Order Seminar
Order Seminar
Order Seminar is organized for the purposes of discussing mathematics. The seminar forcuses on hyperplane arrangements, posets, polytopes, lattice points, enumerations, magnitudes, magnitude homology, periods, holonomic series, etc. Related topics in combinatorics, topology, geometry, analysis, algebra, algebraic geometry, statistics, etc. are also welcome. The talks are supposed to be given in English or Japanese. (Organizer: Masahiko Yoshinaga)
Usually the seminar is held in Department of Math, Osaka University (通常は阪大理学部数学教室で行います)
2025/10/17(Fri) B???, 10;30-12:00, 14:00-15:30
Speaker: 三上陵太（台湾中央研究院）
Title: On Kähler package in tropical geometry
Abstract: Kähler package (hard Lefschetz theorem and Hodge-Riemann relations) for Chow rings of matroids played a central role in Adiprasito-Huh-Katz's proof of the log-concavity of the coefficients of the characteristic polynomials. In the first talk, after a brief introduction to tropical geometry, I will explain another proof of Kähler package independently given by Amini-Piquerez and Braden-Huh-Matherne-Proudfoot-Wang based on tropical modifications, which are ``deletion-contraction''-type operations in tropical geometry. In the second talk, I will discuss Amini-Piquerez's generalization of Kähler package to smooth (i.e., locally matroidal) projective tropical varieties using tropical cohomology. The proof is elementary and conceptually similar to Kähler package for graded quotients of limit mixed Hodge structures of semi-stable degenerations of smooth complex projective varieties. I will explain a slightly different and simpler proof, which is exactly the same as the complex geometric situation.
(cf. Amini-Piquerez ``Hodge theory for tropical varieties'' https://arxiv.org/abs/2007.07826 )
2025/09/22 (Mon) B317, 10:00-
Speaker: Francesco Conti (Inria - CNR)
Title: A representation theorem for linear Group Equivariant Non-Expansive Operators.
Abstract: Group Equivariant Non-Expansive Operators (GENEOs) represent a class of functional operators that preserve specific symmetries while maintaining non-expansivity properties, providing a unifying mathematical framework for many fundamental operators in artificial intelligence, from convolutional neural networks to persistent homology. This seminar presents a complete representation theorem for linear GENEOs through a class of measures called permutant measures. This correspondence allows us to interpret every linear GENEO as a weighted linear combination of elementary operators determined by the orbits of group actions. As a consequence of this representation, we prove that the space of linear GENEOs is compact and forms a polytope.
2025/05/8 (Thu) B317, 10:30-
Speaker: 水野宏亮（名古屋大学）
Title: Spanning trees and their relations in Galois covers
Abstract: We introduce the relations among the complexities of intermediate graphs of a Galois cover 𝑌/𝑋. Here, the complexities refer to the number of spanning trees of graphs. Hammer, Mattman, Sands, and Vallières prove that for a (ℤ/2ℤ)^𝑚-cover 𝑌/𝑋, the complexity of 𝑌 can be expressed in terms of the complexities of the intermediate graphs. We present a generalization of this result to arbitrary finite Galois covers. Our formulas are graph-theoretic analogues of the Brauer-Kuroda relations in algebraic number theory. Finally, we examine in detail the case where the Galois group is cyclic.
2025/04/24 (Thu) B317, 10:30-
Speaker: Marina Salamero Cebollero (Seville)
Title: The Davis and Deligne complexes, and a generalization for Dyer groups.
Abstract: One can deduce nice results for a group that admits proper, cocompact actions by isometries on a locally compact space with non-positive curvature. In this talk we will introduce the construction of actions of groups on cell complexes and give examples for Coxeter (resp. Artin) groups, which act on the Davis (resp. Deligne) complexes. We will also see a generalization of these constructions for Dyer groups due to M. Soergel.
2025/04/17 (Thu) B317, 10:30-
Speaker: Maria Cumplido Cabello (Seville)
Title: Can topological properties of braids be generalized algebraically to Artin groups?
Abstract: Braids can be defined algebraically as finite-type Artin groups or topologically as the mapping class group of the n-punctured disc. Topological methods, particularly those involving the action of the mapping class group on the curve complex of the surface, have proven highly effective in establishing results about braid groups. Meanwhile, finite-type Artin groups possess a rich and well-studied combinatorial framework known as Garside theory. However, many results known for braid groups via topological methods remain unproven in the broader setting of finite-type Artin groups. In this seminar, we will explore possible approaches for translating these topological tools into the language of Garside theory, with the aim of generalizing such results beyond braid groups.
2024/12/06 (Fri) B317, 08:50-
Speaker: 大場亮俊（東京大学）
Title: The g-conjecture over characteristic 2 and beyond.
Abstract: Adiprasito resolved the g-conjecture regarding the characterization of the f-vectors of simplicial spheres. I will explain a proof of this result based on a field of characteristic 2 by Papadakis-Petrotou (and Karu-Xiao). I will also briefly talk about our generalization to balanced simplicial complexes.
Reference: Ryoshun Oba, Multigraded strong Lefschetz property for balanced simplicial complexes. arXiv:2408.17110
2024/08/25 (Sun) One-day workshop "Bundles and lattices" (organized by Takuro Abe, Daniele Faenzi, Masahiko Yoshinaga), at Osaka University (Toyonaka)
会場： 大阪大学全学教育推進機構 総合棟 I (ステューデント・コモンズ 2F) セミナー室C（会場は理学部と同じ豊中キャンパス内ですが、理学部からは徒歩10分程度離れています）Map (Japanese） 、ここに載せると長くなるので、こちらが把握している参加予定者には "practical information" をメールで送りました。必要でしたらご連絡ください。
The Place: is "Seminar Room C" at the 2F of the building 64 in the detailed map of Toyonaka Campus Map(English) . (It is not in the Faculty of Science).
Speakers: Takuro Abe (Rikkyo), Daniele Faenzi (Dijon/Osaka), Kota Yoshioka (Kobe) , Osamu Iyama (Tokyo), Akihiro Kanemitsu (Tokyo Metropolitan), Ryo Uchiumi (Osaka)
Tentative Schedule:
9:10-10:10 Kanemitsu
10:20-11:20 Faenzi
11:30-12:30 Yoshioka
14:00-15:00 Abe
15:10-16:10 Uchiumi
16:20-17:20 Iyama
Title and Abstract:
● Takuro Abe (Rikkyo) : Cokernel of the Euler restriction map of projective line arrangements.
Abstract: When a projective line arrangement in the projective plane is given, we can construct a vector bundle of rank two consisting of polynomial vector fields tangent to each line. To investigate when it splits into a direct sum of line bundle is one of the most important problem, and a famous criterion for splitting is found by Yoshinaga in 2004 based on the cokernel of the restriction map onto its hyperplane section. On the other hand, classically there are a rather combinatorial restriction map of this bundle called the Euler restriction map, and there have been no research on the cokernel of this Euler restriction map. In this talk, we give an upper bound of the dimension of this cokernel in terms of a special value of a characteristic polynomial, and show that this upper bound is attained when the original bundle splits. This is a joint work with Hiraku Kawanoue.
● Daniele Faenzi (Dijon/Osaka): Logarithmic derivations along invariant divisors.
Abstract: Sheaves of logarithmic vector fields, or "derivations", tangent to a given divisor, play an important role in singularities and deformation theory. We will discuss some results about stability of these sheaves and a connection with projective duality. Then we will focus on invariant divisors for the action of a Lie group and examine the case of the usual determinant and of adjoint discriminants. In the first case, we describe étale charts of the moduli space of sheaves of logarithmic vector fields, in the second one we focus on graded free resolutions and projective dimension.
●Osamu Iyama (Tokyo): Coxeter arrangements, Preprojective algebras and cDV singularities
Abstract: A preprojective algebra is defined for each graph. We will explain how the structure of the derived category of a preprojective algebra is encoded in the corresponding Coxeter arrangement. We will apply this picture to give a classification of maximal modifications of cDV singularities. This is a joint work with Michael Wemyss.
● Akihiro Kanemitsu (Tokyo Metropolitan) : Mukai pairs and associated K3 surfaces.
Abstract: A Mukai pair $(X,E)$ is a pair of a smooth projective variety $X$ and an ample vector bundle $E$ with $c_1(X)=c_1(E)$. Such pairs are completely classified when the rank of $E$ is at least $\dim X -2$. When the rank of $E$ is $\dim X -2$, Mukai pairs are related to K3 surfaces; the zero locus of a section of $E$ is a K3 surface. In this talk, we discuss the structure of these K3 surfaces from the viewpoint of their moduli.
● Kota Yoshioka (Kobe) : aCM bundles on a generic K3 surface of degree 2.
Abstract: An arithmetically Cohen-Macaulay (i.e. aCM) vector bundle on a polarized variety $(X,H)$ is a vector bundle $E$ satisfying $h^i(E(nH))=0$ for all $n$ and $0<i<\dim X$. In my talk, I will explain a result on aCM bundles on a polarized K3 surface of degree 2.
● Ryo Uchiumi (Osaka): Exponents of multiarrangements of three lines over a field of positive characteristic.
Abstract: In this talk, I will show how exponents are determined in multiarrangements of three lines on a field of positive characteristic. I will also explain the filling of the connected components of the multiplicity lattice. This is joint work with Shuhei Tsujie (Hokkaido Univ. of Education, Asahikawa campus).
2024/05/24 (Fri) E301, 10:30-
Speaker: Leonie Mühlherr (Bielefeld U)
Title: Graphic hyperplane arrangements and their freeness
Abstract: Graphic hyperplane arrangements are an interesting example of arrangements, since they are the subarrangements of the well-studied braid arrangement and have a strong connection to graph theory. This makes it possible to use graph theoretical tools to study them and specifically their module of logarithmic derivations. This talk will give an introduction to concepts of both fields of study and showcase their connections with a specific focus on freeness and chordality. Some parts of the talk are based on joint work with Takuro Abe, Lukas Kühne and Paul Mücksch.
2023/09/26-27 (大阪大学理学部 E 404,) 組合せ論的ホッジ理論勉強会（第0回）
2023/07/03 (B342-346) 17:00-18:00, 阿部拓郎（立教大学）
Title: 超平面配置の特性多項式の整数根の代数的理解
Abstract: 超平面配置の特性多項式は組み合わせ論的に定義される不変量であるが、それが整数根を持つ場合がある。組み合わせ不変量が整数値であればそれは何かを数えているに違いない、と思いたくなるのが数学者である。本講演ではその代数的な理解について紹介する。
2023/06/02 10:30-11:30 (B342-346), 内海凌(北海道大学)
Title: 符号つきグラフの辺に付随する超八面体群の互換の積
Abstract: グラフの辺に対してその端点を入れ換える互換を対応させるとき，辺集合に全順序を定めることで，グラフの辺に付随する互換の積を得ることができる．グラフがn個の頂点をもつ木であるときには，辺集合の任意の全順序に対して，グラフの辺に付随する互換の積が長さnの巡回置換になることが，Denesによって知られている．
本講演では，符号つきグラフおよび超八面体群に関するアナロジーについて考察し，Denesによる結果の拡張を与える．
2023/06/02 9:20-10:20 (B342-346) 吉永正彦（大阪大学）
Title: ポセットの －１ 倍について
Abstract: Stanley によって示されたポセットの順序多項式の相互律を幾何学的に実現する、というモチベーションと関連付けながら、「ポセットの －１ 倍をどのように定義するべきか？」という問題について論じる。（T. Hasebe, M. Miyatani, T. Yoshida との共同研究に基づく）
2023/05/09 17:00-18:00 (B313) Paul Mucksch (立教大学)
Title: On oriented matroids and their representations
2023/03/29 10:30-12:00 (B342), 中村勇哉(東京大学)
Title: 周期グラフのエルハート理論に向けて
Abstract: 周期グラフとは, 格子Z^Nが自由に作用しているグラフであってその商グラフが有限グラフになるものをいう. 周期グラフは, 数理結晶学における研究対象になっている他, 幾何学的群論においてもvirtually abelian groupのケーリーグラフとして自然に現れる対象である. グラフのgrowth sequence b(n)は, グラフのある頂点からスタートしてグラフ距離n以下の頂点の数え上げとして定義される. 本セミナーではまず,「周期グラフのgrowth sequenceがquasi-polynomial type (十分nが大きい所でquasi-polynomial) になる」(中村-坂本-間瀬-中川) という結果を紹介する. また, 残りの時間で通常のエルハート理論との類似の現象について議論したい.
Reference: https://onlinelibrary.wiley.com/iucr/doi/10.1107/S2053273320016769
2023/03/24 10:30-12:00 (B342), Benoit Guerville-Balle (RIMS)
Title:
Abstract:
2022/12/08 or 09 () 10:30-11:30 (B313-317 Dept. of Math. Osaka University)
Speaker: 山口徹（九州大学）
Title: タイトル: 超平面配置のfree pathについて
Abstract: 超平面配置とはベクトル空間中の超平面の有限集合であり、その対数的ベクトル場のなす加群が自由加群となるような超平面配置を特に自由配置と呼ぶ。 自由配置において、ある自由配置から超平面を一枚除去した配置については研究が進んでおり、その構造についても多くのことが明らかとなっている。その一方で、複数枚の超平面を除去した配置についてはあまり研究が進んでいないのが現状である。
そこで本講演では、そうした研究の足がかりとなる、自由性を保つように複数枚の超平面を除去する道（free path）に関するLukas Kühne氏による予想を紹介し、それに証明を与えることとする。本講演の内容は阿部拓郎氏との共同研究に基づくものである。
2022/11/09 (Wed) 15:00-16:00 (E301 Dept. of Math. Osaka University)
Speaker: Paul Mucksch (Kyushu)
Title: Topology of supersolvable oriented matroids
Abstract: A central result in the topology of complex hyperplane arrangements, due to Falk, Randell and Terao, states that supersolvability of the intersection lattice of such arrangements implies that their complements
are aspherical. The homotopy type of the complement of a complexified real hyperplane arrangement can be modeled by a nice regular CW-complex introduced by Salvetti. The Salvetti complex can be constructed for any oriented matroid -- a combinatorial abstraction of a real hyperplane arrangement.
In my talk, I will present a novel combinatorial way to prove that supersolvability of the geometric lattice of an oriented matroid implies the asphericity of its Salvetti complex. In particular, this extends to the non-realizable case.
2022/11/07 (Mon) 15:30-16:30 (E301 Dept. of Math. Osaka University)
Speaker: Dmitry Feichtner-Kozlov (Bremen/OIST)
Title: Applied and Computational Topology
Abstract: We will give a gentle introduction to the subject of Applied and Computational Topology. The survey of the subject's main ideas and tools will be complemented with applications to discrete mathematics, and, if time allows, theoretical distributed computing.