For a reductive $p$-adic group $G$, we compute the supports of the Hecke algebras for the $K$-types for $G$ lying in a certain frequently-occurring class. When $G$ is classical, we compute the intertwining between any two such $K$-types.

Two-weight inequalities of strong and weak type are obtained in the context of spaces of homogeneous type. Various applications are given, in particular to Cauchy singular integrals on regular curves.

The ‘1-loop partition function’ of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of $\text{S}{{\text{L}}_{2}}(\mathbb{Z})$, and obeying an integrality condition. Classifying these is a clearly defined mathematical problem, and at least for the affine Kac-Moody algebras tends to have interesting solutions. This paper finds for each affine algebra $B_{r}^{\left( 1 \right)}$ and $D_{r}^{(1)}$ all of these at level $k\le 3$. Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2—the $B_{r}^{(1)},D_{r}^{(1)}$ level 2 classification is easily the most anomalous one known and this uniqueness is the primary motivation for this paper. The only level 3 exceptionals occur for $B_{2}^{(1)}\cong C_{2}^{(1)}$ and $D_{7}^{(1)}$. The ${{B}_{2,3}}$ and ${{D}_{7,3}}$ exceptionals are cousins of the ${{\varepsilon }_{6}}$-exceptional and ${{\varepsilon }_{8}}$-exceptional, respectively, in the $\text{A-D-E}$ classification for $A_{1}^{(1)}$, while the level 2 exceptionals are related to the lattice invariants of affine $u(1)$.

We consider the problem

$$\left\{ \begin{matrix} {{\varepsilon }^{2}}\Delta u-u+f(u)=0,u>0 & \text{in}\,\Omega \\ \frac{\partial u}{\partial v}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \text{on}\,\partial \Omega , \\\end{matrix} \right.$$

where $\Omega $ is a bounded smooth domain in ${{R}^{N}},\varepsilon >0$ is a small parameter and $f$ is a superlinear, subcritical nonlinearity. It is known that this equation possesses multiple boundary spike solutions that concentrate, as $\varepsilon $ approaches zero, at multiple critical points of the mean curvature function $H(P),P\in \partial \Omega $. It is also proved that this equation has multiple interior spike solutions which concentrate, as $\varepsilon \to 0$, at sphere packing points in $\Omega $.

In this paper, we prove the existence of solutions with multiple spikes both on the boundary and in the interior. The main difficulty lies in the fact that the boundary spikes and the interior spikes usually have different scales of error estimation. We have to choose a special set of boundary spikes to match the scale of the interior spikes in a variational approach.

In this paper, we use Jacquet module methods to study the problem of classifying discrete series for the classical $p$-adic groups $\text{Sp}(2n,F)\,\,\text{and}\,\text{SO}\,(2n+1,F)$.

This paper treats the moduli space ${{\mathcal{M}}_{g,1}}(\Lambda)$ of representations of the fundamental group of a Riemann surface of genus $g$ with one boundary component which send the loop around the boundary to an element conjugate to exp $\Lambda$, where $\Lambda$ is in the fundamental alcove of a Lie algebra. We construct natural line bundles over ${{\mathcal{M}}_{g,1}}(\Lambda)$ and exhibit natural homology cycles representing the Poincaré dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a particular subring of the cohomology of the moduli space of stable bundles of coprime rank $k$ and degree $d$.

Let ${{\phi }_{1}},...,{{\phi }_{n}}\,(n\,\ge \,2)$ be nonzero integers such that the equation

$$\sum\limits_{i=1}^{n}{{{\phi }_{i}}x_{i}^{2}\,=\,0}$$

is solvable in integers ${{x}_{1}},...,{{x}_{n}}$ not all zero. It is shown that there exists a solution satisfying

$$0\,<\,\sum\limits_{i}^{n}{\left| {{\phi }_{i}}\left| x_{i}^{2}\,\le \,2 \right|{{\phi }_{1}}...{{\phi }_{n}}\, \right|}$$

and that the constant 2 is best possible.

Let ${{A}_{\theta}}$ denote the rotation algebra—the universal ${{C}^{*}}$-algebra generated by unitaries $U,V$ satisfying $VU={{e}^{2\pi i\theta }}UV$, where $\theta $ is a fixed real number. Let $\sigma $ denote the Fourier automorphism of ${{A}_{\theta}}$ defined by $U\mapsto V\text{,}V\mapsto {{U}^{-1}}$, and let ${{B}_{\theta }}={{A}_{\theta }}{{\rtimes }_{\sigma }}\mathbb{Z}/4\mathbb{Z}$ denote the associated ${{C}^{*}}$-crossed product. It is shown that there is a canonical inclusion ${{\mathbb{Z}}^{9}}\hookrightarrow {{K}_{0}}({{B}_{\theta }})$ for each $\theta $ given by nine canonical modules. The unbounded trace functionals of ${{B}_{\theta }}$ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero $\text{H}{{\text{C}}^{0}}({{B}_{\theta }})$ when $\theta $ is irrational. The Chern characters of the nine modules—and more importantly, the Fourier module—are computed and shown to involve techniques from the theory of Jacobi’s theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense ${{\text{G}}_{\delta }}$ set of values of $\theta $ one has ${{K}_{0}}({{B}_{\theta }})\cong {{\mathbb{Z}}^{9}}$ and is generated by the nine classes constructed here.