New perspectives on hyperplane arrangements
The conference "New perspectives on hyperplane arrangements" will take place on May 12-16, 2025 in Bochum.
There will be ten scientific talks:
| Speaker | Title |
|---|---|
| Takuro Abe (Tokyo) | Tame arrangements |
| Xiangying Chen (Bochum) | Conditional Erlangen Program |
| Daniele Faenzi (Bourgogne) | Logarithmic derivations along discriminants |
| Benoît Guerville-Ballé (Paris) | Linking Invariants for Line Arrangements |
| Lukas Kühne (Bielefeld) | Alcoved Polytopes and Arrangements |
| Shota Maehara (Kyushu) | Use of matrix for exponents of 2-dimensional multiarrangements |
| Paul Mücksch (Berlin) | Fibrations for hyperplane arrangements and oriented matroids |
| Leonie Mühlherr (Bielefeld) | Connected hypersubgraph arrangements |
| Piotr Pokora (Krakow) | A new hierarchy for line arrangements |
| Sven Wiesner (Bochum) | Multi-Euler Derivations |
Registration
TBA
Dates and Program
TBA
Venue
The venue for the conference will be on the campus of the Ruhr University Bochum in building IB, level E1 (not 01!), room 103. The directions to the building can be found here. We recommend you make your way to Bochum via train lines (such as RE6, RE1 or S1) and then take the U35 to Hustadt/Querenburg to get off at the stop "Ruhr-Universität".
Funding & Accommodation
TBA
Abstracts
Takuro Abe
Tame arrangements
Tame arrangements were informally introduced by Orlik and Terao for the study of Milnor fibers of arrangements. The definition of tame arrangements are based on projective dimensions, and in general it is very hard to check. However, tame arrangements have played important roles in several areas of arrangements, including freeness, master functions and critical varieties, Bernstein-Sato polynomials, Solomon-Terao algebras, likelihood geometry and so on. On the other hand, the research on tame arrangements themselves was very few, like a sufficient condition for tamenss by Mustata-Schenck.
In this calk, we give several fundamental results for tameness, i.e., addition, deletion and restrictions, Zeigler-Yoshinaga type results for tameness. We also give several examples of tame arrangements by using them.
Xiangying Chen
Conditional Erlangen Program
This talk is based on my PhD thesis. Following Felix Klein's Erlanger Programm, the systematic development of matroid-like structures is initiated by Kung under the name of Combinatorial Erlanger Programm and later studied by Gelfand and Serganova through introducing the Coxeter matroids. In this talk we introduce the "Conditional Erlangen Program", which answers the classic question "What about other Coxeter types" for conditional independence (CI) structures such as semigraphoids, semimatroids and gaussoids. I will present an axiomatization of matroids as CI-structures and, as an application of the "Conditional Erlangen Program", its analoges for types B, C and D, and discuss the geometry of CI-structures, matroids and generalized permutohedra for any root system.
Daniele Faenzi
Logarithmic derivations along discriminants
Sheaves of logarithmic vector fields, or “derivations”, tangent to a given divisor, are studied in the theory of deformations of singularities. 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 concentrate on the study of determinants and discriminants of adjoint groups. We will mention ongoing work on discriminants of polar representations and theta groups. Joint project with Vladimiro Benedetti, Simone Marchesi, Masahiko Yoshinaga.
Benoît Guerville-Ballé
Linking Invariants for Line Arrangements
We will present recent developments in the study of the topology of line arrangements in the complex projective plane. Our focus will be on the gap between the intersection lattice of an arrangement and its embedded topology --ie Zariski pairs. We will discuss the different linking invariants and, through several examples, demonstrate how these invariants have reshaped our understanding of the topology of line arrangements. Lastly, perspectives on future research will be provided.
Lukas Kühne
Alcoved Polytopes and Arrangements
Alcoved polytopes are characterized by the property that all facet normal directions are parallel to the roots e_i - e_j. This fundamental class of polytopes appears in several applications such as optimization, tropical geometry or physics. Symmetric alcoved polytopes are dual to the fundamental polytopes attached to finite metric spaces.
This talk focuses on the type fan of alcoved polytopes which is the subdivision of the metric cone by combinatorial types of alcoved polytopes. The type fan governs when the Minkowski sum of alcoved polytopes is again alcoved. For symmetric alcoved polytopes, this fan is the intersection of the metric cone with the Wasserstein arrangement.
I will discuss both theoretical and computational results for the symmetric and asymmetric type fan of alcoved polytopes.
This talk is based on joint works with Emanuele Delucchi, Nick Early, Leonid Monin, and Leonie Mühlherr.
Shota Maehara
Use of matrix for exponents of 2-dimensional multiarrangements
At once Max Wakefield and Sergey Yuzvinsky utilized some square matrix for the research of exponents of 2-dimensional multiarrangements. It is famous that they proved "for a fixed balanced multiplicity on 2-dimensional arrangement, the exponents is closest in general positions" by using such matrix. In this talk, we introduce a matrix almost similarly as Wakefield and Yuzvinsky do and consider further application for exponents. In fact, exponents of 2-dimensional multiarrangements can be calculated by checking whether the corresponding matrices have full rank or not. Especially, we show some results about exponents of multiarrangements whose underlying arrangements consist of four lines, almost only considering row basic transformations.
Paul Mücksch
Fibrations for hyperplane arrangements and oriented matroids
In the topological study of hyperplane arrangements fibrations play an important role. On the one hand, complex complements of certain arrangements might be realized as fibre bundles over complements of other arrangements leading to solutions of the K(pi,1)-problem in these instances. On the other hand, the Milnor fibration of a complex arrangement is a much studied geometric invariant but there are still many open questions about the topology of its fibres. In my talk I will present new results connecting such fibrations with the combinatorics of oriented matroids. This is partly joint work with Masahiko Yoshinaga.
Leonie Mühlherr
Connected hypersubgraph arrangements
In a recent work, Cuntz and Kühne defined the class of connected subgraph arrangements. This includes the resonance arrangement and some ideal subarrangements of Weyl arrangements. They studied among other things the freeness and simpliciality of these arrangements and found graph theoretical criteria for these properties. In this project, we want to extend the definition of these arrangements to hypergraphs and study the aforementioned properties in order to generalize the characterizations established by Cuntz and Kühne. This talk gives an introduction to the connected subgraph arrangements, explains the generalization idea, presents results pertaining to freeness of this arrangement class and shows an interesting connection to Boolean buildingssets.
Piotr Pokora
A new hierarchy for line arrangements
The main goal of my talk is to explain the notion of type for plane curves, which is equal to the initial degree of the corresponding Bourbaki ideal. After introducing all the necessary definitions, we present basic properties of this invariant and show that it behaves well with respect to the unions. It turns out that the curves of type 0 are exactly the free curves and the curves of type 1 are the plus-one generated curves. The third natural class of curves that appears in our investigations is the class of curves of type 2. There are exactly two subclasses of curves of type 2, called curves of type 2A and 2B, and then we show how to construct infinite families of line arrangements of type 2A and 2B and we show that curves of type 2 show up during the studies on Ziegler pairs. The talk is based on joint work with Takuro Abe and Alex Dimca.
Sven Wiesner
Multi-Euler Derivations
We give a definition of a multi-Euler derivation, which allows us to construct a basis through affine connections and to consider its properties. Also, we give a characterization of multi-Euler derivations. In particular, for 2-multiarrangements, we can give a more explicit one in terms of exponents. As applications, we give a complete classification and another proof of the existence of universal vector fields for Coxeter arrangements. New examples of multi-Euler derivations are given for multi-braid arrangements, as well as for the deleted A_3 arrangement. This is joint work with Takuro Abe, Shota Maehara, and Gerhard Röhrle.
