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SS 2015

21.04.2015 um 16:15 Uhr in 69/125:

Dinh Le Van (Universität Osnabrück)

The broken circuit complex and the Orlik-Terao algebra of a hyperplane arrangement

The Orlik-Terao algebra of an arrangement is a commutative analogue of the well-studied Orlik-Solomon algebra, which is the cohomology ring of the arrangement complement. Recently, it has attracted significant attention, mainly because it encodes subtle information missing in the Orlik-Solomon algebra: it was used by Schenck-Tohaneanu to characterized 2-fomality, a non-combinatorial property which is necessary for the arrangement to be free and for the complement to be aspherical. Nevertheless, the Orlik-Terao algebra is a deformation of a combinatorial object, the broken circuit algebra. Exploiting this strong connection between the two algebras I will give in this talk characterizations for the following properties of each of them: having a linear resolution, being a complete intersection, and being Gorenstein. If time permits, a generalization of Schenck-Tohaneanu's result will also be discussed.

05.05.2015 um 16:15 Uhr in 69/125:

Dr. Mohsen Asgharzadeh (IPM, Tehran, Iran)

On a conjecture of Glaz

Toward solving a conjecture posed by Glaz, we give a generalization of the Hochster-Eagon result on Cohen-Macaulayness of invariant rings, in the context of non-Noetherian rings. Also, we present Cohen-Macaulayness of non-affine normal semigroups (based on joint works with Mehdi  Dorreh and Massoud Tousi).

12.05.2015 um 16:15 Uhr in 69/125:

Vincent Trageser (Freie Universität Berlin)

On stability of syzygy bundles on projective spaces

We will consider cohomological stability and slope-stability of certain syzygy bundles on projective spaces, which are kernels of surjective maps between sums of line bundles.

26.05.2015 um 16:15 Uhr in 69/125:

Juniorprof. Dr. Martina Juhnke-Kubitzke (Universität Osnabrück)

A balanced generalized lower bound Theorem for simplicial polytopes

A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the h-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial d-polytopes, that is simplicial d-polytopes whose underlying graphs are d-colorable, Klee and Novik proposed a balanced analogue of this inequality, that is stronger than just unimodality. In this talk, we will answet this conjecture of Klee and Novik in the affirmative. For this, we also show a Lefschetz property for rank-selected subcomplexes of balanced simplicial polytopes and thereby obtain new inequalities for their h-numbers. At the end of the talk, we will give some hints to what happens if one considers a-balanced simplicial polytopes, as introduced by Stanley. This is joint work with Satoshi Murai and Steve Klee.

09.06.2015 um 16:15 Uhr in 69/125:

Dr. Lukas Katthän (Universität Osnabrück)

Golod rings and A-infinity algebras

Golod rings are local rings where the Betti numbers of the residue field grow as fast as possible. This is equivalent to the vanishing of all Massey products on the Koszulhomology. In this talk, I will discuss both conditions. In the special case of Stanley-Reisner rings, I will give a geometric interpretation of the product on Koszul homology. To understand the higher Massey products, I will discuss multiplicative structures on resolutions. If S is the polynomial ring and R = S/I is a quotient by a homogeneous ideal I, then the (minimal) free resolution of R over S does not always carry the structure of a differential graded algebra. However, it always is an A-infinity algebra, and the Massey products can be defined in terms of this structure.

16.06.2015 um 16:15 Uhr in 69/125:

Prof. Dr. Ngô Việt Trung (University of Hanoi)

Depth and regularity of powers of sums of ideals

(joint work with H.T. Ha and T.N. Trung)

Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum I+J in A ⊗_k B in terms of those of I and J. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.

30.06.2015 um 16:15 Uhr in 69/125:

Dr. Dang Hop Nguyen (Universität Osnabrück)

Linearity defect of edge ideals

(joint work with Thanh Vu)

The linearity defect measures how far a module is from having linear free resolution. It is bounded above by the projective dimension and shares some features with the Castelnuovo-Mumford regularity. On the other hands, there is no combinatorial formula for linearity defect of squarefree monomial ideal and the Betti table alone does not give the exact information about the linearity defect. We will talk about linearity defect of edge ideals, in particular, a generalization of Fröberg's theorem. We classify all graphs whose edge ideals have linearity defect at most
1. The characterization is independent of the characteristic of the base field: the graphs in question are exactly the weakly chordal graphs with induced matching number at most
2. The proof uses the theory of Betti splittings of monomial ideals due to Francisco, Ha and Van Tuyl and the structure of weakly chordal graphs. Along the line, we compute the linearity defect of edge ideals of cycles and weakly chordal graphs.

07.07.2015 um 16:15 Uhr in 69/125:

Prof. Dr. Winfried Bruns (Universität Osnabrück)

Normal lattice polytopes

Normal lattice polytopes can be considered as the discrete analogue of compact convex sets. In algebraic geometry they represent projectively normal toric varieties, and in commutative algebra they correspond to standard graded normal monoid algebras. They provide an ideal testing ground for these areas, both theoretically and experimentally.
We give a survey of challenging solved and open problems on lattice polytopes and report on recent work with Joseph Gubeladze and Mateusz Michalek. The challening problems include unimodular covering and triangulation, the integral Caratheodory property and the normality of smooth polytopes. The recent work deals with the extendability of normal polytopes by elementary "jumps" and the existence of maximal normal polytopes.