| Project/Area Number |
19K14493
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Hokkaido University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2019: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
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| Keywords | Hyperplane arrangements / Lefschetz propterty / Graph theory / Freeness / Lefschetz propety / Falk invariant / Lefschetz properties / Domination theory / Gain graphs |
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| Outline of Research at the Start |
I will study the class of free arrangements and the famous Terao’s conjecture, central lower series and Falk invariants. Specifically, I will study the hyperplane representations of gain graphs and simplicial complexes and characterize combinatorically the graphs and complexes that give rise to free and supersolvable arrangements; I will study inversion hyperplane arrangements associated to permutations, and characterize the hypersolvable ones and the ones for which the characteristic polynomial factors; I will study the Falk invariant of complexified real arrangements.
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| Outline of Final Research Achievements |
I obtained 16 accepted publications. With Bigatti and Palezzato, we found new characterizations of freeness. With Palezzato, we determined when an arrangement and its reduction modulo a prime have isomorphic lattices; we wrote a package for CoCoA to do computations with arrangements; we studied free and plus-one generated arrangements, we described how to compute the associated prime ideals of their Jacobian ideal and we proved that the localization of a plus-one generated arrangement is free or plus-one generated. We studied the k-Lefschetz properties in the non-Artinian case. I studied the relations between freeness over finite fields and the rationals for multiarrangements. With Pahlavsay and Palezzato, we studied domination and k-tuple total domination sets of graphs. With them and Kazemnejad, we described the domination, the total domination and the total dominator coloring numbers of the middle graph associated to a simple graph.
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| Academic Significance and Societal Importance of the Research Achievements |
My research advanced our current knowledge on hyperplane arrangements, in particular on the class of free ones, and our knowledge on domination problems for graphs. In addition, my research connected the study of the Lefschetz properties and of the Jacobian algebra of arrangements.
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