During the summer 2011, some Ph.D. of Stockholm University and KTH have participated to Pragmatic 2011, a summer school in commutative algebra and algebraic geometry. They will present their work on December 12-th at 13.00-17.00. This will take place at the math department of SU in room Room 35, Building 5. The speakers are: Kathrin Vorwerk, Ornella Greco, Ivan Martino, Yohannes Tadesse.

• 13:00, Kathrin Vorwerk: Some results on cones of Hilbert series of squarefree modules
• 13:45, Ornella Greco: The h-vector of the union of two sets of points in the projective plane

• 14:45, Ivan Martino: Vertices Collapsing and Cuts Ideals
• 15:30, Yohannes Tadesse: Poincare series of monomial rings with minimal Taylor resolution

• 13:00, Kathrin Vorwerk: Some results on cones of Hilbert series of squarefree modules 
  We present different generalizations of the notion of squarefreeness for ideals to the more general case of modules. We describe the cones of Hilbert functions for squarefree modules in general and those generated in degree zero. We give their extremal rays and defining inequalities. For squarefree modules generated in degree zero, we compare the defining inequalities of that cone with the classical Kruskal-Katona bound, also asymptotically. 
  
• 13:45, Ornella Greco: The h-vector of the union of two sets of points in the projective plane 
  Given two h-vectors, h and h', we study which are the possible h-vectors for the union of two disjoint sets of points in the projective plane, respectively associated to the given h-vectors and how can they be constructed. We will give some bounds for the resulting h-vector and we will show how to construct the minimal h-vector of the union among all possible ones.

• 14:45, Ivan Martino: Vertices Collapsing and Cuts Ideals 
  In this work we study how the two elementary operations disjoint union and collapsing two vertices modify the cut ideal of a graph. To do that we generalize the definition of cut ideal given in literature, introducing the concepts of edge labelling and edge multiplicity: in fact we state the 'non classical behavior' of the cut ideal. In particular, we observe the invariance of the cut varieties under this operations. Finally, we also present how to obtain information about the cut ideal of a graph, from the knowledge of the cut ideals of two its subgraphes; thus we show an algorithm for computing the cut ideal of a graph. 
  
• 15:30, Yohannes Tadesse: Poincare series of monomial rings with minimal Taylor resolution 
  We give a comparison between the Poincare series of two monomial rings: R = A/I and R_q = A/I_q where I_q is a monomial ideal generated by the q'th power of monomial generators of I. We compute the Poincare series for a new class of monomial ideals with minimal Taylor resolution. We also discuss the structure of a monomial ring with minimal Taylor resolution where the ideal is generated by quadratic monomials.