(A)
The primary goal of the proposed work is to identify broader classses of monomial ideals for which we can prove Stanley's conjecture. Considering the work recentlt done in
[Ad], [Po10], [Po11], [HPV2] we plan to solve in the near future the following first three problems and we expect that until the second year we may use a computer program that will be created for the computation of sdepth by B. Ichim and M. Vladoiu to attach the problems 4 and 5.

1. To prove Stanley's conjecture for monomial ideals with bigsize 3, see [Po11] for the case bigsize 2.
2  To prove Stanley's conjecture for monomial ideals that can be written as intersection of three irreducibible ideals, extending in this way the results of [Ad] via the techniques developed in [HPV2].
3. To prove Stanley's conjecture for the edge ideals, or at least for some large classes of graphs.
4. To prove that with respect to polarization, the sdepth of an arbitrary monomial ideal increases exactly with the number of new added indeterminates, from which one obtains that proving Stanley's conjecture for squarefree monomial ideals implies Stanley's conjecture for arbitrary monomial ideals.
5. To prove that Stanley's conjecture holds (at least) for special classes of squarefree monomial ideals generated in degree greater or equal to 3.

(B)
The secondary goal of this project is to investigate the algebraic invariants of binomial edge ideals and determinantal face ideals associated with pure simplicial complexes. As a starting point for this research we use the papers [DeDiSt], [HHHKR], [EHH], [HHKK1], [HHKK2].
From the investigations we have made so far it appears naturally to consider the folllowing objectives as feasable:

1. Characterize the bipartite graphs whose binomial ideal is unmixed, Cohen-Macaulay, Gorenstein.
2. Find classes of graphs G for which the associated binomial edge ideal J_G and the initial ideal in(J_G) have the same extremal Betti numbers.
3. Describe the simplicial complexes for which the generators of the associated determinantal face ideal form a Groebner basis with respect to a natural monomial order on the ring K[X]. Do they arise as a natural generalization of the so-called closed graphs which appear in the binomial case?
4. Enquire under which conditions on the simplicial complex, the associated deteminantal ideal is prime.
5. Characterize normal or Koszul toric rings generated by the Alexander dual of a generalized Hibi ideal.