1. |
Osamu Saeki, REEB GRAPHS OF SMOOTH FUNCTIONS ON MANIFOLDS, 2156, 2020.05. |

2. |
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.. |

3. |
Osamu Saeki, Reeb spaces of smooth functions on manifolds, *International Mathematics Research Notices*, https://doi.org/10.1093/imrn/rnaa301, 2021.02, 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.. |

4. |
Osamu Saeki, Unlinking singular loci from regular fibers and its application to submersions, *Journal of Singularities*, 22, 92-103, 2020.01. |

5. |
O. Saeki, Linking between singular locus and regular fibers, *Journal of Singularities *, 10.5427/jsing.2020.21n, 21, 234-248, 2020.01. |

6. |
O. Saeki, A signature invariant for stable maps of 3-manifolds into surfaces, *ROMANIAN JOURNAL OF PURE AND APPLIED MATHEMATICS*, 64, 541-563, 2019.10. |

7. |
O. Saeki, Elimination of definite fold II, *Kyushu Journal of Mathematics*, 73, 239-250, 2019.10. |

8. |
H. Hamada, S. Matsutani, J. Nakagawa, O. Saeki, M. Uesaka, An algebraic description of screw dislocations in SC and BCC crystal lattices, *Pacific Journal of Mathematics for Industry*, https://pacific-mathforindustry.springeropen.com/articles/10.1186/s40736-018-0037-8, 10, 3, 2018.08. |

9. |
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。. |

10. |
O. Saeki and T. Yamamoto, Singular fibers of stable maps of manifold pairs and their applications, *"Singularities and Foliations. Geometry, Topology and Applications", Araújo dos Santos, R.N., Menegon Neto, A., Mond, D., Saia, M.J., Snoussi, J. (Eds.), Springer Proceedings in Mathematics & Statistics*, 222, 259-294, 2018.04. |

11. |
Osamu Saeki, Theory of singular fibers and Reeb spaces for visualization, * “Topological Methods in Data Analysis and Visualization IV - Theory, Algorithms, and Applications”, H. Carr, C. Garth, T.Weinkauf (Eds.), Proc. Topology-Based Methods in Visualization 2015, Springer*, 3-33, 2017.04. |

12. |
O. Saeki and T. Yamamoto, Cobordism group of Morse functions on surfaces with boundary, *REAL AND COMPLEX SINGULARITIES, Proc. XIII International Workshop on Real and Complex Singularities, São Carlos, 2014, Contemporary Mathematics*, 10.1090/conm/675/13597, 675, 279-297, 2016.04. |

13. |
Osamu Saeki, Topology of manifolds and global theory of singularities, *RIMS Kôkyûroku Bessatsu*, B55, 185-203, 2016.04. |

14. |
Osamu Saeki, T. Yamamoto, Singular fibers of stable maps of 3-manifolds with boundary into surfaces and their applications, *Algebraic & Geometric Topology*, 10.2140/agt.2016.16.1379 , 16, 1379-1402, 2016.08. |

15. |
Osamu Saeki, R. Araújo dos Santos, M. A. B. Hohlenwerger, T. O. Souza, New examples of Neuwirth--Stallings pairs and non-trivial real Milnor fibrations, * Annales de l'Institut Fourier (Grenoble) *, 66, 83-104, 2016.04, We use topology of configuration spaces to give a characterization of Neuwirth--Stallings pairs (S^5, K) with dim K = 2. As a consequence, we construct polynomial map germs (R^6,0) -> (R^3,0) with an isolated singularity at the origin such that their Milnor fibers are not diffeomorphic to a disk, thus putting an end to Milnor's non-triviality question.. |

16. |
Osamu Saeki, A. Chattopadhyay, H. Carr, Zhao Geng, D. Duke, Multivariate topology simplification, *Computational Geometry*, 58, 1-24, 2016.04. |

17. |
Osamu Saeki, D. Sakurai, H. Carr, Hsiang-Yun Wu, T. Yamamoto, D. Duke, S. Takahashi, Interactive visualization for singular fibers of functions f : R^3 -> R^2, *IEEE Transactions on Visualization & Computer Graphics*, 22, 1, 945-954, 2016.01. |

18. |
Osamu Saeki, S. Takahashi, Visual data mining based on differential topology: a survey, *Pacific Journal of Mathematics for Industry*, 2014.04. |

19. |
Osamu Saeki, S. Takahashi, D. Sakurai, Hsiang-Yun Wu, K. Kikuchi, H. Carr, D. Duke, T. Yamamoto, Visualizing multivariate data using singularity theory, *The Impact of Applications on Mathematics, Proceedings of Forum “Math-for-Industry” 2013, Springer, 2014 *, 10.1007/978-4-431-54907-9, 51-65, 2014.04, This is a survey article on recent developments in visualization of large data, especially that of multivariate volume data. We present two essential ingredients. The first one is the mathematical background, especially the singularity theory of differentiable mappings, which enables us to capture topological features of given multivariate data in a mathematically rigorous way. The second one is a new development in computer science, called the joint contour net, which can encode topological structures of a given set of multivariate data in an efficient and robust way. Some applications to real data analysis are also presented.. |

20. |
Osamu Saeki, J. T. Hiratuka, Connected components of regular fibers of differentiable maps, *Topics on Real and Complex Singularities, Proceedings of the 4th Japanese-Australian Workshop (JARCS4), Kobe 2011, World Scientific *, 61-73, 2014.01. |

21. |
Osamu Saeki, Masamichi Takase, Desingularizing special generic maps, *Journal of Gökova Geometry Topology*, 7, 1-24, 2013.12, Let f : M → R^p be a special genericmap of a closed n-dimensional manifold M with n \geq p \geq 1. We study the condition for f to be factorized as f = \pi \circ \eta for an immersion (or an embedding) \eta : M → R^{n+1} and an orthogonal projection \pi : R^{n+1} → R^p. For various dimension pairs (n,p) we give answers to such a lifting problem. In particular, for the cases where p=1 and 2 we obtain complete results.. |

22. |
Osamu Saeki, J. T. Hiratuka, Triangulating Stein factorizations of generic maps and Euler characteristic formulas, *RIMS Kôkyûroku Bessatsu*, 38, 61-89, 2013.04. |

23. |
OSAMU SAEKI, Cobordism of exact links, *Algebraic and Geometric Topology*, 10.2140/agt.2012.12.1443, 12, 3, 1443–1455, 2012.07, In this paper we introduce the notion of exact links, which constitute a broad class of high dimensional links. . |

24. |
L. A. Lucas and O. Saeki, Fox property for codimension one embeddings of product of three spheres into spheres, *Algebraic and Geometric Topology*, 10.2140/agt.2011.11.3043, 11, 5, 3043-3064, 2011.12, Fox has shown that for every closed connected surface smoothly embedded in the 3-sphere, the closure of each component of its complement is diffeomorphic to the closure of the complement of a handlebody embedded in the 3-sphere. In this paper, we study a similar “Fox property” for smooth embeddings of Sp × Sq × Sr in Sp+q+r+1.. |

25. |
V. Blanlœil and O. Saeki, Cobordism of algebraic knots defined by Brieskorn polynomials, *Tokyo J. Math. *, 34, 429-443, 2011.12, In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only if the Brieskorn polynomials have the same exponents.. |

26. |
V. M. do Nascimento and O. Saeki, Curves in homogeneous spaces and their contact with 1-dimensional orbits, *Geometriae Dedicata*, 154, 117-131, 2011.01, Let α be a C∞ curve in a homogeneous space G/H. For each point x on the curve, we consider the subspace Sk of the Lie algebra of G consisting of the vectors generating a one parameter subgroup whose orbit through x has contact of order k with α. In this paper, we give various important properties of the sequence of subspaces G ⊃ S1 ⊃ S2 ⊃ S3 ⊃ · · ·. In particular, we give a stabilization property for certain well-behaved curves.We also describe its relationship to the isotropy subgroup with respect to the contact element of order k associated with α.. |

27. |
Fumiya Morishita and Osamu Saeki, Height functions on surfaces with three critical values, *Journal of the Mathematical Society of Japan*, 63, 153-162, 2011.01. |

28. |
Yasutaka Masumoto and Osamu Saeki, A smooth function on a manifold with given Reeb graph, *Kyushu Journal of Mathematics*, 65, 75-84, 2011.01. |

29. |
Osamu Saeki, Singular fibers and 4-dimensional cobordism group, *Pacific Journal of Mathematics*, 248, 233–256, 2010.04, 可微分写像の特異ファイバーを用いることにより，4次元可微分多様体の同境群を決定し，その生成元として複素射影平面が取れることを幾何的に明確に導出し，さらに既存の符号数定理に新しい証明を与えた．. |

30. |
Osamu Saeki, Special generic maps on open 4-manifolds, *Journal of Singularities*, 1, 1-12, 2010.01. |

31. |
Kazuichi Ikegami and Osamu Saeki, Cobordism of Morse maps and its application to map germs, *Mathematical Proceedings of the Cambridge Philosophical Society*, doi: 10.1017/S0305004109002321, 2009.02. |

32. |
O. Saeki and Y. Takeda, Surface links and their generic planar projections, *J. Knot Theory Ramifications*, 18 (2009), 41 - 66, 2009.01. |

33. |
J. T. Hiratuka and O. Saeki, Number of singularities of stable maps, *Journal of Geometry*, 89 (2008), 53-69 , 2008.09. |

34. |
O. Saeki and Y. Takeda, On 2-knots with total width eight, *Illinois Journal of Mathematics*, 52, 825-838, 2008.01. |

35. |
Topology of singular fibers of differentiable maps. |

36. |
V. Blanloeil and O. Saeki, Cobordism of fibered knots and related topics, *Advanced Stud. in Pure Math.
*, 46, pp. 1-47, 2007.02. |

37. |
O. Saeki and T. Yamamoto, Singular fibers and characteristic classes, *Topology Appl.
*, Vol. 155, 112–120., 2007.01. |

38. |
Concordance of four dimensional knots. |

39. |
M. Rachidi and O. Saeki, Extending generalized Fibonacci sequences and their Binet type formula, *Advances in Difference Equations *, 2006, Article ID 23849, 11 pages, 2006.07. |

40. |
Osamu Saeki, Stable mapping class groups of 4-manifolds with boundary, *Trans. Amer. Math. Soc. *, 358, pp. 2091-2104 , 2006.05. |

41. |
Osamu Saeki, Elimination of definite fold, *Kyushu J. Math. *, 60, pp. 363-382, 2006.05. |

42. |
O. Saeki and T. Yamamoto, Singular fibers of stable maps and signatures of 4-manifolds, *Geometry and Topology *, 10, pp. 359-399, 2006.04. |

43. |
Osamu Saeki, Cobordism of Morse functions on surfaces, universal complex of singular fibers, and their application to map germs, *Algebraic and Geometric Topology*, 6, pp. 539-572, 2006.04. |

44. |
S. Massago, O. M. Neto and O. Saeki, Open book structures on (n-1)-connected (2n+1)-manifolds, *J. Math. Sci. Univ. Tokyo*, 13, pp. 439-523 , 2006.04. |

45. |
Osamu Saeki, Morse functions with sphere fibers, *Hiroshima Math. J. *, 36, pp. 141-170, 2006.03. |

46. |
L. A. Lucas and O. Saeki, Codimension one embeddings of product of three spheres, *Topology Appl.*, 10.1016/j.topol.2003.06.005, 146, 409-419, 146-147, pp. 409-419
, 2005.01. |

47. |
O. Saeki and K. Suzuoka, Generic smooth maps with sphere fibers, *J. Math. Soc. Japan*, 10.2969/jmsj/1158241939, 57, 3, 881-902, 57, pp. 881-902, 2005.01. |

48. |
Osamu Saeki, Topology of singular fibers of differentiable maps, *Lecture Notes in Mathematics, Springer*, Vol.1854, 2004.01. |

49. |
B. Bernoussi, M. Rachidi and O. Saeki, Extending Bernoulli-Euler's method for finding zeros of holomorphic functions, *Fibonacci Quarterly*, 42, pp. 55-65, 2004.01. |

50. |
B. Bernoussi, M. Rachidi and O. Saeki, Factorial Binet formula and distributional moment formulation of generalized Fibonacci sequences, *Fibonacci Quarterly*, 42, 4, 320-329, 42, pp. 320-329, 2004.01. |

51. |
O. Saeki and Y. Takeda, Canceling branch points and cusps on projections of knotted surfaces in 4-space, *Proc. Amer. Math. Soc.*, 10.1090/S0002-9939-04-07487-8, 132, 10, 3097-3101, 132, pp. 3097-3101 , 2004.01. |

52. |
B. Bernoussi, W. Motta, M. Rachidi and O. Saeki, On periodic infinitely generalized Fibonacci sequences, *Fibonacci Quarterly*, 42, 4, 361-367, 42, pp. 361-367, 2004.01. |

53. |
V. Blanloeil, Y. Matsumoto and O. Saeki, Pull back relation for non-spherical knots, *J. Knot Theory Ramifications*, 10.1142/S0218216504003378, 13, 5, 689-701, 13, pp. 689-701
, 2004.01. |

54. |
W. Motta, M. Rachidi, and O. Saeki, Generalized Fibonacci sequences and Ostrowski's theorem, *Journal of Interdisciplinary Mathematics*, 7, pp. 221-231, 2004.01. |

55. |
K. Ikegami and O. Saeki, Cobordism group of Morse functions on surfaces, *J. Math. Soc. Japan*, 10.2969/jmsj/1191418765, 55, 4, 1081-1094, 55, 1081-1094, 2003.01. |

56. |
T. Ohmoto, O Saeki, and K. Sakuma, Self-intersection class for singularities and its application to fold maps, *Trans. Amer. Math. Soc.*, 10.1090/S0002-9947-03-03345-2, 355, 9, 3825-3838, 355, 3825-3838, 2003.01. |

57. |
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. |

58. |
L. A. Lucas and O. Saeki, Diffeomorphisms of a product of spheres and embedded spheres, *Topology Appl.*, 123, 471-478, 2002.01. |

59. |
L. A. Lucas and O. Saeki, Embeddings of S^p X S^q X S^r into S^{p+q+r}, *Pacific J. Math.*, 207, 447-462, 2002.01. |

60. |
O. Saeki, A. Szucs, and M. Takase, Regular homotopy classes of immersions of 3-manifolds into 5-space, *Manuscripta Math.*, 108, 13-32, 2002.01. |

61. |
Osamu Saeki, Open books on 5-dimensional manifolds, *Hiroshima Math. J.*, 32, 189-205, 2002.01. |

62. |
O. Saeki and M. Takase, Spin structures and codimension two embeddings of 3-manifolds up to regular homotopy, *Trans. Amer. Math. Soc.*, 10.1090/S0002-9947-02-03070-2, 354, 12, 5049-5061, 354, 5049-5061, 2002.01. |

63. |
V. Blanloeil and O. Saeki, Theory of concordance for non-spherical 3-knots, *Trans. Amer. Math. Soc.*, 354, 3955-3971, 2002.01. |

64. |
Osamu Saeki, Cobordism groups of special generic functions and groups of homotopy spheres, *Japanese J. Math.*, 28, 287-297, 2002.01. |

65. |
V. L. Carrara, M. A. S. Ruas, and O. Saeki, Maps of manifolds into the plane which lift to standard embeddings in codimension two, *Topology Appl.*, 110, 265-287, 2001.01. |

66. |
J. J. Nuno Ballesteros and O. Saeki, Euler characteristic formulas for simplicial maps, *Math. Proc. Camb. Phil. Soc.*, 130, 307-331, 2001.01. |

67. |
C. Biasi, J. Daccach and O. Saeki, A primary obstruction to topological embeddings for maps between generalized manifolds, *Pacific J. Math.*, 197, 275-289, 2001.01. |

68. |
B. Bernoussi, W. Motta, M. Rachidi and O. Saeki, Approximation of infinitely generalized Fibonacci sequences and their asymptotic Binet formula, *Fibonacci Quart.*, 39, 168--180, 2001.01. |

69. |
C. Biasi, J. Daccach and O. Saeki, A primary obstruction to topological embeddings and its applications, *Manuscripta Math.*, 104, 97-110, 2001.01. |

70. |
Osamu Saeki, Real Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in C^3, *J. Math. Soc. Japan*, 52, 409-431, 2000.01. |

71. |
W. Motta and O. Saeki, A two colour theorem and the fundamental class of a polyhedron, *"Real and complex singularities", ed. J. W. Bruce and F. Tari, Proceedings of the 5th Workshop on Real and Complex Singularities, Brazil, CHAPMAN & HALL/CRC Research notes in Math.*, 412, pp.94-109, 2000.01. |

72. |
W. Motta, M. Rachidi, and O. Saeki, Convergent infinitely generalized Fibonacci sequences, *Fibonacci Quarterly*, 38, 326-333, 2000.01. |

73. |
C. Biasi, J. Daccach and O. Saeki, On R-bordism of maps and obstruction to topological embeddings, *Osaka J. Math.*, 37, 527-535, 2000.01. |

74. |
T. Kobayashi and O. Saeki, Rubinstein-Scharlemann graphic of 3-manifold as the discriminant set of a stable map, *Pacific J. Math.*, 195, 101-156, 2000.01. |

75. |
O. Saeki and K. Sakuma, Stable maps between 4-manifolds and elimination of their singularities, *J. London Math. Soc.*, (2) 59, 1117-1133, 1999.01. |

76. |
C. Biasi, A. K. M. Libardi, and O. Saeki, On the Betti number of the union of two generic map images, *Topology Appl.*, 95, 31-46, 1999.01. |

77. |
O. Saeki and K. Sakuma, Elimination of singularities: Thom polynomial and beyond, *Singularity Theory*, Proceedings of the European Singularities Conference, Liverpool, August 1996, Dedicated to C. T. C. Wall on the occasion of his 60th birthday, London Math. Soc. Lect Note Series 263, Cambridge University Press, pp.291-304, 1999.01. |

78. |
W. Motta, M. Rachidi, and O. Saeki, On infinitely generalized Fibonacci sequences, *Fibonacci Quarterly*, 37, 223-232, 1999.01. |

79. |
Osamu Saeki, On algebraic unknotting numbers of knots, *Tokyo J. Math.*, 22, 425-443, 1999.01. |

80. |
Osamu Saeki, On punctured 3-manifolds in 5-sphere, *Hiroshima Math. J.*, 29, 255-272, 1999.01. |

81. |
A. Katanaga, O. Saeki, M. Teragaito, and Y. Yamada, Gluck surgery along a 2-sphere in a 4-manifold is realized by surgery along a projective plane, *Michigan Math. J.*, 46, 555-571, 1999.01. |

82. |
O. Saeki and K. Sakuma, Special generic maps of 4-manifolds and compact complex analytic surfaces, *Math. Ann.*, 313, 617-633, 1999.01. |

83. |
Osamu Saeki, Theory of fibered 3-knots in S^5 and its applications, *J. Math. Sci. Univ. Tokyo*, 6, 691-756, 1999.01. |

84. |
Osamu Saeki, On topological invariance of weights for quasihomogeneous polynomials, *Pitman Research Notes in Mathematics Series*, 381, pp.207-214, 1998.01. |

85. |
J. J. Nuno Ballesteros and O. Saeki, On the number of singularities of a generic surface with boundary in a 3-manifold, *Hokkaido Math. J. 27*, 27, 517-544., 1998.01. |

86. |
O. Saeki and K. Sakuma, Maps with only Morin singularities and the Hopf invariant one problem, *Math. Proc. Camb. Phil. Soc.*, 124, 501-511., 1998.01. |

87. |
O. Saeki and K. Sakuma, On special generic maps into R^3, *Pacific J. Math.*, 184, 175-193, 1998.01. |

88. |
C. Biasi and O. Saeki, Transversality with deficiency and a conjecture of Sard, *Trans. Amer. Math. Soc.*, 350, 5111-5122, 1998.01. |

89. |
A. Katanaga and O. Saeki, Embeddings of quaternion space in S^4, *J. Austral. Math. Soc.*, Ser. A 65, 313-325, 1998.01. |