Serre's Modularity Conjecture

These are the lecture notes from a five-hour mini-course given at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. Their aim is to give an overview of Serre's modularity conjecture and of its proof by Khare, Wintenberger, and Kisin, as well as of the results of other mathematicians that played an important role in the proof. Along the way we remark on some recent (as of 2012) work concerning generalizations of the conjecture

Normal zeta functions of the Heisenberg groups over number rings II -- the non-split case

We compute explicitly the normal zeta functions of the Heisenberg groups $H(R)$, where $R$ is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form $H(\mathcal{O}_K)$, where $\mathcal{O}_K$ is the ring of integers of an arbitrary number field~$K$, at the rational primes which are non-split in~$K$. We show that these local zeta functions satisfy functional equations upon the inversion of the prime.Comment: 19 pages; to appear in Israel J. Mat

Semigroups of tolerance relations

AbstractWe give an abstract algebraic characterization of semigroups of tolerance relations and semigroups of symmetric binary relations

West Virginia and Regional History Collection Newsletter Twenty-Year Index, Volume 1-Volume 20, Spring 1985-Spring 2005

Index of articles and photographs appearing in the WVRHC Newsletter over a 20 year period