2017年度秋季総合分科会(於:山形大学)

SUMMARY: Necessary conditions are obtained for certain types of rational delay differential equations to admit a transcendental meromorphic solution of hyper-order less than one. The equations obtained include delay Painlevé equations and equations solved by elliptic functions. Difference analogue of Nevanlinna theory is a central tool in the proofs of the main results. An overview of this theory, as well as some of its applications to difference Painlevé equations, are also presented.

SUMMARY: We consider a parabolic operator \(L^{(\alpha )}=\partial _t+(-\Delta )^\alpha \) on \({\boldsymbol R}^{n+1}\) for \(0<\alpha \leq 1\) and \(n\in {\boldsymbol N}\). When \(\alpha = 1\), \(L^{(1)}\) is the heat operator, and otherwise, \(L^{(\alpha )}\) is a non-local operator. When \(\alpha =1/2\), the operator \(L^{(\alpha )}\) is called the Poisson operator and closely related with harmonic functions on \({\boldsymbol R}^{n+1}\). In this talk, after recalling the potential theory for \(L^{(\alpha )}\) to define \(L^{(\alpha )}\)-harmonic functions, we discuss function spaces of \(L^{(\alpha )}\)-harmonic functions, called the parabolic Bergman space and the parabolic Bloch space.

SUMMARY: In this talk, we will generalize distortion theorems for normalized holomorphic functions on the unit disc in \(\mathbb C\) to normalized holomorphic mappings on bounded symmetric domains in a higher dimensional complex Banach space.

SUMMARY: In this talk, we will see spirallikenss (including starlikeness) of the shifted hypergeometric function \(F(z)=z_2F_1(a,b;c;z)\) with complex parameters \(a,b,c.\) First, we observe the asymptotic behaviour of the hypergeometric function around the point \(z=1\) to obtain necessary conditions for \(F\) to be \(\lambda \)-spirallike for a given \(\lambda \) with \(- \pi /2< \lambda <\pi /2.\) We next give sufficient conditions for \(F\) to be \(\lambda \)-spirallike. More general results will also be given in the talk if time permits.

SUMMARY: Let \(\Omega \) be a domain in the complex plane with hyperbolic metric \(\lambda _\Omega (z)|dz|\) of Gaussian curvature \(-4.\) Mejia and Minda proved that \(\Omega \) is (Euclidean) convex if and only if \(d(z,\partial \Omega )\lambda _\Omega (z)\ge 1/2\) for \(z\in \Omega ,\) where \(d(z,\partial \Omega )\) denotes the Euclidean distance from \(z\) to the boundary \(\partial \Omega .\) In the present talk, we give spherical and hyperbolic counterparts of this result in terms of the spherical/hyperbolic density of the hyperbolic metric \(\lambda _\Omega (z)|dz|.\) A key idea is to obtain a geometric characterization of such convex domains relative to the spherical/hyperbolic metric.

SUMMARY: Let \(R\) be an open Riemann surface of finite genus \(g (\geq 1)\) and \(\chi _R\) be a canonical homology basis of \(R\) modulo dividing cycles. A closing of \((R, \chi _R)\) is, roughly speaking, a triplet \([S, \chi _S, \iota ]\) consisting of a closed Riemann surface \(S\) of genus \(g\), a canonical homology basis \(\chi _S\), and a conformal mapping \(\iota : R \rightarrow S\) which induces the prescribed corespondence between \(\chi _R\) and \(\chi _S\). Denote by \(C\) the set of closings of \((R, \chi _R)\), and let \(\frak M\) be the set of the period matrices \((\tau _{jk})\) of \((S, \chi _S)\), \([S, \chi _S, \iota ] \in C\). For any \((a_1, a_2,\ldots , a_g) \in {\mathbb R}^g\) with \(\sqrt {a_1^2 + a_2^2 + \cdots + a_g^2} \neq 0\) the set \(\{\sum a_j a_k \tau _{jk} \mid (\tau _{jk}) \in {\frak M} \} \) is a closed disk in \(\mathbb H\). We show among other things that \(\partial {\frak M}\) is described by the generalized period matrices derived from holomorphic differentials with hydrodynamically specific boundary behavior.

SUMMARY: In this talk, I will describe the infinitesimal deformations of singular flat structures defined from generic holomorphic quadratic differentials under the de Rham theoretic framework.

SUMMARY: I will give a formula of the Levi form of the Teichmüller distance on the Teichmüller space.

SUMMARY: Let \(\Gamma \) be the (a, b, c)-hyperbolic triangle group acting on \(\hat {\bf {D}}\), the unit disk \( \bf {D}\) with the set of cusps of \(\Gamma \). Then the quotient space \(\hat {\bf {D}}/\Gamma \) is isomorphic to the Riemann sphere \(P^1(\bf {D})\) which induces a meromorphic function on \(\bf {D}\). It is called the Schwarz automorphic function, and we write it down explicitly in terms of a, b and c.

SUMMARY: The Bers embedding of the Teichmüller space is a map into the Banach space of corresponding holomorphic quadratic differentials. This induces a complex Banach manifold structure to the Teichmüller space. If we take a subspace of the universal Teichmüller space, we can usually project down the Bers embedding to a well-define map from the quotient Teichmüller space to the quotient Banach space. We call this the quotient Bers embedding but its injectivity is not a trivial matter. In this talk, we consider several cases where the injectivity holds true.

SUMMARY: We consider polynomial solution to Dirichlet problems for the heat equation, where polynomials are in two variables \(x\) and \(t\) with real coefficients. Our interest is to determine a polynomial \(\psi (x,t)\) such that for any polynomial \(f(x,t)\) there exists a heat polynomial \(u(x,t)\) which is equal to \(f(x,t)\) on the curve \(\psi (x,t)=0\) in the \(xt\)-plane. In our previous work we determined \(\psi \) of degree at most two and showed that there exist no such \(\psi \) of degree 3. In this talk we show that there exist no such \(\psi \) of degree greater than 3.

SUMMARY: We show that the limit of the lower capacity density is equal either to 0 or to 1.

SUMMARY: Polyharmonic functions are solutions of the iterated Laplace equation. In this talk, we discuss spaces of polyharmonic functions together with iterated parabolic operators on the upper half space of the Euclidean space. After explaining some basic properties of polyharmonic functions and parabolic operators of fractional order, we introduce weighted polyharmonic and polyparabolic Bergman spaces, and shall discuss their relations and reproducing properties.

SUMMARY: Caloric morphisms are transformations preserving solutions of heat equation. Bateman mappings are conformal in semi-euclidean spaces. In this talk, we shall discuss problems whether there exist caloric morphisms with Bateman space mapping for radial semi-riemannean metrics.

SUMMARY: We show that for any finite collection of quadratic Julia sets, there exist a polynomial and a transcendental entire function whose Julia sets include copies of the given quadratic Julia sets. In order to prove the result, we construct quasiregular maps with required dynamics and employ the quasiconformal surgery to obtain the desired functions.

SUMMARY: We show that there are quasiconformal copies of the Cantor Julia sets embedded in the boundary of the Mandelbrot set, whose dilatations are arbitrarily close to one. Indeed, these embeddings are also close to complex affine maps. It implies that these copies are “superfine”.

SUMMARY: For a holomorphic skew product with a superattracting fixed point, we construct a Böttcher coordinate on an invariant open set whose closure contains the fixed point.

SUMMARY: We consider a class of rigid holomorphic germs on \({\bf C}^2\) at super saddle fixed points. Their formal normal forms are given by Ruggiero. We investigate the convergence/divergence of their formal conjugacies to the normal forms. It turns out that, in most cases, the formal conjugacies diverge. We also show convergence result under some assumptions.

SUMMARY: Annihilators in the ring of analytic linear partial differential operators associated with a \(\mu \)-constant deformation of isolated hypersurface singularities are considered. Algorithmic methods of computing annihilators and \(b\)-functions are described for semi-quasihomogeneous singularities. Key ingredients of the proposed methods are local cohomology classes and integral dependence relations.

SUMMARY: We present new algorithms for computing integral numbers w.r.t. an ideal in a ring of convergent power series. The problems of solving the integral numbers can be regarded as the ideal membership problems in the ring of convergent power series. In this talk, we give two methods for solving the membership problems. One is utilizing Gröbner bases and the another is utilizing local cohomology classes. We also address the question of how to generalize the methods to parametric cases.

SUMMARY: We give an algorithm for computing Grothendieck local residues via transformation law. Actually, we need syzygy, standard bases, ideal quotient, local cohomology etc, to get Grothendieck local residues. Thus, we give the relations in the talk. Furthermore, we give some computation examples.

SUMMARY: Let \(p\) be any prime number. The toroidal group defined by \(Q(\sqrt [5]{p})\) has no non-constant meromorphic functions on it. On the other hand, the toroidal group defined by \(Q(\sqrt [6]{p})\) is a quasi-Abelian variety.

SUMMARY: We show the existence of a complex \(K3\) surface \(X\) which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such \(X\) by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.

SUMMARY: Let \(n \geq 4\) and let \(\Omega \) be a bounded hyperconvex domain in \(\mathbb {C}^{n}\). Let \(\varphi \) be a negative exhaustive smooth plurisubharmonic function on \(\Omega \). We show that any holomorphic function defined on a connected open neighborhood of the support of \((i\partial \overline {\partial }\varphi )^{n-3}\) can be extended to the holomorphic function on \(\Omega \).

SUMMARY: In this talk, we will show that the shearing process recently introduced by Bracci can be generalized to \({\mathcal N}_A(\mathbb {B}^2)\) and \({\mathcal M}_g(\mathbb {B}^2)\), where \(A\) is a diagonal matrix whose diagonal elements are \(\lambda \) and \(1\) with \(\lambda \in [1,2)\) and \(g\in H(\mathbb {U})\) is a convex \((\)univalent\()\) function with real coefficients such that \(g(0)=1\), \(\Re g(\zeta )>0\) for all \(\zeta \in \mathbb {U}\) and \(\mathbb {U}(1,a_0)\subseteq g(\mathbb {U})\), where \(a_0\) is a constant defined by \(g\). We also give the results for \({\mathcal M}_g(\mathbb {U}^2)\).

SUMMARY: In this talk, we will show that the shearing process recently introduced by Bracci can be generalized to \(S_A^0(\mathbb {B}^2)\), \(S_g^0(\mathbb {B}^2)\), where \(A\) is a diagonal matrix whose diagonal elements are \(\lambda \) and \(1\) with \(\lambda \in [1,2)\) and \(g\in H(\mathbb {U})\) is a convex \((\)univalent\()\) function with real coefficients such that \(g(0)=1\), \(\Re g(\zeta )>0\) for \(\zeta \in \mathbb {U}\) and \(\mathbb {U}(1,a_0)\subseteq g(\mathbb {U})\). As a corollary, we obtain bounded suppot points for these families. This result is in contrast to the one dimensional case, where all support points of \(S\) are unbounded. Also, our result shows the existence of bounded support points for various subclasses of \(S^*(\mathbb {B}^2)\) and that \(S_A^0(\mathbb {B}^2)\neq S^0(\mathbb {B}^2)\). We also give a result for \(S_g^0(\mathbb {U}^2)\) and \(S_g^*(\mathbb {U}^2)\).

SUMMARY: In this talk, we show the relation of the reachable families and the support points of \(S_A^0(\mathbb {B}^2)\)(or, \(S_g^0(\mathbb {B}^2)\)) and apply it to show that \( \tilde {\mathcal R}_{\log M}({\rm id}_{\mathbb {B}^2}, {\mathcal N}_A(\mathbb {B}^2))\neq S_A^0(\mathbb {B}^2,M)\) and \( \tilde {\mathcal R}_{\log M}({\rm id}_{\mathbb {B}^2}, {\mathcal M}_g(\mathbb {B}^2))\neq S_g^0(\mathbb {B}^2,M) \), where \(A\) is a diagonal matrix whose diagonal elements are \(\lambda \) and \(1\) with \(\lambda \in [1,2)\) and \(g\in H(\mathbb {U})\) is a convex \((\)univalent\()\) function with real coefficients such that \(g(0)=1\), \(\Re g(\zeta )>0\) for all \(\zeta \in \mathbb {U}\) and \(\mathbb {U}(1,a_0)\subseteq g(\mathbb {U})\). This result provides a basic difference between the theory of bounded univalent mappings on the unit disc \(\mathbb {U}\) and that on the unit ball \(\mathbb {B}^n\), \(n\geq 2\).

SUMMARY: Let \(T_\Omega \) be a tube domain in \(\bold C^n\) with polynomial infinitesimal automorphisms and \(\frak g(T_{\Omega })\) the Lie algebra of all complete holomorphic vector fields on \(T_\Omega \). By definition, every element of \(\frak g(T_{\Omega })\) has the form of a polynomial vector field. The investigation into the tube domain \(T_\Omega \) such that \(\frak g(T_{\Omega })\) is solvable has significance to the general study of tube domains with polynomial infinitesimal automorphisms. We have made an experimental investigation into such a case previously. In this talk, we disucuss the general structure of solvable \(\frak g(T_{\Omega })\), which gives a development to the previous investigation.

SUMMARY: The proofs of Oka’s Coherence Theorems are based on Weierstrass’ Preparation (division) Theorem. Here we observe that a Weak Coherence of Oka proved without Weierstrass’ Preparation (division) Theorem, but only with power series expansions is sufficient to prove Oka’s Jôku-Ikô and hence Cousin I, II, holomorphic extensions, and Levi’s Problem, as far as the domain spaces are non-singular. The proof of the Weak Coherence of Oka is almost of linear algebra. We will present some new or simplified arguments in the proofs.