## MEMBERS

### Program Members

It is about one hundred years ago that my main objects of study, the Painlevé equations, were discovered. It is around the same time as H. Poincaré initiated the qualitative study of dynamical systems and discovered chaotic phenomena in celestial mechanics. While having discovered his famous equations, P. Painlevé acted as a specialist of celestial mechanics and left what is now called the Painlevé conjecture for many-body problems. So there should have been qualitative studies of the Painlevé equations as dynamical systems from the beginning of their discovery, but history did not turn this way. Even now the mainstream research in the field focuses on the aspects of integrable systems. Against this background, I have been conducting qualitative studies of the Painlevé equations based on algebraic geometry, dynamical system theory and ergodic theory.

My research is built on two foundations. One is the algebraic geometry of the Painlev´e equation, or more specifically its moduli-theoretical formulation based on geometric invariant theory. The other is the ergodic theory of birational maps on algebraic varieties. The most recent achievement is the uncovering of a chaotic nature of the Painlev´e VI equation by combining these two foundations via a transcendental mapping, called the Riemann-Hilbert correspondence. The main results consist of the construction of a mixing, hyperbolic, invariant probability measure of saddle type with maximal entropy for the monodromy map of the Painlev´e equation, the establishment of an algorithm to calculate the entropy explicitly, and the proof of the exponential growth of the number of periodic solutions as the period tends to infinity. It is interesting that behind this work there is pluripotential theory in several complex variables.

Currently, I am also working on the periodic solutions and algebraic solutions to the Painlev´e equation. Generally, for the problems of periodic solutions, important roles are played by the Lefschetz-type theorems such as the Atiyah-Bott fixed point formula as well as by the Shub-Sullivan theorem on the behavior of local indices under iterations of the mapping. However, these conventional theorems cannot be applied to the dynamics of the Painlev´e equation which is area-preserving. Therefore my collaborator and I are developing a general theory of periodic points for area-preserving birational maps on an algebraic surface, by applying a more powerful fixed point formula due to Shuji Saito, and also by generalizing the Shub-Sullivan theorem to mappings which allow the presence of periodic curves.

Algebraic solutions to Painlev´e VI have not been fully understood yet. My research on this issue aims to clarify the geometrical structures of the algebraic solutions by employing such tools as the Riemann-Hilbert correspondence, the dynamics of mapping class group actions on character varieties, resolutions of singularities, power geometry (the method of Newton polyhedra), and the concept of nonlinear regular singularities. My philosophy in dealing with this problem is “to begin with algebraic geometry and end up with elementary geometry”.

Figure : The monodromy map of the Painlevé equation can be described as a discrete dynamical system on the moduli space of stable parabolic connections (K. Iwasaki and T. Uehara, Math. Ann. 338 (2007), no. 2, 295-345, Fig. 9)

My main goal through these studies is to make Poincare´’s 100-year-old dream coming true (as much as I can). In order to understand the continuos dynamical system defined by a differential equation, one should proceed to the discrete dynamical system of its Poincar´e return mappings, so that a complete understanding of the global behaviors of the former can be extracted from the analysis of the latter. Many branches of mathematics, including dynamical system theory, algebraic geometry and topology, have evolved out of this program. However, Poincare´’s original program itself has remained to be a kind of slogan. Virtually, it does not appear that the original program has been realized globally and completely by incorporating the development of mathematics over the last hundred years since his time. Moreover, it would be too difficult for the contemporary mathematics to achieve his program even for general (nonlinear) ordinary differential equations, let alone for partial differential equations. Thus my motivation, modest, for working on the Painlev´e equations is to realize this program for at least one meaningful nonlinear ordinary differential equation.

However, it is interesting that as my target becomes narrower, the range of mathematics I employ becomes broader and deeper. I often try to learn a new area of mathematics, not knowing if it is of use or not for my purpose. Very rarely it happens to have an unexpected application to my mathematics, which makes me excited. I should be happy if I could have any contribution from the side of pure mathematics to the Math-for-Industry as a member of the Fundamental Mathematics Unit for the GCOE.