We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each p, d, r, explicit bounds on the p-th Betti numbers of affine and projective subvarieties of Ck and PC, defined by r polynomials of degrees at most d as a function of p, d and r. Unlike previous bounds these bounds are independent of k, the dimension of the ambient s...

We calculate Betti numbers of the framed moduli space of instantons on Ĉ/Z2, under the assumption that the corresponding torsion free sheaves E have vanishing properties(Hom(E, E(−l∞)) = Ext (E, E(−l∞)) = 0). Moreover we derive the generating function of Betti numbers and obtain closed formulas. On the other hand, we derive a universal relation between the generating function of Betti numbers o...

We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR X, in the presence of a continuous real valued function on X.The refinement of Betti numbers consists of finite configurations of points with multiplicities in the complex plane whose total cardinality are the Betti numbers and the refinement of homology consists of configurations of v...

Topological complexity of semialgebraic sets in R has been studied by many researchers over the past fifty years. An important measure of the topological complexity are the Betti numbers. Quantitative bounds on the Betti numbers of a semialgebraic set in terms of various parameters (such as the number and the degrees of the polynomials defining it, the dimension of the set etc.) have proved use...

The 1-skeleton of a G-manifold M is the set of points p ∈ M , where dimGp ≥ dimG − 1, and M is a GKM manifold if the dimension of this 1-skeleton is 2. M. Goresky, R. Kottwitz, and R. MacPherson show that for such a manifold this 1-skeleton has the structure of a “labeled” graph, ( , α), and that the equivariant cohomology ring ofM is isomorphic to the “cohomology ring” of this graph. Hence, if...

‎Let $I$ be an ideal in a regular local ring $(R,n)$‎, ‎we will find‎ ‎bounds on the first and the last Betti numbers of‎ ‎$(A,m)=(R/I,n/I)$‎. ‎if $A$ is an Artinian ring of the embedding‎ ‎codimension $h$‎, ‎$I$ has the initial degree $t$ and $mu(m^t)=1$‎, ‎we call $A$ a {it $t-$extended stretched local ring}‎. ‎This class of‎ ‎local rings is a natural generalization of the class of stretched ...

We give a leisurely account of the relationship between volume and L2-Betti numbers on closed, aspherical manifolds based on the results in [5] – albeit with a different point of view. This paper grew out of a talk presented at the first colloquium of the Courant Center in Göttingen in October 2007. 1. Review of L-Betti numbers The L-Betti numbers of a closed Riemannian manifold, as introduced ...

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of t-minors of a generic pluri-circulant matrix is a transversal monomial ideal . Using a Gröbner basis for this ideal, it is sho...

We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation t...