Non-Abelian gauge fields end Painleve functions

该研究开始提出“Painleve方程是超几何方程的非阿贝尔类似物”的指导原则。在研究的最后，我们得到了痛点分析的一个新方向：“单可解痛点函数”。这类解将在今后的painlevel分析研究中发挥重要作用。在这三年中，在其他许多领域都取得了各种进展。痛级分析并不局限于数学物理，而是与数论、组合论、概率论等各个领域都有关系。现在，来自这些领域的反馈是相反的。单可解性是这种相互作用中的一个新关键词。也就是说，在应用中起重要作用的painlevel方程（或一元保存变形方程）的大多数解都不是Umemura意义上的经典解。但是与这些解相对应的线性方程的单调性是……更可解的。一元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品，我提出了两个重要的问题。利用非结合代数得到了4阶Darboux-Halphen方程的可解性条件。在阶数为3的情况下，已知了Darboux-Halphen方程化简为超几何方程的类似条件。在排名4的情况下，这个新结果打开了对painlevel分析的应用程序。其次，我们确定了D_7型Painleve方程的特殊解，从而完全分类了Painleve方程Umemura意义的超越经典解。painlevel VI代数解的分类仍然存在。一个关于painleevi方程不可约性的不完美证明被宣布后，它走过了100年。我们无法将Umemura意义的所有经典解进行分类，但我认为我们已经找到了painlevel分析的新方向。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