Non-Abelian gauge fields end Painleve functions

这项研究开始发展“Painleve方程是超几何方程的非阿贝尔类比”的指导原则。在研究的最后，我们得到了Painleve分析的一个新方向：“单调可解的Painleve函数”。这类解将在未来Painleve分析的研究中发挥重要作用。在这三年中，许多其他领域都取得了不同的进展。Painleve分析不再停留在数学物理中，而是涉及到多个领域，如Number理论、组合理论、概率理论等。现在，这些场的反馈是互换的。单调可解性是这种相互作用中的一个新的关键词。也就是说，在应用中起重要作用的Painleve方程(或保持单调变形的方程)的解大多不是Uemura意义下的经典解。但与这些解对应的线性方程的单行性是…更具可解性。单调函数的可解性可以发展为Painleve函数本身的可解性。作为这项研究的衍生，我提出了两个重要的事情。利用非结合代数得到了四阶Darboux-Halphen方程的可解性条件。在秩为3的情况下，已知Darboux-Halphen方程化为超几何方程的相似条件。第二，我们确定了D_7型Painleve方程的特解，从而完全分类了Painleve方程的Uemura意义的超越经典解。Painleve VI代数解的分类仍然存在，在宣布了Painleve I方程不可约的一个不完善的证明后，它已经过去了100年。我们不能对内村意义上的所有经典解进行分类，但我认为我们已经找到了Painleve分析的新方向。较少

This research started to develop the guiding principle "The Painleve equations are non-Abelian analogue of the hypergeometric equations". At the end of this research we obtain a new direction in Painleve analysis : "monodromy-solvable Painleve functions". This class of solutions will play an important role in future study of the Painleve analysis.In this three years, various progress was developped in many other fields. The Painleve analysis does not remain in mathematical physics, but has a relation on various fields, such as nuber theory, a combination theory, probability theory and more. Now feedbach from those fields is performed conversely.The monodromy solvablity is a new keyword in such interaction. Namely, most of solutions of the Painleve equations (or the equations of monodromy preservation deformations) which play the important role in applications are not classical solutions in Umemura's meaning. But the monodromy of the linear equations corresponding to these solutions is … More solvable. The solvablity of the monodromy may be developped into the solvablity of the Painleve functions themselves.As research derived from this research, I raise two important things. We obtain the conditions of the solvablity of the Darboux-Halphen equations of rank 4 using non-associative algebras. In the case of the rank 3, the similar conditions when the Darboux-Halphen equations reduce to the hypergeometric equations were known. An application to Painleve analysis have opened by this new result in the case of the rank 4. In the second, we determined specials solutions of the Painleve equation of D_7 type, Thus the transcendent classical solutions of Umemura's meaning of the Painleve equations were completely classified. Classification of algebra solutions of the Painleve VI still remains.After an imperfect proof on the irreducibility of the Painleve I equation is announced, it passed 100 years. We could not classify all of classical solutions of Umemura's meaning, but I think that we have found out a new direction of the Painleve analysis. Less