Title: Points in product of two projective spaces
Abstract: I will report on ongoing joint work with Navid Nemati about Hilbert function, regularity and free resolutions of points in a product of two projective spaces. This includes some general results and more specific ones about « complete intersection » sets of points, that I will illustrate on specific examples.
Title: The categorical Poincaré-Birkhoff-Witt theorem.
Abstract: The usual Poincaré-Birkhoff-Witt theorem describes explicitly the vector space underlying the
Title: An algorithm for volumes of polytopes with applications to social choice
Abstract: e discuss a fast algorithm for the computation of the volume
of rational polytopes with few (nonsimplicial) facets. It is based on a
natural recursive approach, originally suggested by Lasserre, that uses descent in the face lattice. For efficient computations in high
dimensions it needs a sophisticated implementation that has now been
realized in Normaliz.
Probabilities in social choice that are based on the Impartial Anonymous
Culture can often be interpreted as volumes of rational polytopes. For 4
candidates these polytopes have dimension 24, and the computation is a
challenge. Before the new algorithm had been implemented, Normaliz had
to use triangulations. Descent in the face lattice makes all these
computations very easy and gives access to many more that hitherto had
Title: Wilf's conjecture by multiplicity
Abstract: Let $S$ be a numerical semigroup. Its embedding dimension
$e(S)$ is the minimal number of generators, the Frobenius number $F(S)$
is the largest integer $\notin S$ , and $n(S)$ counts the elements in
$S$ that are $<F(S)$. Wilf's conjecture states that $F(S) < e(S)n(S)$.
It has been proved in many cases, but remains a major open problem in
the combinatorial theory of numerical semigroups. We will show that for
fixed multiplicity $m=m(S)$, the smallest nonzero element of $S$, the
conjecture can be decided algorithmically by polyhedral methods using
the parametrization of multiplicity $m$ semigroups by the lattice points
of the Kunz polyhedron $P_m$. With them we have verified the conjecture
for $m\le 17$.
Title: Measuring the non-Gorenstein locus of Hibi rings and
normal affine semigroup rings
Abstract: In a recent paper, together with Takayuki Hibi and Dumitru Sta-
mate, we considered the trace of the canonical module ωR of a local
(or standad graded) Cohen-Macaulay ring (R, m). The significance of
this trace is that it describes the non-Gorenstein locus of R. Thus R
is Gorenstein if and only if tr(ωR) = R, and R is Gorenstein on the
punctures spectrum if and only if tr(ωR) is an m-primary ideal.
We apply this technique to classify the Hibi ring and normal sim-
plicial affine semigroup rings which are Gorenstein on the punctured spectrum.
Title: Normal simplicial affine semigroups rings which are almost Gorenstein
Abstract: In 2015, Goto, Takahashi and Taniguchi introduced almost Goren-
stein rings. These a Cohen-Macaulay rings which admit a canonical
module ωR and for which there exists an exact sequence
0 → R → ωR → E → 0,
where E is an Ulrich module.
We characterize all normal simplicial affine semigroups rings which
are almost Gorenstein and give a very explicit classification of such
rings in dimension 2.
Santiago Zarzuela, University of Barcelona
Title: Generating Frobenius algebras of Stanley-Reisner rings
G. Lyubeznik and K. Smith introduced in  the Frobenius algebra of a module over a Noetherian local ring containing a ﬁeld of positive characteristic. It is known that for the injective hull of the residue ﬁeld its Frobenius algebra is in general not ﬁnitely generated, with a ﬁrst example constructed by M. Katzman in  over a complete Stanley-Reisner ring. In fact, by  such an algebra can be only principally generated or inﬁnitely generated as algebra over its degree zero component. Moreover, by  this property can be characterized in terms of the corresponding simplicial complex by means of the existence of the so-called free faces. In this talk, we shall review the above results and exhibit, in the inﬁnitely generated case, a more precise 1−1 correspondence between the minimal generators in each graded component of the Frobenius algebra of the injective hull of the residue ﬁel of a complete Stanley-Reisner ring and the maximal free pairs of the associated simplicial complex, as shown in . This new notion of maximal free pair is a strict generalization of the concept of free face.
Title: Cohen-Macaulay simplicial complexes in arbitrary codimension
Abstract: A simplicial complex of dimension d − 1 is said to be CohenMacaulay in codimension t, 0 ≤ t ≤ d−1, if it is pure and the link of any face with cardinality at least t is Cohen-Macaulay. This generalizes several concepts on simplicial complexes such as Cohen-Macaualy-ness, Buchsbaum property, Sr condition of Serre, and locally Cohen-Macaulay-ness. Most results on the simplicial complexes with aforementioned properties naturally extend to the case of Cohen-Macaulay simplicial complexes in codimension t. In particular, the Eagon-Reiner theorem, the local behavior, and the homological vanishing properties are suitably retained. Furthermore, characterizations of certain families of Cohen-Macaulay simplicial complexes carry over characterizations of these families of simplicial complexes which are Cohen-Macaulay in codimension t. In particular, Cohen-Macaulay-ness of a simplicial complex in an arbitrary codimension is a topological invariant.