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- Hisashi Ishida

# Professor Hisashi Ishida

Area and Subject Taught | Complex Analysis |
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Research Theme(s) | (1) Moduli problem of Riemann surfaces (2) Complex dynamical systems |

Academic Degrees | Doctor of Science, Kyoto University |

Keywords for Research Field | Riemann surfaces, conformal mappings, extremal length, complex dynamical systems |

Office Phone Number | 81-75-705-1620 |

## Research Overview

- Two plane regions are conformally equivalent if there exists a univalent holomorphic function (conformal mapping) that maps one region to the other one.

It is one of the fundamental problems of complex analysis to classify all plane regions(or Riemann surfaces) into conformally equivalent classes.

Among simply connected regions, consider all rectangles. We say that two rectangles are equivalent if there exists a conformal mapping which maps one to the other so that the vertices of one rectangle to the vertices of the other. In this case, two rectangles belong to the same class if one rectangle is similar to the other.

Now consider classification of triply connected plane regions. To decide whether two regions are conformally equivalent, we use the extremal length which corresponds to the ratio of sides of rectangles above. The extremal lengthes of three curve families allow us to completely classify triply connected plane regions. - I have interest in complex dynamical systems, particularly the properties of Julia sets of polynomials and polynomial semigroups. Computers are very useful in this branch.

## Notable Publications and Works in the Last Three Years

- Real cross section of the connectedness locus of the family of polynomials (z
^{2n+1}+ a)^{2n+1}+b (with Kamei, T. and Takahashi, Y.) Acta Human. Sci. Univ. Sangio Kyotiensis (2014)