2019 Fiscal Year Annual Research Report
| Project/Area Number |
19J12024
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| Research Institution | Hokkaido University |
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Principal Investigator |
TRAN NHAT TAN 北海道大学, 理学院, 特別研究員(DC2)
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| Project Period (FY) |
2019-04-25 – 2021-03-31
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| Keywords | hyperplane arrangement / quasi-polynomial / root system / matroid / Tutte polynomial |
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| Outline of Annual Research Achievements |
1. With M. Yoshinaga (Hokkaido) and A. U. Ashraf (Western Ontario), we introduce the notion of A-Eulerian polynomial producing an Eulerian-like polynomial for any subarrangement of a Weyl arrangement A. This polynomial describes how the characteristic quasi-polynomial of a certain class of subarrangements containing ideal subarrangements of A can be expressed in terms of the Ehrhart quasi-polynomial of the fundamental alcove.
2. After obtaining my PhD, I continue to exploit the Tutte-related polynomials of matroids with multiplicity. I have recently found new interpretations of the multivariate Tutte polynomial in terms of the expectation of functions of random restriction and contraction.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
1. Finding subarrangements of a Weyl arrangement that have the characteristic quasi-polynomials given by the Ehrhart quasi-polynomials of rational polytopes arising from the corresponding root system is a wide open problem. Our result is a good progress towards it. Interestingly, the ideal subarrangements belong to the computable list.
2. The results reveal a connection between Tutte polynomials of matroids with non-trivial multiplicity (e.g., arithmetic Tutte and G-Tutte polynomials) and probability theory. This is highly expected to generate considerable interest to the community (I was already informed by some researchers).
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| Strategy for Future Research Activity |
With the main focus on the G-Tutte polynomials and G-arrangements, I plan to work on the following problems:
Problem 1 (Topology). Is every abelian Lie group arrangement minimal, i.e., its complement has the homotopy type of a CW-complex in which the number of k-cells equals the k-th Betti number? I am working with M. Yoshinaga (Hokkaido) and E. Delucchi (Fribourg) on this problem. I am invited to take part in a research period by Delucchi at U. Fribourg, Switzerland over the period May-July, 2020.
Problem 2 (Matroid theory). What matroidal structure can the G-Tutte polynomial be associated with? I am recently working on this problem with I. Martino (KTH Stockholm). I am invited by him to give a talk at the conference Tropical and Algebraic Encounter at U. Catania, Italy in December 2020.
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Research Products
(10 results)
