to other problems in differential geometry.

Research problems in geometry and topology were investigated using methods from the theory of integrable systems. These included research on the classification of constant mean curvature (CMC) surfaces, isoparametric manifolds, Lagrangian submanifolds, harmonic maps, and quantum cohomology. The principal investigator obtained results on the D-module structure of quantum cohomology, and began a systematic study of relations between quantum cohomology and integrable systems.Computer experiments played an important role in this project. MAPLE was used for computations with differential operato...

Results were obtained on the geometry and topology of harmonic maps and spaces of harmonic maps, especially in the case where the domain is a Riemann surface and the target space is a compact Lie group or symmetric space. Guest used a generalization of the Weierstrass representation for minimal surfaces to study harmonic maps from the two-dimensional sphere (or, more generally, harmonic maps of finite uniton number, from any Riemann surface) to the unitary group. Earlier results of Uhlenbeck, Segal, Dorfmeister-Pedit-Wu, Burstall-Guest were developed into an effective tool for describing su...

The main activity supported by this grant was research collaboration with Prof. Nan-Kuo Ho (National Tsing Hua University, Taiwan) on the topic stated in the title of the project. We studied a certain space of solutions of the tt* equations, originating in previous joint research of M. Guest, A. Its, and C.-S. Lin. We understood how this space can be regarded as a subspace of a moduli space of flat bundles or Higgs bundles. This gives a link with the Hitchin-Kobayashi correspondence, which is an important and active topic of current research spanning the boundary of geometry and mathematical physics. In particular we used the framework of P. Boalch, "Stokes matrices, Poisson Lie groups and Frobenius manifolds" Invent. Math. 146 (2001) 479–506. Further work in this direction is in progress and a joint article is in preparation.Prof. Nan-Kuo Ho visited Waseda University for the period 12-16 February 2014. A workshop "Symplectic geometry of moduli spaces of connections" was held on 14 February 2014 at Waseda University. The speakers and titles were: Tosiaki Kori (Waseda University) "A canonical pre-symplectic structure on the space of connections over a four-manifold and an induced pre-symplectic structure on the space of connections over a three-manifold"; Yuji Hirota (Keio University) "On prequantization of Dirac manifolds"; Hokuto Konno (Waseda University) "The moduli space of flat SU(2)-connections on a surface with boundary: an example"; Martin Guest (Waseda University) "Linear and nonlinear convexity, and the relation with singular connections on surfaces"; Nan-Kuo Ho (National Tsing Hua University, Taiwan) "On the moduli space of singular connections: a survey".

Research was carried out on several sub-projects related to differential geometry and integrable systems. This involved researchers in Japan as well as researchers in foreign countries, especially Germany and Taiwan.Guest made two visits to Mannheim University (Germany) in order to work with Claus Hertling on the tt* equations. Progress was made on describing the moduli space of solutions in the simplest nontrivial case, namely the 3rd Painleve equation, and a joint article is in preparation. As part of the second visit, Guest spent 2 days at the Technical University of Berlin in order to collaborate with Alexander Its, as part of an ongoing project on the Riemann-Hilbert approach to the tt* equations.A workshop on "Isomonodromic deformations and related topics" was held at Waseda University on 22-23 November 2013. The speakers and titles were Shinobu Hosono (Tokyo University) "Differential rings over the moduli spaces of Calabi-Yau manifolds II"; Martin Guest (Waseda University) "On the Riemann-Hilbert problem for the tt*-Toda equations"; Masa-Hiko Saito (Kobe University) "Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line"; Daisuke Yamakawa (Tokyo Institute of Technology) "Fourier-Laplace transform and isomonodromic deformations"; Kazunori Miyazaki (Kobe University) "On compactifications of moduli of unramified irregular singular connections and Okamoto-Painleve pairs"; Makoto Miura (Tokyo University) "Hibi toric varieties and mirror symmetry". The workshop was attended by approximately 20 people. Web page: http://www.f.waseda.jp/martin/conf/2013isomonodromy.htmlAn informal workshop was held at Waseda University on 2 August 2013, with talks by M. Guest, H. Iritani (Kyoto University), A. Strangeway (Imperial College, London). Guest attended and gave an invited talk at the 11th Pacific Rim Geometry Conference, held at Fudan University, Shanghai, on 10-13 December 2013. Cooperation with researchers in Taiwan is a significant aspect of this project, and Ting-Jung Kuo (National Taiwan University) and Shu-Cheng-Chang (National Taiwan University) visited Waseda University for the period 2-9 November 2013. They gave informal talks at Waseda, and seminar talks at Tokyo Metropolitan University on 8 November 2013 organised by Takashi Sakai (Tokyo Metropolitan University). Research and secretarial assistance in connection with this project and proposals for Kakenhi applications was also supported by this project.