OSAMU SAEKI | Last modified date：2020.05.20 |

Graduate School

Undergraduate School

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Academic Degree

Doctor of Science

Country of degree conferring institution (Overseas)

No

Field of Specialization

Topology

Outline Activities

My main research interest is the global singularity theory of differentiable mappings. It has been known that differentiable functions on a manifold can be well used to study its global geometric structures. In the 1950's Thom began to try to generalize such a theory to that of differentiable mappings between manifolds. However, because of the difficulty in controling local singularities, the theory has not been well developed until recently. So I am studying differentiable mappings between manifolds with only mild singularities or those between low dimensional manifolds.

This kind of global study of singularities is fairly new and my recent results have shown that the singularities of differentiable mappings play an essential role in the study of geometric structures of manifolds. In this way, it has been recognized that such a study is important in Topology.

Other than the above mentioned research, I am also interested in the following vast area of Topology and related fields: primary obstruction to topological embeddings, separation properties of codimension 1 maps, topology of complex isolated hypersurface singularities, fibered knots, 4-dimensional manifolds, codimension 1 embeddings, differential geometric invariants of space curves, unknotting numbers of knots, etc. I am also interested in the asymptotic behavior of generalized Fibonacci sequences. Furthermore, I am interested in the application of Topology to other areas in Science and Industry, such as DNA knots, visual data analysis for multivariate functions, analysis of materials from microscopic levels, etc.

This kind of global study of singularities is fairly new and my recent results have shown that the singularities of differentiable mappings play an essential role in the study of geometric structures of manifolds. In this way, it has been recognized that such a study is important in Topology.

Other than the above mentioned research, I am also interested in the following vast area of Topology and related fields: primary obstruction to topological embeddings, separation properties of codimension 1 maps, topology of complex isolated hypersurface singularities, fibered knots, 4-dimensional manifolds, codimension 1 embeddings, differential geometric invariants of space curves, unknotting numbers of knots, etc. I am also interested in the asymptotic behavior of generalized Fibonacci sequences. Furthermore, I am interested in the application of Topology to other areas in Science and Industry, such as DNA knots, visual data analysis for multivariate functions, analysis of materials from microscopic levels, etc.

Research

**Research Interests**

- Mathematical Descriptions of Figures from the Viewpoints of Topology and Differential Geometry

keyword : Topology, Differential Geometry, Material Science

2013.04～2014.03. - Application of Pure Mathematics to Materials Science

keyword : Microstructure of Materials, Geometric Features, Mathematical Modeling

2011.04Research on DNA recombinations by enzymes by using DNA knots and tangles.. - Low Dimensional Topology, Morse Theory and Computer Graphics

keyword : low dimensional topology, computer graphics, Morse theory, singularities of differentiable maps

2011.04Research on DNA recombinations by enzymes by using DNA knots and tangles.. - Topology-based visual data analysis for multivariate functions

keyword : multivariate function, data analysis, visualization, differential topology

2010.10Research on DNA recombinations by enzymes by using DNA knots and tangles.. - Research on DNA knots

keyword : DNA recombination, knot theory, tangle, site-specific recombination enzyme, topoisomerase, cyclic surgery theorem

2007.06Research on DNA recombinations by enzymes by using DNA knots and tangles.. - Research on separation property of codimension 1 maps, Betti number of generic map images, and primary obstruction to topological embeddings.

keyword : codimension 1 map, separation property, generic map, Betti number, primary obstruction to topological embeddings

1991.04～2001.03Research on separation properties of codimension 1 maps, Betti numbers of generic maps, and primary obstruction to topological embeddings.. - Research on the topology of stable maps

keyword : stable map, elimination of singularities, manifold, characteristic class, differentiable structure

1991.04Research on topology of stable maps.. - Research on generalized Fibonacci sequences.

keyword : Fibonacci sequence, recurrence, asymptotic behavior, Binet formula, holomorphic function

1994.04Research on generalized Fibonacci sequences.. - Research on high dimensional knots

keyword : knot, codimension 1 embedding, product of spheres, fibered knot, Milnor fibration

1994.04Research on high dimensional knot theory.. - Research on contact between curves and 1-parameter orbits in homogeneous spaces

keyword : homogeneous space, 1-parameter subgroup, contact, orbit, Lie algebra

1997.04Research on contact between curves and 1-parameter orbits in homogeneous spaces.. - Research on regular homotopy classes of immersions and embeddings of 3-manifolds into 5-space

keyword : immersion, embedding, regular homotopy, spin structure, 3-manifold

2000.04Research on regular homotopy classes of immersions and embeddings of 3-manifolds into 5-space..

**Current and Past Project**

- Visual data analysis for multivariate functions

**Academic Activities**

**Books**

**Papers**

**Presentations**

1. | 佐伯修, Simplified broken Lefschetz fibrations and trisections of 4-manifolds, 研究集会 Intelligence of Low-dimensional Topology, 2018.05. |

2. | 佐伯 修, Topology of singular fibers for visualization, Topology-Based Methods in Visualization 2015, 2015.05, [URL]. |

3. | 佐伯 修, Broken Lefschetz fibrations and their moves, Geometry and topology of smooth 4-manifolds, 2013.06. |

4. | Cobordism of Morse maps and its application to map germs. |

5. | Singularities and Characteristic Classes -- important roles played by explicit examples . |

6. | On 2-knots with total width less than or equal to 8. |

7. | Morse functions with sphere fibers. |

8. | Singular fibers of differentiable maps and characteristic classes of surface bundles. |

9. | Elimination of definite fold. |

10. | Generic smooth maps with sphere fibers. |

11. | Theory of singular fibers of differentiable maps and characteristic classes of surface bundles I. |

12. | Introduction to singular fibers of differentiable maps: theory and examples. |

13. | Topology of manifolds and singularities of differentiable maps. |

14. | Universal complex of singular fibers and cobordism of singular maps. |

**Membership in Academic Society**

- Asia Pacific Consortium of Mathematics for Industry
- Australian Mathematical Society
- The Japan Society for Industrial and Applied Mathematics
- The Mathematical Society of Japan

**Awards**

- Stable maps and Topology of Manifolds

Educational

**Educational Activities**

The numbers of recent master course students that I supervised were 1 (2019), 1 (2018), 2 (2015), 1 (2014), 4 (2013), 2 (2012), 1 (2011), 2 (2010), 1 (2009), 4 (2008), 2 (2007), 4 (2006), 2 (2005), 1 (2004), 4 (2001), 2 (2000), 2 (1998), 1 (1996) and 1 (1993).

For undergraduate students, I am teaching general topology and algebraic topology. The numbers of students that I supervised were 5 (2019), 1 (2017), 1 (2016), 1 (2014), 1 (2012), 4 (2011), 4 (2010), 3 (2009), 2 (2008), 4 (2007), 3 (2006), 4 (2005), 3 (2004), 2 (2003), 2 (2001), 3 (2000), 6 (1999), 3 (1998), 3 (1996), 1 (1995), 3 (1993), 3 (1992), 2 (1991) and 4 (1988).

Other than the above mentioned activities, I have given 14 mini-courses in other universities. Furthermore, I have supervised 3 Brazilian students for their PhD.

For undergraduate students, I am teaching general topology and algebraic topology. The numbers of students that I supervised were 5 (2019), 1 (2017), 1 (2016), 1 (2014), 1 (2012), 4 (2011), 4 (2010), 3 (2009), 2 (2008), 4 (2007), 3 (2006), 4 (2005), 3 (2004), 2 (2003), 2 (2001), 3 (2000), 6 (1999), 3 (1998), 3 (1996), 1 (1995), 3 (1993), 3 (1992), 2 (1991) and 4 (1988).

Other than the above mentioned activities, I have given 14 mini-courses in other universities. Furthermore, I have supervised 3 Brazilian students for their PhD.

**Other Educational Activities**

- 2020.05, Kyushu University Leading PhD Program in Mathematics for Key Technologies, Program Coordinator.
- 2010.08, Joint Lecture with Pusan National University for Graduate Students.

Social

**Professional and Outreach Activities**

I have written the following educational essays (in Japanese).

1. Let us enjoy books on Mathematics

2. Let us look at 4-dimensional spaces by using maps

3. Differential topology and singularities

4. Various ways to topology

5. When I encountered with Mathematics --- fascinated by the mystery

6. Mathematics is interesting

7. Book Review "Introduction to Topology (in Japanese)" by T. Tanaka and H. Murakami.

8. Differential topology and singularities

9. Mathematics, Industry, and Mathematics

10. Mathematics helps in an unexpected way -- Singularity Theory and Data Visualization.

1. Let us enjoy books on Mathematics

2. Let us look at 4-dimensional spaces by using maps

3. Differential topology and singularities

4. Various ways to topology

5. When I encountered with Mathematics --- fascinated by the mystery

6. Mathematics is interesting

7. Book Review "Introduction to Topology (in Japanese)" by T. Tanaka and H. Murakami.

8. Differential topology and singularities

9. Mathematics, Industry, and Mathematics

10. Mathematics helps in an unexpected way -- Singularity Theory and Data Visualization.

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