2019年度年会(於:東京工業大学)

SUMMARY: Loewner Theory, which goes back to the parametric representation of univalent functions introduced by Loewner in 1923, has recently undergone significant development in various directions, including Schramm’s stochastic version of the Loewner differential equation and the new intrinsic approach suggested by Bracci, Contreras, Diaz-Madrigal and Gumenyuk. The relation between Loewner’s equation and the quasiconformal extensions of conformal mappings has been indicated by Becker in 1972. Recently His result was recently extended to the framework of modern Loewner theory by Gumenyuk, Prause and Hotta. In this talk, we firstly review the theory of Loewner equations in classical and modern treatments. Then we discuss some recent results on this topics.

SUMMARY: The entropy measures the complexity of a dynamical system, and if the entropy is positive then the system exhibits a chaotic behavior. When a dynamical system with positive entropy is given by an automorphism of a compact complex surface, it is known from the Enriques–Kodaira classification that the surface is either a complex torus, an Enriques surface, a K3 surface or a rational surface. In this talk, we will restrict our attention to K3 surfaces and rational surfaces, and give a survey on complex dynamical systems.

SUMMARY: In this talk, we will introduce beautiful horn torus models by Puha and Däumler for the Riemann sphere in complex analysis from the viewpoint of the division by zero.

Contents: Division by zero calculus, Horn torus models by Puha and Däumler, Properties of horn torus models and Conformal mapping from the plane to the horn torus.

SUMMARY: We first find the Schiffar span of a \(2n\)-ply connected horizontal slit region which is symmetric in the real and the imaginary axis, Then we let the mutually symmetric two slits converge to a single slit on the real axis, and obtain a \((2n - 1)\)-ply connected region. We compare the Schiffer spans of these regions We also give a numerical example for the case of doubly connected regions.

SUMMARY: The difference equation \((*) (\Delta f(z))^2=A(z)(f(z)f(z+1)-B(z))\) where \(A(z)\) and \(B(z)\) are meromorphic functions, is investigated. If (*) possesses two distinct transcendental meromorphic solutions, it is shown that these solutions satisfy an algebraic relation, and that their growth behaviors are almost same in the sense of Nevanlinna under some conditions.

SUMMARY: We show that the quotient of the BMO Teichmüller space by the VMO Teichmüller space has a natural complex structure modeled on the quotient Banach space. Then, we introduce a certain complete translation-invariant metric on the BMO Teichmüller space and its quotient. By using these structures, we investigate the space of chord-arc curves in the BMO Teichmüller space.

SUMMARY: We prove: For a generic hyperbolic rational map \(f\) of degree \(d\) whose Julia set is Cantor, there is a simply connected domain \(U\) such that \(f^{-1}(U)\subset U\) and \(f^{-1}(U)\) consists of \(d\) connected components. Using this, we show that the shift locus is connected.

SUMMARY: C. Bishop has introduced a technique called quasiconformal folding, and as an application, constructed a transcendental entire function of class \(\mathcal B\) (with bounded singular values) having an oscillating wandering domain. We propose a new and simplified construction which uses quasiconformal mappings but not Bishop’s quasiconformal folding. With our method, we can show that the obtained function has finite order.

SUMMARY: When a family of Beltrami differentials is given, the corresponding quasiconformal mappings are differentiable with respect to the parameter, under a suitable condition. We propose a new approach for the proof with a weaker hypothesis. The proof is related to the deformation of cross-ratio of 4 points and double covering torus. As a by-product, we obtain an integro-differential equation for the elliptic modular function.

SUMMARY: We study proper holomorphic mappings between generalized complex pseudo-ellipsoids, especially different dimensions. Under some regularity condition on the boundary, such mapping is classified into special kind of mapping whose components are monomials.

SUMMARY: Let \(\mathbb {B}_X\) be a bounded symmetric domain realized as the unit ball of a JB\(^*\)-triple \(X\). Recently, Chu, Hamada, Honda and Kohr introduced the concept of a Bloch function on a bounded symmetric domain which can be infinite dimensional and derived some basic properties of the Bloch space \(\mathcal {B}(\mathbb {B}_X)\) and composition operators in this setting, generalising several finite dimensional results. In this talk, we refine and develop some results in Chu, Hamada, Honda and Kohr by extending finite dimensional results on Bloch spaces and composition operators, as well as answering two open questions by Allen and Colonna.

SUMMARY: In this talk, we continue the work related to embedding univalent mappings in Loewner chains on the unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\). We always assume that \(n\geq 2\) and we obtain approximation properties of various families of normalized univalent mappings \(f\) on \(\mathbb {B}^n\) by automorphisms of \(\mathbb {C}^n\) whose restrictions to \(\mathbb {B}^n\) have the same geometric property of \(f\).

SUMMARY: In this talk, we always assume that \(n\geq 2\) and we show that every automorphism restricted to a bounded domain in \(\mathbb {C}^n\) extends to a quasiconformal diffeomorphism of class \(C^\infty \) from \(\mathbb {C}^n\) onto \(\mathbb {C}^n\), which is equal to the identity mapping outside some ball. Next, we give a variational result for \(A\)-normalized univalent subordination chains, for \(A\in L(\mathbb {C}^n)\) with \(m(A)>0\). Using these results, we shall deduce that every mapping which has \(A\)-parametric representation on \(\mathbb {B}^n\) can be approximated locally uniformly on \(\mathbb {B}^n\) by mappings which have \(A\)-parametric representation on \(\mathbb {B}^n\) and admit extensions, on one hand, to automorphisms of \(\mathbb {C}^n\) and, on the other hand, to smooth quasiconformal diffeomorphisms of class \(C^\infty \) from \(\mathbb {C}^n\) onto \(\mathbb {C}^n\), which are equal to the identity mapping outside some ball, when \(m(A)>0\).

SUMMARY: We consider Bruce–Roberts Milnor numbers, on isolated hypersurece singularities, in the context of symbolic computation. We describe, as an application of logarithmic vector fields, an algorithm for computing Bruce–Roberts Milnor numbers. We give a relation between a Bruce–Roberts Milnor number and a generic linear function.

SUMMARY: We introduce an algorithm for computing Chern–Schwartz–MacPherson classes in the context of symnolic computation. The key idea of the approach is the use of parametric Gröbner bases.

SUMMARY: For the maximal ideal cycle \(M\) and the fundamental cycle \(Z\) of the resolution of nomal complex surface singularities, \(M\geq Z\) hols. H. Laufer showed a example \(M\) does not coincident with \(Z\) in any resolution of the form \(z^2=y(x^4+y^6)\). We interested in the condition \(M\) equals \(Z\) of normal complex surface singularities. In this talk, we report the condition that \(M\) equals \(Z\) in the minimal good resolution of the normal hypersurface singularity of the form \(z^n=y(x^a+y^b)\).

SUMMARY: The following finiteness theorems holds by recent progress of MMP theorem and our studies of singularities by filtered blowing-ups;

Theorem 1. Let \(f\) be a weighted homogeneous \(d\)-dimensional complex singularity with \(a(f) = a(C[x]/f) \geq 0\). Here \(a(C[x]/f)\) is the Goto–Watanabe’s a-invariant for graded ring. We assume \(\{ f = 0\} -\{0\}\) has rational singularities. Then for a fixed value \(\alpha = a(f)\), the possible weight types are finite.

Theorem 2. Let \(R\) be a Gorenstein normal graded complex analytic singularity with geometric genus \(p_g(R) \geq 1\). We assume \(Spec(R) - V(R_+)\) has rational singularites. For fixed embedding dimension and \(p_g\), \(a(R)\) is bounded from above.

SUMMARY: Fixing a topological type of a Brieskorn complete intersection surface singularity, we study the singularities which realizes the maximal geometric genus or minimal maximal ideal cycle.

SUMMARY: In this talk, we study deflation type identities of the Bergman kernels, which was initiated by Boas, Fu and Straube in 1999. We establish two variations of deflation type identities for a certain class of domains which is so-called the Roos domains. Our main result includes deflation type identities due to Boas–Fu–Straube and Beberok as special cases.

SUMMARY: Let \(D\) be a pseudoconvex domain in \(\mathbb {C}^n\) and let \(f:D\to \mathbb {C}\) be a continuous function. F. Hartogs (1909) proved that \(f\) is holomorphic if and only if the complement of its graph \(G_f:=\{(z,f(z)); z\in D\}\) in \(D\times \mathbb {C}\) is pseudoconvex. N. Shcherbina (2005) has shown that \(f\) is holomorphic if and only if \(G_f\) is pluripolar. It will be proved that \(f\) is holomorphic if and only if the complement of \(G_f\) admits a complete Kähler metric.