Let R be a local, Noetherian ring and I ⊆ R an ideal. A question of Kodiyalam asks whether for fixed i > 0, the polynomial giving the ith Betti number of In has degree equal to the analytic spread of I minus one. Under mild conditions on R, we show that the answer is positive in a number of cases, including when I is divisible by m or I is an integrally closed m-primary ideal.

The result of Siegel that the Tamagawa number of SLr over a function field is 1 has an expression purely in terms of vector bundles on a curve, which is known as the Siegel formula. We prove an analogous formula for vector bundles with quasi-parabolic structures. This formula can be used to calculate the betti numbers of the moduli of parabolic vector bundles using the Weil conjucture.

We review topological properties of Kähler and symplecticmanifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b1 = 1 case (in the coKähler/cosymplectic situation). MSC: 53C15, 55S30, 53D35, 55P62, ...

In this paper, we prove a number of inequalities between the signature and the Betti numbers of a 4–manifold with even intersection form and prescribed fundamental group. Furthermore, we introduce a new geometric group invariant and discuss some of its properties.

We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply connected four-manifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature.

Abstract: Let X be a building of uniform thickness q+1. L-Betti numbers of X are reinterpreted as von-Neumann dimensions of weighted L-cohomology of the underlying Coxeter group. The dimension is measured with the help of the Hecke algebra. The weight depends on the thickness q. The weighted cohomology makes sense for all real positive values of q, and is computed for small q. If the Davis comp...

This is an attempt to define a notion of Betti structure with nice functorial property for algebraic holonomic D-modules which are not necessarily regular singular.

A degeneration of compact Kähler manifolds gives rise to a monodromy action on Betti moduli space H (X,G) = Hom(π1(X), G)/G over smooth fibres with a complex algebraic structure group G being either abelian or reductive. Assume that the singularities of the central fibre is of normal crossing. When G = C, the invariant cocycles arise from the global cocycles. This is no longer true in general. ...

This paper determines the representation type of the IwahoriHecke algebras of type B when q 6= ±1. In particular, we show that a single parameter non–semisimple Iwahori–Hecke algebra of type B has finite representation type if and only if q is a simple root of the Poincaré polynomial, confirming a conjecture of Uno’s [17].

We present an extension of the Delsarte linear programming method for spherical codes. For several dimensions it yields improved upper bounds including some new bounds on kissing numbers. Musin’s recent work on kissing numbers in dimensions three and four can be formulated in our framework.