Eigenvolues of random matrices and soliton equations

我们与M.Adler和P.van Moerbeke一起研究了一个称为Pfaff格子的可积系统(通过对二维Toda格子施加一定的对称条件并将时间演化限制在轨迹S=-t上而得到的，这是JIMBO和MIWA的D‘_∞-族的一个子系统，并且与Hirota和Kakei的耦合KP谱系密切相关)，并构造了它的tau函数、双线性恒等式、Fay恒等式和一个拟周期解。结合Adler，Horosov，van Moerbeke等人关于Pfaff晶格的Lax表示等工作，给出了Pfaff晶格的一般理论。对称和辛矩阵积分给出了Pfaff格子的tau函数。我们还研究了这些特解所满足的Virasoro条件，并与E.霍洛佐夫一起证明了系数为自变量有理函数的秩1交换的常微分算子环的谱曲线是只有尖点的有理曲线。这改进了G.Wilson的一个类似的结果，其中假设了环的双谱。

Together with M.Adler and P.van Moerbeke, we studied an integrable system called Pfaff lattice (obtained by imposing certain symmetry condition to the 2-dimensional Toda lattice and restricting the time evolutions to the locus s=-t ; this is a subsystem of the D'_∞-hierarchy of Jimbo and Miwa, and is closely related to the coupled KP hierarchy of Hirota and Kakei), and constructed its tau function, bilinear identity, Fay identities, and a quasiperiodic solution. Combined with the work of Adler, Horozov, van Moerbeke on the Lax representation of the Pfaff lattice, etc., this work shows the general theory of the Pfaff lattice. Symmetric and symplectic matrix integrals give tau functions of the Pfaff lattice. We also studied the Virasoro conditions satisfied by those special solutions.Together with E.Horozov, we proved that the spectral curve of a rank-1 commutative ring of ordinary differential operators, whose coefficients are rational functions of the independent variable, is a rational curve with only cusps. This refines a similar result of G.Wilson in which the bispectrality of the ring was assumed.