We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normalcrossing quasi-projective orbifold groups. Finally, we extend Sakuma’s formula for the first Betti number of abelian cove...

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.

We show that the L-Betti numbers of equivalence relations defined by R. Sauer coincide with those defined by D. Gaboriau.

We classify the cohomology spaces H(g,K) for all filiform nilpotent Lie algebras of dimension n ≤ 11 over K and for certain classes of algebras of dimension n ≥ 12. The result is applied to the determination of affine cohomology classes [ω] ∈ H(g,K). We prove the general result that the existence of an affine cohomology class implies an affine structure of canonical type on g, hence a canonical...

Let ∆ (k) n−1 denote the k-dimensional skeleton of the (n − 1)-simplex ∆n−1 and consider a complex ∆ (k−1) n−1 ⊂ X ⊂ ∆ (k) n−1. Let K be a field and let 0 ≤ l < k. It is shown that if H̃k−l−2(lk(X, τ);K) = 0 for all l-dimensional faces τ of X then dim H̃k−1(X ;K) ≤ ( n−1 l )( n−l−2 k−l )

We characterize all graphs whose binomial edge ideals have a linear resolution. Indeed, we show that complete graphs are the only graphs with this property. We also compute some graded components of the first Betti number of the binomial edge ideal of a graph with respect to the graphical terms. Finally, we give an upper bound for the Castelnuovo-Mumford regularity of the binomial edge ideal of...

We calculate the Betti numbers of moduli spaces of representations of a universal central extension of a surface group in the groups U(2, 1) and SU(2, 1). In order to obtain our results we use the identification of this space with an appropriate moduli space of Higgs bundles and Morse theory, following Hitchin’s programme [11]. This requires a careful analysis of critical submanifolds which tur...

Recall that the (Mumford-Castelnuovo) regularity of M is the least integer ρ such that for each i all free generators of Fi lie in degree ≤ i + ρ, that is βi,j = 0, for j > i + ρ. In terms of Macaulay [Mac] regularity is the number of rows in the diagram produced by the “betti” command. A Betti number βi,j 6= 0 will be called extremal if βl,r = 0 for all l ≥ i and r ≥ j + 1, that is if βi,j is ...

Erdős conjectured that there are x1−o(1) Carmichael numbers up to x, whereas Shanks was skeptical as to whether one might even find an x up to which there are more than √ x Carmichael numbers. Alford, Granville and Pomerance showed that there are more than x2/7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in person-to-person discussions) that Erdős must be correct...