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

我们已经通过一些关键的例子取得了进展，这些例子展示了有趣和不平凡的现象。在《Cecotti和Vafa的TT*方程的非线性偏微分方程组》(M.Guest和C.-S.Lin，J.Reine Angew)中。数学，2012，出版)建立了TT*-Toda方程的一族光滑解的存在性。这是一项技术突破：P.D.E。方法非常适合非紧凑的情况，在这种情况下，标准的循环组方法失败。在《Cecotti和Vafa I.Stokes数据的TT*方程的等长单调方面》(M.Guest，A.ITS和C.-S.Lin，ARXIV：1209.2045)中，通过将解的全局光滑性与相关线性方程的单调数据(Stokes数据)联系起来，取得了第二个技术突破。这个Stokes数据是针对TT*-Toda方程的所有全局光滑解进行显式计算的。我们期望这些技巧也能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.