| 研究課題/領域番号 |
23K03108
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| 研究種目 |
基盤研究(C)
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| 配分区分 | 基金 |
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| 応募区分 | 一般 |
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| 審査区分 |
小区分11020:幾何学関連
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| 研究機関 | 東京工業大学 |
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研究代表者 |
KALMAN Tamas 東京工業大学, 理学院, 准教授 (00534041)
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| 研究期間 (年度) |
2023-04-01 – 2028-03-31
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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
4,680千円 (直接経費: 3,600千円、間接経費: 1,080千円)
2027年度: 780千円 (直接経費: 600千円、間接経費: 180千円)
2026年度: 780千円 (直接経費: 600千円、間接経費: 180千円)
2025年度: 780千円 (直接経費: 600千円、間接経費: 180千円)
2024年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
2023年度: 1,430千円 (直接経費: 1,100千円、間接経費: 330千円)
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| キーワード | Knot invariants / Algebraic combinatorics / 代数的組み合わせ論 / knot invariants |
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| 研究開始時の研究の概要 |
We intend to incorporate quantum knot invariants into the Floer-style theory of low-dimensional manifolds. We plan to do so by investigating discrete structures, such as lattice point arrangements, found within certain Floer homology groups. The necessary combinatorics will be developed as well.
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| 研究実績の概要 |
Lilla Tothmeresz of Eotvos University, Hungary and I published the paper “h*-vectors of graph polytopes using activities of dissecting spanning trees” in Algebraic Combinatorics vol. 6 (2023), no. 6, pp. 1637--1651. This gives an easily computer-implementable algorithm for computing these h*-vectors or polynomials, which we have already relied on to collect data and formulate conjectures about the behavior of these objects. Our preprint "Degrees of interior polynomials and parking function enumerators" is very near publication (pending revision). In it we prove one such conjecture, on the length of the h*-vector and how it can be expressed in terms of the graph structure in many cases. We have other manuscripts in preparation. The most important one is titled "Ehrhart theory of symmetric edge polytopes via ribbon structures". This contains another approach to computing the h*-vector in perhaps its most important instance, one that is better suited to potentially prove properties such as gamma-positivity. All of these combinatorial advancements are closely related to knot invariants as well.
I have also conducted joint research with Soheil Azarpendar and Andras Juhasz of Oxford University on the trapezoidal conjecture of Fox, about the Alexander polynomial of an arbitrary alternating link, which we establish in a variety of special cases. The preprint on this work is near completion.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
All four collaborations that I described are well within my area of expertise, at the intersection of knot theory and algebraic combinatorics. One has already produced a number of publications, and in two others the writing is already underway.
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| 今後の研究の推進方策 |
Jointly with Remi Avohou of the Okinawa Institute of Science and Technology, we are working on two projects about graphs on surfaces. One aims at developing a version of Conway's polynomial for bipartite ribbon graphs, via counts of certain quasitrees of various genera. The goal of our other project is to associate a polymatroid to these same objects. We hope to achieve this by extending the notion and properties of delta-matroids to a new theory of delta-polymatroids.
I am also working on a joint preprint with Karola Meszaros of Cornell University, USA and Alexander Postnikov of the Massachusetts Institute of Technology, USA. This examines a graph-theoretical generalization of the Alexander polynomial and proves that its sequence of coefficients is log-concave.
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