Exploitation of new relations between differential geometry and quantum cohomology in the context of integrable systems

我们在一些关键的例子上取得了进展，这些例子展示了有趣的和不平凡的现象。在“Cecotti和Vafa的tt*方程的非线性偏微分方程方面”(M. Guest和C.-S.；林J.雷恩·安纽。数学。证明了tt*-Toda方程光滑解族的存在性。这是一个技术上的突破：p.d.e.方法非常适合非紧化情况，而标准环路群方法则无法实现。在“Cecotti和Vafa I. Stokes数据的tt*方程的同构方面”(M. Guest， A. Its和c . s .；Lin, arXiv:1209.2045)，通过将解的全局光滑性与相关线性方程的一元数据（Stokes数据）联系起来，取得了第二个技术突破。对于tt*-Toda方程的所有全局光滑解，明确地计算了Stokes数据。我们期望这些技术将适用于微分几何中的其他问题。

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.