For X a metric space and r> 0 a scale parameter, the Vietoris–Rips simplicial complex
VR<(X; r)(resp. VR≤(X; r)) has X as its vertex set, and a finite subset σ⊆ X as a simplex …

We study monomial ideals with linear presentation or partially linear resolution. We give
combinatorial characterizations of linear presentation for square-free ideals of degree 3, and …

Let G be a finite simple graph on the vertex set V (G)={x 1,…, xn} and I (G)⊂ K [V (G)] its
edge ideal, where K [V (G)] is the polynomial ring in x 1,…, xn over a field K with each deg⁡ …

Let I (G) be the edge ideal of a simple graph G. We prove that I (G) 2 has a linear free
resolution if and only if G is gap-free and reg I (G)≤ 3. Similarly, we show that I (G) 3 has a …

We prove a sharp bound for the regularity of a squarefree monomial ideal with a linear
presentation. This result also answers in positive a question on the cohomological …

In this paper we survey what is known about the maximal degrees of minimal syzygies of
graded ideals over polynomial rings. Subadditivity is one such property that is conjectured to …

We show that the virtual cohomological dimension of a Coxeter group is essentially the
regularity of the Stanley–Reisner ring of its nerve. Using this connection between geometric …

We build a new bridge between geometric group theory and commutative algebra by
showing that the virtual cohomological dimension of a Coxeter group is essentially the …

Mathematisches Forschungsinstitut Oberwolfach Asymptotic Phenomena in Local Algebra
and Singularity Theory Page 1 Mathematisches Forschungsinstitut Oberwolfach Report No …