Pragmatic 2008

Scientific activities

During the courses the teachers introduced the basic notions and techniques from commutative algebra and combinatorics for studying the polynomials h_A(t), for the use of Alexander duality and the analysis of vertex covers and Stanley decompositions. During the seminars they also exhibited the basic techniques on important examples and example classes and related the subject to neighboring fields. In particular, they addressed the following topics with related open research problems:

Generalities on Hilbert-series, rational generating functions and real rooted polynomials.
Generalities on simplicial complexes and their Stanley-Reisner rings.
Alexander duality, multigraded modules and finite free resolutions.
Koszul-algebras and infinite free resolutions.
Lefschetz-algebras and g-theorems.
Vertex cover algebras.
Groebner bases with squarefree initial ideals.
Transformations preserving real rootedness and applications to algebra.
Stanley decompositions, partionable simpicial complexes and prime filtrations.

Calendars of seminars

July 14 2008

Free Resolutions (Herzog)
Hilbert series (Welker)

July 15 2008

Resolutions of powers of ideals (Herzog)
Kruskal-Katona snd Macaulay Theorems (Welker)

July 17 2008

Resolutions of powers of ideals II (Herzog)
Koszul algebras (Welker)

July 18 2008

Edge ideals (Herzog)
Real rooted polynomials (Welker)

July 21 2008

Vertex cover algebras (Herzog)
The Neggers-Stanley conjecture and Koszula algebras (Welker)

July 22 2008

Stanley depth I (Herzog)
Lefschetz algebras I (Welker)

July 24 2008

Stanley depth II (Herzog)
Lefschetz algebras II (Welker)

July 25 2008

Algebraic shifting I (Herzog)
Algebras with special Groebner basis I (Welker)

July 29 2008

Algebraic shifting II (Herzog)
Algebras with special Groebner basis II (Welker)

July 31 2008 and August 1 2008

Presentations by the participants

(Anda Georgiana Olteanu - Loredana Sorrenti) When are lexsegment ideals componentwise linear or Gotzmann?
(Le Din Nam) When are vertex cover algebras of quasi forests standard graded?
(Benjamin Nill) The linear ordering polytope is Gorenstein.
(Benjamin Nill - Katherin Vorwek) The Stanley depth of certain monomial ideals.
(Eric Emtander - Veronica Crispin Quinonez) When are symbolic powers of certain classes of monomial ideals componentwise linear?
(Matteo Varbaro - Alexandru Constantinescu) The Hilbert series of the algebra of essential vertex covers for a graph.
(Giancarlo Rinaldo) Find and implement a "good" algorithm for determining the Stanley depth of a monomial ideal.
(Bruno Benedetti - Alexandru Constantinescu - Katherin Vorwek) Give a term order of the defining ideal of the Grassmannian that is Stanley-Reisner ideal of a polytope.
(Gesa Kempf - Martina Kubitzke) The behavior of t-uples of generic linear forms on graded modules over the exterior algebra.
(Vincenzo Micale - Cristina Bertone) The Hilbert series and the ohen Macaulay type of the algebra of essential vertex covers for unmixed graphs..
(Oscar Fernandez) Characterization of the graphs G whose corresponding edge ideal I(G) is Stanley-Reisner of a shellable simplicial complex.
(Anargyros Katsabekis) The Groebner basis of the defining ideal of the vertex cover algebra of higher dimensional trees.
(Anargyros Katsabekis) Upper bounds on the triangle density of graphs using Bar resolution.
(Mircea Cimpoeas) The relation of Stanley of a module M and its depth.
(Gioia Failla) The polynomial growth of Betti numbers of powers of ideals is independent on the homological degree.
(Leila Sharifan - Matteo Varbaro) The linear quotients ideals are componentwise linear and computation of their Betti numbers.