In recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel sum by taking a special Cauchy data.

A generalization of the Landen identity, in the form of an inequality, is proved for hypergeometric functions. Some well-known asymptotic formulas are refined.

We describe the branch locus of a proper holomorphic mapping between two smoothly bounded pseudoconvex domains of finite type in $\mathbb{C}^2$ under the assumption that the first domain admits a transversal holomorphic action of the unit circle. As an application we show that any proper holomorphic self-mapping of a smoothly bounded pseudoconvex complete circular domain of finite type in $\mathbb{C}^2$ is biholomorphic.

In this paper, we shall consider the birational structure of moduli of stable sheaves on elliptic surfaces, which is a generalization of Friedman's results to higher rank cases. As applications, we show that some moduli spaces of stable sheaves on ${\Bbb P}^2$ are rational. We also compute the Picard groups of those on Abelian surfaces.

In this paper, some qualitative properties of trajectories inside an isolating block for planar differential equations are obtained.

We study moduli spaces $M(c_1,c_2,d,r)$ of isomorphism classes of algebraic 2-vector bundles with fixed numerical invariants $c_1,c_2,d,r$ over a ruled surface. These moduli spaces are independent of any ample line bundle on the surface. The main result gives necessary and sufficient conditions for the non-emptiness of the space $M(c_1,c_2,d,r)$ and we apply this result to the moduli spaces ${\cal M}_L(c_1,c_2)$ of stable bundles, where $L$ is an ample line bundle on the ruled surface.

We show the twisted cohomology on $\mathbb{P}^{1}$ has a natural polarized Hodge structure and hence derive the analogues of Riemann's equality and inequailty.

Applying the methods of Serre duality in the setting of $CR$ manifolds we prove approximation theorems and we study the Hartogs-Bochner phenomenon in 1-concave $CR$ generic manifolds.

In this paper, for three dimensional compressible Euler equations with small perturbed initial data which are axisymmetric, we prove that the classical solutions have to blow up in finite time and give a complete asymptotic expansion of lifespan.

A compact Riemannian symmetric space admits a canonical complexification. This so called adapted complex manifold structure $J_{A}$ is defined on the tangent bundle. For compact rank-one symmetric spaces another complex structure $J_S$ is defined on the punctured tangent bundle. This latter is used to quantize the geodesic flow for such manifolds. We show that the limit of the push forward of $J_{A}$ under an appropriate family of diffeomorphisms exists and agrees with $J_S$.

We prove a general embedding theorem for Cohen-Macaulay curves (possibly nonreduced), and deduce a cheap proof of the standard results on pluricanonical embeddings of surfaces, assuming vanishing $H^{1}(2K_{X}) = 0$.