14:15-14:35 Uhr Lehrprobe zum Thema The Theorem of Carathéodory (auf Englisch)

ca. 14:45-15:25 Uhr Fachvortrag:

Constructing Koszul Gorenstein algebras from Cohen-Macaulay simplicial complexes

My main area of interest is combinatorial commutative algebra, a topic that sits at the crossroads between algebra, combinatorics and topology. The main aim of this talk is to discuss a joint project with Lorenzo Venturello (KTH Stockholm) relating Koszul Gorenstein algebras and Cohen-Macaulay simplicial complexes.
Koszul algebras are quadratic algebras satisfying desirable homological properties and arising naturally in many geometric and combinatorial contexts: for instance, the coordinate rings of Veronese, Segre and Grassmannian varieties (in their natural embeddings) are all Koszul, and so is the Stanley-Reisner ring of any flag simplicial complex. However, the Koszul property is hard to control and to check in general, and many conjectures about the general behaviour of Koszul algebras are currently open.
Starting from a flag pure simplicial complex Δ, we propose a construction of a (non-monomial) Gorenstein ring R_Δ which is Koszul if and only if Δ is Cohen-Macaulay, thus providing a bridge between these two worlds. On a more combinatorial level, the very same correspondence also yields that R_Δ has a Gröbner basis of quadrics if and only if Δ is shellable.
As an application, we provide counterexamples to an algebraic generalization of a conjecture by Charney and Davis about flag homology spheres.