Fakultäten der RUB » Fakultät für Mathematik » Lehrstühle » Algebraische Kombinatorik

Oberseminar "Arrangements and Symmetries"

an der Fakultät für Mathematik der Ruhr-Universität Bochum (KVV 150926)

Organisatoren: Paul Mücksch, Gerhard Röhrle, Christian Stump

Vorträge Sommersemester 2021



Montag, 3. Mai 2021, 16:15-17:45

Tan Nhat Tran, RUB

CHARACTERISTIC QUASI-POLYNOMIALS OF INTEGRAL HYPERPLANE ARRANGEMENTS (1/3)

Abstract



Vorträge Wintersemester 2020/2021



Montag, 8. März 2021, 16:00-17:30

Misha Feigin, Glasgow

Quasi-invariants and free multiarrangements

Abstract: Quasi-invariant polynomials are associated to a reflection arrangement and an invariant multiplicity function on it. They first appeared in the work of Chalykh and Veselov on quantum integrable systems in 1990 in the Coxeter case. Freeness results for quasi-invariants can be related with freeness of modules of logarithmic vector fields for the reflection arrangements. This is due to the observation that components of invariant logarithmic vector fields are given by certain quasi-invariants. This has useful consequences both for quasi-invariants and for logarithmic vector fields. This relation can also be extended to finite complex reflection groups and to affine settings in which cases it gives new free (multi-)arrangements. The talk is based on joint work with T. Abe, N. Enomoto and M. Yoshinaga.



Montag, 8. Februar 2021, 16:15-17:45, per Zoom

Volkmar Welker, Philipps-Universität Marburg

tba



Montag, 18. Januar 2021, 16:15-17:45

Paul Mücksch, Ruhr-Universität Bochum

On Yuzvinsky's lattice sheaf cohomology for hyperplane arrangements

Abstract: In my talk, I will establish the exact relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space respectively projective space associated to the derivation module of the arrangement. I will derive a Künneth formula connecting the cohomology theories, answering a question posed by Yoshinaga. This, in turn, gives a new proof of Yuzvinsky’s freeness criterion and yields a stronger form of the latter.



Montag, 18. Januar 2021, 12:15-13:45

Masahiko Yoshinaga, Hokkaido University Sapporo

A geometric realization of combinatorial reciprocity of order polynomials

Abstract: The Euler characteristic of topological space can be considered as a generalization of the cardinality of a finite set. In previous work with Hasebe and Miyatani (2017), we generalized Stanley's combinatorial reciprocity for order polynomials to an equality of Euler characteristics of certain spaces of homomorphisms of posets. In this talk, we discuss recent development of geometric realization of the combinatorial reciprocity. The main result asserts that certain spaces of poset homomorphisms are actually homeomorphic which clearly implies the Euler characteristics. The proof is based on the detailed analysis of upper semicontinuous functions on metrizable topological spaces. This is joint work with Taiga Yoshida.



Montag, 7. Dezember 2020, 16:15-17:45

René Marczinzik, Universität Stuttgart

Distributive lattices and Auslander regular algebras

Abstract: We show that the incidence algebra of a finite lattice L is Auslander regular if and only if L is distributive. As an application we show that the order dimension of L coincides with the global dimension of its incidence algebra when L has at least two elements and we give a categorification of the rowmotion bijection for distributive lattices. At the end we discuss the Auslander regular property for other objects coming from combinatorics. This is joint work with Osamu Iyama.



Montag, 30. November 2020, 16:15-17:45, per Zoom

Lukas Kühne, Max-Planck-Institut Leipzig

The Resonance Arrangement

Abstract: The resonance arrangement is the arrangement of hyperplanes which has all nonzero 0/1-vectors in R^n as normal vectors. It is also called the all-subsets arrangement. Its chambers appear in algebraic geometry, in mathematical physics and as maximal unbalanced families in economics.

In this talk, I will present a universality result of the resonance arrangement. Subsequently, I will report on partial progress on counting its chambers. Along the way, I will review some of the combinatorics of general hyperplane arrangements. If time permits I will also touch upon the related threshold arrangement which encodes Boolean functions that are linearly separable.



Montag, 23. November 2020, 16:15-17:45, per Zoom

Theo Douvropoulos, University of Massachusetts Amherst

Recursions and proofs in Coxeter-Catalan combinatorics

Abstract: The noncrossing partition lattice NC(W) associated to a finite Coxeter group W has become a central object in Coxeter-Catalan combinatorics during the last 25 years. We focus on two recursions on the simple generators of W; the first due to Deligne (and rediscovered by Reading) determines the chain number of NC(W) and the second, more general, due to Fomin-Reading recovers the whole zeta polynomial. The resulting formulas have nice product structures and are key players in the field, but are still not well understood; in particular, they are derived by the (case-free) recursions separately for each type.

A uniform derivation of the formulas from these recursions requires proving certain identities between the Coxeter numbers and invariant degrees of a group and those of its parabolic subgroups. In joint work with Guillaume Chapuy, we use the W-Laplacian (for W of rank n, this is an associated nxn matrix that we introduced in earlier work and which generalizes the usual graph Laplacian) to prove the required identities for the chain number of W. We give a second proof by using the theory of multi-reflection arrangements and the local-to-global identities for their characteristic polynomials. This latter approach is in fact applicable to the study of the whole zeta polynomial of NC(W) although it, currently, falls short of giving a uniform derivation of Chapoton's formula for it.



Montag, 2. November 2020, 16:15-17:45, Gebäude HIB
Montag, 9. November 2020, 16:15-17:45, Gebäude HIB
Montag, 16. November 2020, 16:15-17:45, Gebäude HIB
Montag, 14. Dezember 2020, 16:15-17:45, per Zoom
Montag, 21. Dezember 2020, 16:15-17:45, per Zoom
Montag, 11. Januar 2021, 16:15-17:45, per Zoom

Georges Neaime, Ruhr-Universität Bochum

Garside Theory I - VI


Abstract