Kyushu University [Toshio Sumi (Professor) Faculty of Arts and Science, Division for Theoretical Natural Science]

My research focuses on transformation theories. I am currently interested in the interactions between finite group actions on spheres and representations of finite groups.
Also I am studying properties of 2-knots by twisted Alexander polynomials and the rank of real tensors.
I contribute to the KIKAN education for the way to active learners and focusing on mathematical approaches to developing media environments.

Research

Research Interests

Classification of knots by twisted Alexander polynomials
keyword : ribbon 2-knots, twisted Alexander polynomial
2018.02.
On Borsuk-Ulam groups
keyword : representation spaces
2017.10.
Relationship between plural typical ranks of tensors and existence of a nonsingular multilinear map
keyword : typical rank
2014.01.
Rank, Typical rank of higher order tensors
keyword : tensor, arrays, rank, typical rank, generic rank
2008.07I am studying about a relationship between two tangential representations over fixed point on a sphere with smooth finite group action. .
Finite group action on a sphere with two fixed points and tangential representation spaces
keyword : group action, sphere, tangential representation
2006.08I am studying about a relationship between two tangential representations over fixed point on a sphere with smooth finite group action. .

Academic Activities

Books

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Toshio Sumi, Toshio Sakata, Mitsuhiro Miyazaki, Algebraic and Computational Aspects of Real Tensor Ranks, Springer, SpringerBriefs in Statistics, 2016.03, Recently, multi-way data or tensor data have been employed in various applied fields. We consider the decomposition of a tensor datum into a sum of rank-1 tensors, where rank-1 tensors are considered to be the simplest tensors. The minimal length of the rank-1 tensors in the sum is called the rank of the tensor. The objective of rank determination is to answer the question, “How many rank-1 tensors are required to express the given tensor?” In this book, we focus on the rank over the real number field R, which is particularly interesting for statisticians..

Reports

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1.	Toshio Sumi, Alternating groups and Borsuk-Ulam groups, RIMS講究録 2135, 2019.11, [URL].
2.	角　俊雄, Finite groups with Smith equivalent, nonisomorphic representations, 変換群論シンポジューム2006報告集, 2006.11.

Papers

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1.	Taizo Kanenobu, Toshio Sumi, Extension of Takahashi’s ribbon 2knots with isomorphic groups, Journal of Knot Theory and Its Ramifications, 10.1142/S021821652350013X, 32, 2, 2350013 (12 pages), 2023.03, We give infinitely many pairs of ribbon 2-knots of 1-fusion in S4 with isomorphic knot groups, which extend Takahashi’s examples. They are distinguished by the trace sets, which are calculated by using SL(2, C)-representations of the knot groups..
2.	Taizo Kanenobu, Toshio Sumi, Twisted Alexander polynomial of a ribbon 2-knot of 1-fusion, Osaka Journal of Mathematics, 57, 4, 789-803, 2020.10.
3.	Taizo Kanenobu, Toshio Sumi, Suciu's ribbon 2-knots with isomorphic group, Journal of Knot Theory and Its Ramifications, https://doi.org/10.1142/S0218216520500534, 29, 7, 2050053 (9 pages), 2020.09.
4.	Taizo Kanenobu, Toshio Sumi, Classification of ribbon 2-knots presented by virtual arcs with up to four crossings, Journal of Knot Theory and Its Ramifications, http://dx.doi.org/10.1142/S0218216519500676, 28, 10, 1950067 (18 pages), 2019.10.
5.	Toshio Sumi, Mitsuhiro Miyazaki, Toshio Sakata, Typical ranks of semi-tall real 3-tensors, Linear and Multilinear Algebra, 10.1080/03081087.2019.1637811, 2019.07.
6.	Taizo Kanenobu, Toshio Sumi, Classification of a family of ribbon 2-knots with trivial Alexander polynomial, Communications of the Korean Mathematical Society, http://doi.org/10.4134/CKMS.c170222, 33, 2, 591-604, 2018.04.
7.	Mitsuhiro Miyazaki, Toshio Sumi, Toshio Sakata, Typical ranks of certain 3-tensors and absolutely full column rank tensors, Linear and Multilinear Algebra, http://www.tandfonline.com/doi/abs/10.1080/03081087.2017.1292994, 66, 1, 193-205, 2017.03, In this paper, we study typical ranks of 3-tensors and show that there are plural typical ranks for m x n x p tensors over the real number field in the following cases: (1) 3≦m≦h(n) and (m-1)(n-1)+1≦p≦(m-1)n, where h is the Hurwitz-Radon function, (2) m=3, n=3 (mod 4) and p=2n-1, (3) m=4, n=2 (mod 4), n≧6 and p=3n-2, (4) m=6, n=4 (mod 8), n≧12 and p=5n-4, (5) m=10, n=24 (mod 32) and p=9n-8..
8.	Toshio Sumi, Mitsuhiro Miyazaki, Toshio Sakata, Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals, Journal of Algebra, http://dx.doi.org/10.1016/j.jalgebra.2016.09.028, 471, 1, 409-453, 2017.02, Let m,n≥3, (m−1)(n−1)+2≤p≤mn, and u=mn−p. The set R^u×n×m of all real tensors with size u×n×m is one to one corresponding to the set of bilinear maps R^m×Rⁿ→R^u. We show that Rm×n×p has plural typical ranks p and p+1 if and only if there exists a nonsingular bilinear map R^m×Rⁿ→R^u. We show that there is a dense open subset O of R^u×n×m such that for any Y∈O, the ideal of maximal minors of a matrix defined by Y in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal. Further, we show that there is a dense open subset T of R^n×p×m and continuous surjective open maps ν:O→R^u×p and σ:T→R^u×p, where R^u×p is the set of u×p matrices with entries in R, such that if ν(Y)=σ(T), then rank T=p if and only if the ideal of maximal minors of the matrix defined by Y is a real prime ideal..
9.	Toshio Sumi, Richness of Smith equivalent modules for finite gap Oliver groups, Tohoku Journal of Mathematics, doi:10.2748/tmj/1474652268, 68, 3, 457-469, 2016.09, [URL].
10.	Toshio Sumi, Mitsuhiro Miyazaki, Toshio Sakata, Typical rank of mxnx(m-1)n tensors with 3Linear and Multilinear Algebra, http://dx.doi.org/10.1080/03081087.2014.910206, 63, 5, 940-955, 2015.07, Let 3\rho(n), then the set of m x n x (m-1)n tensors has only one typical rank (m-1)n..
11.	Toshio Sumi, Centralizers of gap groups, Fundamenta Mathematicae, 10.4064/fm226-2-1, 226, 101-121, 2014.06, A finite group G is called a gap group if there exists an RG-module which has no large isotropy groups except at zero and satisfies the gap condition. The gap condition facilitates the process of equivariant surgery. Many groups are gap groups and also many groups are not. In this paper, we clarify the relation between a gap group and the structures of its centralizers. We show that a nonsolvable group which has a normal, odd prime power index proper subgroup is a gap group..
12.	角俊雄, Smith sets of non-solvable groups whose nilquotients are cyclic groups of order 1,2, or 3, RIMS Kokyuroku Bessatsu, B39, 149-165, 2013.06, Let $G$ be a finite group. Two real $G$-modules $U$ and $V$ are called Smith equivalent if there exists a smooth action of $G$ on a sphere with two fixed points at which tangential representations are isomorphic to $U$ and $V$ respectively. The Smith set of $G$ is the subset of the real representation ring of $G$ consisting differences of Smith equivalent $G$-modules. We discuss when the Smith set of an Oliver group becomes an abelian group and give several examples of non-solvable groups of which the Smith sets are groups. .
13.	Krzysztof Pawalowski, Toshio Sumi, The Laitinen Conjecture for finite non-solvable groups, Proceedings of the Edinburgh Mathematical Society , http://dx.doi.org/10.1017/S0013091512000223, 56, 1, 303-336, 2013.02, For any finite group G, we impose an algebraic condition, the G^{nil}-coset condition, and prove that any finite Oliver group G satisfying the G^{nil}-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A_{6}) or PΣL(2, 27), the G\sup{nil}-coset condition holds if and only if r_{G} ≥ 2, where r_{G} is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A_{6})..
14.	Toshio Sumi, Mitsuhiro Miyazaki, Toshio Sakata, Typical ranks for m x n x (m-1)n tensors with mLinear Algebra and its Applications, 10.1016/j.laa.2011.08.009, 438, 2, 953-958, 2013.01, In various application fields, tensor type data are used recently and then a typical rank is important. There may be more than one typical ranks over the real number field. It is well known that the set of 2 × n × n tensors has two typical ranks n, n + 1 for n>=2, that the set of 3 × 4 × 8 tensors has two typical ranks 8, 9, and that the set of 4×4×12 tensors has two typical ranks 12, 13. In this paper, we show that the set of m×n×(m−1)n tensors with mtypical ranks (m − 1)n, (m − 1)n + 1 if mHurwitz–Radon function..
15.	Toshio Sumi, The gap hypothesis for finite groups which have an abelian quotient group not of order a power of 2, Journal of the Mathematical Society of Japan, 10.2969/jmsj/06410091, 64, 1, 91-106, 2012.01.
16.	Toshio Sumi and Toshio Sakata, Connectivity for 3x3xK contingency tables, Journal of Algebraic Statistics, 2, 1, 54-74, 2011.08, [URL].
17.	Toshio Sumi and Toshio Sakata, 2-neighborhood theorem for 3x3x3 contingency tables, Journal of the Indian Society for Probability and Statistics, 12, 66-84, 2010.12.
18.	Toshio Sumi and Toshio Sakata, The Set of 3x4x4 Contingency Tables has 3-Neighborhood Property, Proceedings of COMPSTAT'2010, Electronic Supplementary Material (e-book) 19th International Conference on Computational Statistics, Paris France, August 22-27, 2010, 1629-1636, 2010.09.
19.	Toshio Sumi, Mitsuhiro Miyazaki, and Toshio Sakata, About the maximal rank of 3-tensors over the real and the complex number field, Ann. Inst. Stat. Math., 10.1007/s10463-010-0294-5, 62, 807-822, 2010.09.
20.	Toshio Sumi, Mitsuhiro Miyazaki, and Toshio Sakata, Rank of 3-tensors with 2 slices and Kronecker canonical forms, Linear Algebra and its Applications, vol. 431, 1858-1868, 2009.10.
21.	Krzysztof Pawalowski and Toshio Sumi, The Laitinen Conjecture for finite solvable Oliver groups, Proceedings of American Mathematical Society, Proceedings of the American Mathematical Society 137 (6), 2147-2156, 2009, 2009.06.

Works, Software and Database

Presentations

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1.	Toshio Sumi, Construction of gap modules, The 41st Symposium on Transformation Groups, 2014.11.
2.	Toshio Sumi, Note on tangential representations on a sphere, Joint Meeting of the German Mathematical Society and the Polish Mathematical Society, 2014.09, Our target is Smith sets for Oliver groups. A solvable Oliver group possessing non-trivial Smith set is not determined, but we know completely for non-solvable groups. In this talk, we give many non-solvable groups of which Smith set are additive groups. .
3.	The Smith equivalence problem and Smith sets of Oliver groups.
4.	Toshio Sumi, The Smith equivalence problems for finite Oliver groups, Geometry of manifolds and group actions, 2012.09, [URL], Let G be a finite group. A real G-module V is a finite dimensional real vector space with a linear G-action. Two real G-modules U and V are called Smith equivalent if there exists a smooth G-action on a homotopy sphere with exactly two fixed points, at which the tangential G-modules are isomorphic to U and V, respectively. The Smith equivalence problem is stated as follows. Is it true that two Smith equivalent G-modules are isomorphic? In this talk, I would like to introduce a history of this problem and recent results..
5.	Nonsolvable groups of which the Smith sets are groups.
6.	Existence and construction of absolutely nonsingular tensors.

Educational

Educational Activities

My Education activities includes KIKAN education and Calulus/Linear Algebra for first year undergraduated students.

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