We prove that the non-ordinary component is connected in the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This was conjectured by Kisin. As an application to global Galois representations, we prove a theorem on the modularity comparing a deformation ring and a Hecke ring.

Given a graph G on n vertices with maximal degree d, form a random subgraph Gp by choosing each edge of G independently with probability p. Strengthening a classical result of Margulis we prove that if the edge connectivity k(G) satisfies k(G) À d/ log n, then the connectivity threshold in Gp is sharp. This result is asymptotically tight.

In [Many disjoint dense subgraphs versus large k-connected subgraphs in large graphs with given edge density, Discrete Math. 309 (2009), 997–1000.], Böhme and Kostochka showed that every large enough graph with sufficient edge density contains either a kconnected subgraph of order at least r or a family of r vertex-disjoint k-connected subgraphs. Motivated by this, in this note we explore the l...

On the prime-top part of the groups of connected components of Néron models.

We prove orthogonality formulas for the characteristic functions of certain complexes on a connected component of a reductive group.