Betti numbers can be used as a means for feature detection to aid in the exploration of complex large-scale data sets. We present a fast algorithm for the calculation of Betti numbers for triangulated isosurfaces, along with examples of their use. Once an isosurface is extracted from a data set, calculating Betti numbers only requires time and space proportional to the isosurfaces, not the data...

We show that very weak topological assumptions are enough to ensure the existence of a Hellytype theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b, d) such that the following holds. If F is a finite family of subsets of R such that β̃i ( ⋂G) ≤ b for any G ( F and every 0 ≤ i ≤ dd/2e−1 then F has Helly number at most h(b, d). Here β̃i denotes t...

We consider the open problem of determining the graded Betti numbers for fat point subschemes Z supported at general points of P. We relate this problem to the open geometric problem of determining the splitting type of the pullback of ΩP2 to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one fo...

An important application of the algebraic theory of L2-Betti numbers [10] (see Farber [8] for an alternative approach) is that the L2-Betti numbers β i (Γ ) of a group Γ vanish if it has a normal subgroup whose L2-Betti numbers vanish. With regard to the first L2-Betti number, one can significantly relax the normality condition to obtain similar vanishing results [14]. Peterson and Thom prove i...

This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded R-module, where R = k[x1, . . . , xm] is the polynomial ring over a field k in m variables. The bound is given in terms of the rank and the first two Betti numbers of the module. An example is given which achieves these bounds simultaneously in each homological degree. Using Alexander duality, a bound...