Understanding and computing volumes of hyperbolic manifolds and orbifolds is a rich and fascinating subject. There are for instance deep connections with number theory, more specifically special values of arithmetic functions such as Dedekind ζ-functions, Dirichlet L-functions and polylogarithms (see [Za], [K2], Prasad’s volume formula from [P] as used in the hyperbolic case in [Be] and [BE]). ...

The generalized Euler number En|k counts the number of permutations of {1, 2, . . . , n} which have a descent in position m if and only if m is divisible by k. The classical Euler numbers are the special case when k = 2. In this paper, we study divisibility properties of a q-analog of En|k. In particular, we generalize two theorems of Andrews and Gessel [3] about factors of the q-tangent numbers.

Let X be a compact Kähler manifold with strictly pseudoconvex boundary, Y. In this setting, the SpinC Dirac operator is canonically identified with ∂̄ + ∂̄∗ : C∞(X ; Λ) → C∞(X ; Λ). We consider modifications of the classical ∂̄-Neumann conditions that define Fredholm problems for the SpinC Dirac operator. In part 2, [7], we use boundary layer methods to obtain subelliptic estimates for these bound...

Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x) :::: (c + o( I)) logx loglogx log log log 10gx(loglog logx) -2, where c = eY and y is Euler's constant. Here, this result is shown with c = coeY where Co = 1.31256... is the solution of the equation 4/ Co e -4/co = 3 . The principal new tool ...

We introduce the index i(v) = 1− χ(S−(v)) for critical points of a locally injective function f on the vertex set V of a simple graph G = (V,E). Here S−(v) = {w ∈ S(v) | f(w) − f(v) < 0 } is the subgraph of the unit sphere S(v) generated by the vertices {w ∈ V | (v, w) ∈ E }. We prove that the sum ∑ v∈V i(v) of the indices always is equal to the Euler characteristic χ(G) of the graph G. This is...

Euler’s polyhedron formula asserts for a polyhedron p that

The aim of this note is to study the behavior of intersection homology Euler characteristic under morphisms of algebraic varieties. The main result is a direct application of the BBDG decomposition theorem. Similar formulae for Hodge-theoretic invariants of algebraic varieties were announced by the first and third authors in [4, 11].

We calculate the E-polynomials of certain twisted GL(n,C)-character varieties Mn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,Fq) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PG...

Given a finite group G and a G-space X, we show that a direct sum FG(X) = ⊕ n≥0KGn(X n) ⊗ C admits a natural graded Hopf algebra and λ-ring structure, where Gn denotes the wreath product G ∼ Sn. FG(X) is shown to be isomorphic to a certain supersymmetric product in terms of KG(X) ⊗ C as a graded algebra. We further prove that FG(X) is isomorphic to the Fock space of an infinite dimensional Heis...

We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.