We find and study a 3-parameter family of coupled Painlevé III systems in dimension four, which can be obtained by a degeneration from the system of type $A_{5}^{(1)}$. We also give the phase space for this system.

We study Toeplitz operators on the pluriharmonic Bergman spaces $b^{p}$ for $1<p<\infty$. We give characterizations of Toeplitz operators with positive symbols to be bounded, compact and in the Schatten classes. Also, we describe the essential spectra of Toeplitz operators with uniformly continuous symbols.

Characters of wreath products $G=\mathfrak{S}_{\infty}(T)$ of any compact groups $T$ with the infinite symmetric group $\mathfrak{S}_{\infty}$ are studied. It is proved that the set $E(G)$ of all normalized characters is equal to the set $F(G)$ of all normalized factorizable continuous positive definite class functions. A general explicit formula of $f_{A} \in E(G)$ is given corresponding to a parameter $A=\left( \left( \alpha_{\zeta ,\epsilon} \right)_{({\zeta ,\epsilon})\in \hat{T}\times\{0,1\}} ; \mu \right)$. Similar results are obtained for certain canonical subgroups of $G$.

We determine the L-S category of stunted quasi-projective space $Q_{n,m}=Q_{n}/Q_{m}$ for $n\leq 4m+3$. As a special case of our main result, the L-S category of $Q_{3}$ is determined to satisfy cat $Q_{3}=3$, which answers the question raised by Fernández-Suárez, Gómez-Tato and Tanré [4].

We give a generalization of the Penrose transform on Hermitian manifolds with metrics locally conformally equivalent to Bochner-Kähler metrics. We also give an explicit formula for the inverse transform.

We introduce an algorithm which transforms a finitely presented group $G$ into another one $G_{\Psi}$. By using this, we can get many finitely presented groups whose group homology with coefficients in the group von Neumann algebra vanish, that is, many counterexamples to an algebraic version of the zero-in-the-spectrum conjecture. Moreover we prove that the Baum-Connes conjecture does not imply the algebraic version of the zero-in-the-spectrum conjecture for finitely presented groups. Also we will show that for any $p\geq 3$ the $p$-th group homology of $G_{\Psi}$ coming from free groups has infinite rank.

We give three examples of non-hyperelliptic curves of genus 4 whose Jacobian varieties are isomorphic to products of four elliptic curves. Two of the examples belong to one-parameter families of curves whose Jacobian varieties are isomorphic to products of two 2-dimensional complex tori. By constructing analogous families, we prove that for each $n>1$, there is a one-parameter family of non-hyperelliptic curves of genus $2n$ whose Jacobian varieties are isomorphic to products of two $n$-dimensional tori.

Two functional limit theorems for occupation times of Lamperti’s stochastic processes are established. One is a generalization of Lamperti’s result in 1957 in the null recurrent case, and the other is a limit theorem for the fluctuation in the positively recurrent case. The proofs are based on a limit theorem for i.i.d. random variables with common distribution function belonging to the domain of attraction of a stable law.

We determine the mod 2 cohomology algebra over the Steenrod algebra $\mathcal{A}_{2}$ of the classifying space of loop groups $LG$ where $G=Spin(7)$, $Spin(8)$, $Spin(9)$, $F_{4}$ and $DI(4)$. Then we show they are isomorphic as algebras over $\mathcal{A}_{2}$ to the mod 2 cohomology of the 2-compact groups of type $G$.