| Toshio Sumi | Last modified date：2024.01.15 |
Professor /
Division for Theoretical Natural Science /
Faculty of Arts and Science
Presentations
| 1. | Toshio Sumi, Progress toward the Laitinen Conjecture, Transformation Group Theory -The path we have walked along, and the way to progress-, 2021.02, [URL], Let G be a finite Oliver 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 Laitinen-Smith equivalent if there exists a smooth G-action on a homotopy sphere Σ with exactly two fixed points, at which the tangential G-modules are isomorphic to U and V , respectively, and in addition Σh is connected for any 2-element h of order ≥ 8. The Laitinen Conjecture is stated as follows. “ If the number of real conjugacy classes of elements of G not of prime power order is greater than 1, then there exist nonisomorphic Laitinen-Smith equivalent G-modules. ” Although there exist counterexamples of the Laitinen conjecture, infinitely many Oliver groups meet Laitinen’s expectations. In this talk, I would like to explain why it is not easy to get a counterexample.. |
| 2. | Toshio Sumi, Smith Problem and Laitinen's Conjecture, Conference celebrating the 70th birthday of Prof. Krzysztof Pawałowski, 2021.01, [URL], Professor Krzysztof Pawa lowski is studying the Smith problem. Corresponding to the Smith problem, Laitinen conjectured that if a finite Oliver group G has the property that the number of real conjugacy classes of elements not of prime power order is greater than or equal to 2, there exist nontrivial Laitinen–Smith equivalent G-modules. Here, two G-modules are called Laitinen–Smith equivalent if they are tangential representations of a sphere with a smooth G-action having exactly two fixed points under mild connectivity condition. First, Morimoto pointed out that Aut(A6) is a counterexample for the Laitinen’s conjecture. After that, Pawa lowski and I concluded that it is a unique counterexample among unsolvable groups. Although there exist a few counterexamples among solvable Oliver groups, many groups satisfy the Laitinen’s conjecture. I will talk about a sufficient condition for a group that the Laitinen’s conjecture is true. This is a joint work with professor Krzysztof Pawalowski.. |
| 3. | Let G be a finite group. A group G acts smoothly on a smooth manifold M with fixed point x. By the derivative, the tangent space Tx(M) at x regards as a representation G-space. Wasserman studied whether an isovariant G-map f : V → W between representation G-spaces V and W has dim V − dim VG ≤ dim W − dim WG. We say G is a Borsuk-Ulam group if the above question is always true. We will discuss Borsuk-Ulam groups from the viewpoint of computer-aided experiments.. |
| 4. | , [URL]. |
| 5. | Toshio Sumi, SL(3, F_q) is a BUG or ..., Workshop on Geometric Discrete Mathematics II, 2018.12. |
| 6. | Toshio Sumi, Sufficient condition to be a Borsuk-Ulam group, UMI – SIMAI – PTM, 2018.09. |
| 7. | Toshio Sumi, Mitsuhiro Miyazaki, Toshio Sakata, Monomial preorder, ideals of minors, and plural typical ranks, Innovation in statistics and related mathematics through computational algebraic statistics, 2015.03, Let m, n The set of all real tensors with size m x n x p is one to one corresponding to the set of bilinear maps from R^m x R^n to R^p. We show that if there exists a non-singular bilinear map for (m-1)(n-1)+1 As an approach for the converse problem, we consider monomial preorders and ideals of minors. This is a joint work with Prof. Mitsuhiro Miyazaki and Prof. Toshio Sakata.. |
| 8. | Toshio Sumi, Construction of gap modules, The 41st Symposium on Transformation Groups, 2014.11. |
| 9. | 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. . |
| 10. | Toshio Sumi, Toshio Sakata, Mitsuhiro Miyazaki, Rank of tensors with size 2 x ... x 2, Computational Algebraic Statistics, Theories and Applications (CASTA 2014), 2014.01. |
| 11. | Toshio Sumi, Nonsolvable groups possessing gap modules, Knots, Manifolds, and Group Actions, 2013.09, For a finite group G, a gap G-module is a finite dimensional real vector space with linear G-action satisfying the suitable condition. A finite group not of prime power order is called a gap group if there exists a gap module. The purpose of this talk is to show a necessary and sufficient condition for a group G to be a gap group and to give a gap module. We also discuss when a nonsolvable group is a gap group.. |
| 12. | The Smith equivalence problem and Smith sets of Oliver groups. |
| 13. | 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.. |
| 14. | On rank of tensors. |
| 15. | Nonsolvable groups of which the Smith sets are groups. |
| 16. | Existence and construction of absolutely nonsingular tensors. |
| 17. | On dimensions of vector spaces of consisting of nonsingular n×n matrices except zero. |
| 18. | Dimensions of the fixed point sets of representation spaces and the Smith problem . |
| 19. | On absolutely nonsingular tensors -characterization, properties and construction-. |
| 20. | , [URL]. |
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