Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the br...

For n ≥ 3, let Ωn be the set of line segments between the vertices of a convex n-gon. For j ≥ 2, a j-crossing is a set of j line segments pairwise intersecting in the relative interior of the n-gon. For k ≥ 1, let ∆n,k be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of Ωn not containing any (k + 1)-crossing. The complex ∆n,k has been the ...

A weaker form of the multiplicity conjecture of Herzog, Huneke, and Srinivasan is proven for two classes of monomial ideals: quadratic monomial ideals and squarefree monomial ideals with sufficiently many variables relative to the Krull dimension. It is also shown that tensor products, as well as Stanley-Reisner ideals of certain unions, satisfy the multiplicity conjecture if all the components...

In this paper we consider graded ideals in a polynomial ring over a field and ask when such an ideal has the property that all of its powers have a linear resolution. It is known [7] that polymatroidal ideals have linear resolutions and that powers of polymatroidal ideals are again polymatroidal (see [2] and [8]). In particular they have again linear resolutions. In general however, powers of i...

We show that Jn, the Stanley-Reisner ideal of the n-cycle, has a free resolution supported on the (n−3)-dimensional simplicial associahedron An. This resolution is not minimal for n > 6; in this case the Betti numbers of Jn are strictly smaller than the f -vector of An. We show that in fact the Betti numbers βd of Jn are in bijection with the number of standard Young tableaux of shape (d + 1, 2...

When do syzygies depend on the characteristic of field? Even for well-studied families examples, very little is known. For a family random monomial ideals, namely Stanley--Reisner ideals flag complexes, we prove that Betti numbers asymptotically almost always characteristic. Using this result, also develop heuristic dependence asymptotic algebraic varieties.