Kyushu University [OSAMU SAEKI (Professor) Institute of Mathematics for Industry, Division of Fundamental mathematics]

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.

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

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1.	Osamu Saeki, Topology of Singular Fibers of Differentiable Maps, Springer Verlag, Lecture Notes in Math., Vol. 1854, Springer-Verlag, 2004. , 2004.01.
2.	Singular Points of Complex Hypersurfaces, J. W. Milnor, Translated into Japanese by O. Saeki and K. Sakuma, Springer Verlag Tokyo, 2003..
3.	Characteristic Classes, J. W. Milnor and J. D. Stasheff, Translated into Japanese by O. Saeki and K. Sakuma, Springer Verlag Tokyo, 2001..
4.	Geometry and Singularities (in Japanese), S. Izumiya, T. Sano, O. Saeki and K. Sakuma, Kyoritsu Publ., 2001..

Papers

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1.	R.I. Baykur, O. Saeki, Simplifying indefinite fibrations on 4-manifolds, Trans. Amer. Math. Soc., https://doi.org/10.1090/tran/8325, The main goal of this article is to connect some recent perspectives in the study of 4–manifolds from the vantage point of singularity theory. We present explicit algorithms for simplifying the topology of various maps on 4–manifolds, which include broken Lefschetz fibrations and indefinite Morse 2–functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1–parameter families. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations and Morse 2–functions on general 4–manifolds, and a theorem of Auroux–Donaldson–Katzarkov on the existence of certain broken Lefschetz pencils on near-symplectic 4–manifolds. We moreover establish a correspondence between broken Lefschetz fibrations and Gay–Kirby trisections of 4–manifolds, and show the existence and stable uniqueness of simplified trisections on all 4–manifolds. Building on this correspondence, we also provide several new constructions of trisections, including infinite families of genus–3 trisections with homotopy inequivalent total spaces, and exotic same genera trisections of 4–manifolds in the homeomorphism classes of complex rational surfaces..
2.	Osamu Saeki, Reeb spaces of smooth functions on manifolds, International Mathematics Research Notices, https://doi.org/10.1093/imrn/rnaa301, 2022, 8740-8768, 2022.06, The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy..
3.	R.I. Baykur, O. Saeki, Simplified broken Lefschetz fibrations and trisections of 4-manifolds, PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, 10.1073/pnas.1717175115, 115, 43, 10894-10900, 2018.10, 特異点論の観点から、特異シンプレクティック構造に付随した特異Lefschetz構造の存在や、単純化されたtrisectionの存在を、具体的かつ構成的に証明することに成功した。.
4.	Osamu Saeki, Special generic maps on open 4-manifolds, Journal of Singularities, 1, 1-12, 2010.01.
5.	O. Saeki and T. Yamamoto, Singular fibers of stable maps and signatures of 4-manifolds, Geometry and Topology , 10, pp. 359-399, 2006.04.
6.	Osamu Saeki, Topology of singular fibers of differentiable maps, Lecture Notes in Mathematics, Springer, Vol.1854, 2004.01.
7.	Osamu Saeki, Fold maps on 4-manifolds, Comment. Math. Helv., 10.1007/s00014-003-0758-9, 78, 3, 627-647, 78, 627-647, 2003.01.
8.	O. Saeki and K. Sakuma, Special generic maps of 4-manifolds and compact complex analytic surfaces, Math. Ann., 313, 617-633, 1999.01.

Presentations

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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 2 (2021), 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 have been teaching general topology and algebraic topology. The numbers of students that I supervised were 1(2021), 1 (2020), 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

2022.04.
2021.04, The MEXT WISE program, Graduate Program of Mathematics for Innovation, has been approved and I am running the program as coordinator..
2021.04, Kyushu University Leading PhD Program in Mathematics for Key Technologies, Program Coordinator.
2020.10, The MEXT WISE program, Graduate Program of Mathematics for Innovation, has been approved and I am running the program as coordinator..
2020.04, 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.

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