We will construct a finite union of finite quotients of the affine building of the group $\text{G}{{\text{L}}_{3}}$ over the field of $p$ -adic numbers ${{\mathbb{Q}}_{p}}$ . We will view this object as a hypergraph and estimate the spectrum of its underlying graph.

We describe the embedded resolution of an irreducible quasi-ordinary surface singularity $\left( V,\,p \right)$ which results from applying the canonical resolution of Bierstone-Milman to $\left( V,\,p \right)$ . We show that this process depends solely on the characteristic pairs of $\left( V,\,p \right)$ , as predicted by Lipman. We describe the process explicitly enough that a resolution graph for $f$ could in principle be obtained by computer using only the characteristic pairs.

A simple ${{C}^{*}}$ -algebra is constructed for which the Murray-von Neumann equivalence classes of projections, with the usual addition—induced by addition of orthogonal projections—form the additive semigroup

$$\left\{ 0,\,2,\,3,\ldots\right\}.$$

(This is a particularly simple instance of the phenomenon of perforation of the ordered ${{K}_{0}}$ -group, which has long been known in the commutative case—for instance, in the case of the four-sphere—and was recently observed by the second author in the case of a simple ${{C}^{*}}$ -algebra.)

Let $G$ be a connected reductive $p$ -adic group and let $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{O}$ be any $G$ -orbit in $\mathfrak{g}$ . Then the orbital integral ${{\mu }_{\mathcal{O}}}$ corresponding to $\mathcal{O}$ is an invariant distribution on $\mathfrak{g}$ , and Harish-Chandra proved that its Fourier transform ${{\hat{\mu }}_{\mathcal{O}}}$ is a locally constant function on the set ${\mathfrak{g}}'$ of regular semisimple elements of $\mathfrak{g}$ . If $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ , and $\omega $ is a compact subset of $\mathfrak{h}\,\cap \,{\mathfrak{g}}'$ , we give a formula for ${{\hat{\mu }}_{\mathcal{O}}}\left( tH \right)$ for $H\,\in \,\omega $ and $t\,\in \,{{F}^{\times }}$ sufficiently large. In the case that $\mathcal{O}$ is a regular semisimple orbit, the formula is already known by work of Waldspurger. In the case that $\mathcal{O}$ is a nilpotent orbit, the behavior of ${{\hat{\mu }}_{\mathcal{O}}}$ at infinity is already known because of its homogeneity properties. The general case combines aspects of these two extreme cases. The formula for ${{\hat{\mu }}_{\mathcal{O}}}$ at infinity can be used to formulate a “theory of the constant term” for the space of distributions spanned by the Fourier transforms of orbital integrals. It can also be used to show that the Fourier transforms of orbital integrals are “linearly independent at infinity.”

A nest representation of a strongly maximal $\text{TAF}$ algebra $A$ with diagonal $D$ is a representation $\pi $ for which $\text{Lat}\,\pi \left( A \right)$ is totally ordered. We prove that $\ker \,\pi$ is a meet irreducible ideal if the spectrum of $A$ is totally ordered or if (after an appropriate similarity) the von Neumann algebra $\text{ }\!\!\pi\!\!\text{ }{{\left( D \right)}^{\prime \prime }}$ contains an atom.

There is a well-known correspondence (due to Mehta and Seshadri in the unitary case, and extended by Bhosle and Ramanathan to other groups), between the symplectic variety ${{M}_{h}}$ of representations of the fundamental group of a punctured Riemann surface into a compact connected Lie group $G$ , with fixed conjugacy classes $h$ at the punctures, and a complex variety ${{\mathcal{M}}_{h}}$ of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For $G\,=\,\text{SU}\left( 2 \right)$ , we build a symplectic variety $P$ of pairs (representations of the fundamental group into $G$ , “weighted frame” at the puncture points), and a corresponding complex variety $\mathcal{P}$ of moduli of “framed parabolic bundles”, which encompass respectively all of the spaces ${{M}_{h}}$ , ${{\mathcal{M}}_{h}}$ , in the sense that one can obtain ${{M}_{h}}$ from $P$ by symplectic reduction, and ${{\mathcal{M}}_{h}}$ from $\mathcal{P}$ by a complex quotient. This allows us to explain certain features of the toric geometry of the $\text{SU(2)}$ moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system.

We study extensions $L/K$ of complete discrete valuation fields $K$ with residue field $\bar{K}$ of characteristic $p\,>\,0$ , which we do not assume to be perfect. Our work concerns ramification theory for such extensions, in particular we show that all classical properties which are true under the hypothesis “the residue field extension $\bar{L}/\bar{K}$ is separable” are still valid under the more general hypothesis that the valuation ring extension is monogenic. We also show that conversely, if classical ramification properties hold true for an extension $L/K$ , then the extension of valuation rings is monogenic. These are the “well ramified” extensions. We show that there are only three possible types of well ramified extensions and we give examples. In the last part of the paper we consider, for the three types, Kato’s generalization of the conductor, which we show how to bound in certain cases.

For every integer $n\,>\,1$ and infinite field $F$ we construct a spectral sequence converging to the homology of $\text{G}{{\text{L}}_{n}}\left( F \right)$ relative to the group of monomial matrices $\text{G}{{\text{M}}_{n}}\left( F \right)$ . Some entries in ${{E}^{2}}$ -terms of these spectral sequences may be interpreted as a natural generalization of the Bloch group to higher dimensions. These groups may be characterized as homology of $\text{G}{{\text{L}}_{n}}$ relatively to $\text{G}{{\text{L}}_{n-1}}$ and $\text{G}{{\text{M}}_{n}}$ . We apply the machinery developed to the investigation of stabilization maps in homology of General Linear Groups.