Many questions on special values of L-functions are tied to the study of period integrals of automorphic forms. A notable example is the formula of Waldspurger ([22]) that relates the toric period on GL2 or its inner form to some central critical L-value. The conjecture of Gan, Gross and Prasad ([4]) started with the Gross-Prasad conjecture ([5]), as well as the refinement of Ichino and Ikeda (...

We study Ruan’s cohomological crepant resolution conjecture [39] for orbifolds with transversal ADE singularities. In the An-case we compute both the orbifold cohomology ring H ∗ orb([Y ]) and the quantum corrected cohomology ring H(Z)(q1, ..., qn). The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between H orb([Y ])...

We prove that if R is a Hensel local ring with infinite residue field k, the natural map Hi(GLn(R),Z/p) → Hi(GLn(k), Z/p) is an isomorphism for i ≤ 3, p 6= char k. This implies rigidity for Hi(GLn), i ≤ 3, which in turn implies the Friedlander–Milnor conjecture in positive characteristic in degrees ≤ 3. A fundamental question in the homology of linear groups is that of rigidity: given a smooth ...

Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the maximum degree of a vertex of H. We conjecture that for every > 0, if D is sufficiently large as a function of t, k and , then the chromatic index of H is at most (t − 1 + 1/t + )D. We prove this conjecture for the special case of intersecting hypergraphs in the following stronger form:...

Interval digraphs were introduced by West et all. They can be recognized in polynomial time and admit a characterization in terms of incidence matrices. Nevertheless, we do not have a forbidden structure characterization nor a low-degree polynomial time algorithm. We introduce a new class of ‘adjusted interval digraphs’, obtained by a slight change in the definition. By contrast, these digraphs...

We prove that for every $n$, there is a graph $G$ with $\chi(G) \geq n$ and $\omega(G) \leq 3$ such induced subgraph $H$ of $\omega(H) 2$ satisfies $\chi(H) 4$. This disproves well-known conjecture. Our construction digraph bounded clique number, large dichromatic no directed cycles odd length at least 5.

We further generalise a construction – the fibre construction – that was developed in an earlier paper of the first two authors. The extension in this paper gives a polynomial-time reduction of CSP(H) for any relational system H to CSP(P ) for any relational system P that meets a certain technical partition condition, that of being K3-partitionable. Moreover, we define an equivalent condition o...

Suppose m and n are integers such that 1 m n, and H is a subgroup of the symmetric group S m of degree m. Deene the generalized matrix function associated with the principal character of the group H on an m m matrix B = (b ij) by d H (B) = X 2H m Y j=1 b j(j) ; and deene the generalized numerical range of an n n matrix A associated with d H by It is known that W H (A) is convex if m = 1 or if m...

The aim of this note is to disprove a conjecture of H. Kierstead 5].

Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the maximum degree of a vertex of H. We conjecture that for every > 0, if D is sufficiently large as a function of t, k and , then the chromatic index of H is at most (t − 1 + 1/t + )D. We prove this conjecture for the special case of intersecting hypergraphs in the following stronger form:...