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.

Let $\left( {{M}_{n}},\,g \right)$ be a strictly convex riemannian manifold with ${{C}^{\infty }}$ boundary. We prove the existence of classical solution for the nonlinear elliptic partial differential equation of Monge-Ampère: $\det (-u\delta _{j}^{i}\,+\,\nabla _{j}^{i}u)\,=\,F(x,\,\nabla u;\,u)$ in $M$ with a Neumann condition on the boundary of the form $\frac{\partial u}{\partial v}=\varphi (x,u)$, where $F\in {{C}^{\infty }}(TM\times \mathbb{R})$ is an everywhere strictly positive function satisfying some assumptions, $v$ stands for the unit normal vector field and $\varphi \in {{C}^{\infty }}(\partial M\times \mathbb{R})$ is a non-decreasing function in $u$.

We study the “Brownian snake” introduced by Le Gall, and also studied by Dynkin, Kuznetsov, Watanabe. We prove that Itô’s formula holds for a wide class of functionals. As a consequence, we give a new proof of the connections between the Brownian snake and super-Brownian motion. We also give a new definition of the Brownian snake as the solution of a well-posed martingale problem. Finally, we construct a modified Brownian snake whose lifetime is driven by a path-dependent stochastic equation. This process gives a representation of some super-processes.

In this article, using 3-orbifolds singular along a knot with underlying space a homology sphere ${{Y}^{3}}$, the question of existence of non-trivial and non-abelian $\text{SU(2)}$-representations of the fundamental group of cyclic branched covers of ${{Y}^{3}}$ along a knot is studied. We first use Floer Homology for knots to derive an existence result of non-abelian $\text{SU(2)}$-representations of the fundamental group of knot complements, for knots with a non-vanishing equivariant signature. This provides information on the existence of non-trivial and non-abelian $\text{SU(2)}$-representations of the fundamental group of cyclic branched covers. We illustrate the method with some examples of knots in ${{S}^{3}}$.

This paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals—an expression which is ideally suited for the study of the stability of those characters. Building on work of F. Murnaghan, our proof involves Lusztig’s Generalised Springer Correspondence in a fundamental way, and also makes use of some results on elliptic orbital integrals proved elsewhere by the author using Moy-Prasad filtrations of $p$-adic Lie algebras. Two applications of the main result are considered toward the end of the paper.

A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that ${{\ell }^{1}},{{c}_{0}}$ and ${{\ell }^{\infty }}$ are the only symmetric sequence spaces with the Dunford- Pettis property, and that in the class of symmetric spaces on $(0,\,\alpha ),\,0\,<\,\alpha \,\le \,\infty$, the only spaces with the Dunford-Pettis property are ${{\text{L}}^{1}},{{\text{L}}^{\infty }},{{\text{L}}^{1}}\cap {{\text{L}}^{\infty }},{{\text{L}}^{1}}+{{\text{L}}^{\infty }},{{({{\text{L}}^{\infty }})}^{\text{o}}}$ and ${{({{\text{L}}^{1}}+{{\text{L}}^{\infty }})}^{\text{o}}}$, where ${{\text{X}}^{\text{o}}}$ denotes the norm closure of ${{\text{L}}^{1}}\cap {{\text{L}}^{\infty }}$ in $X$. It is also proved that all Banach dual spaces of ${{\text{L}}^{1}}\cap {{\text{L}}^{\infty }}$ and ${{\text{L}}^{1}}+{{\text{L}}^{\infty }}$ have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces ${{({{\text{L}}^{1}}+{{\text{L}}^{\infty }})}^{\text{o}}}$ and ${{({{\text{L}}^{\infty }})}^{\text{o}}}$ have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some Köthe-Bochner spaces.

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

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.

Errors to a previous paper (Canad. J. Math. (2) 49(1997), 232–262) are corrected. A non-standard regularisation of the auxiliary operator $A$ appearing in Mourre theory is used.

It is proved that a “typical” $n$-dimensional quotient ${{X}_{n}}$ of $l_{1}^{m}$ with $n={{m}^{\sigma }},0<\sigma <1$, has the property

$$\text{Average}\int_{G}{||Tx|{{|}_{{{X}_{n}}}}d{{h}_{G}}(T)\ge \frac{c}{\sqrt{n{{\log }^{3}}n}}\left( n-\int_{G}{|trT|d{{h}_{G}}(T)} \right)},$$

for every compact group $G$ of operators acting on ${{X}_{n}}$, where ${{d}_{G}}(T)$ stands for the normalized Haar measure on $G$ and the average is taken over all extreme points of the unit ball of ${{X}_{n}}$. Several consequences of this estimate are presented.

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 $d$.

J. Arthur put the trace formula in invariant form for all connected reductive groups and certain disconnected ones. However his work was written so as to apply to the general disconnected case, modulo two missing ingredients. This paper supplies one of those missing ingredients, namely an argument in Galois cohomology of a kind first used by D. Kazhdan in the connected case.

Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety. We show that $X$ can be transformed, by a sequence of blowups with smooth $G$-equivariant centers, into a $G$-variety ${{X}^{'}}$ with the following property: the stabilizer of every point of ${{X}^{'}}$ is isomorphic to a semidirect product $U\rtimes A$ of a unipotent group $U$ and a diagonalizable group $A$.

As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.

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.

Suppose $G$ is a countable, Abelian group with an element of infinite order and let $\text{ }\!\!\chi\!\!\text{ }$ be a mixing rank one action of $G$ on a probability space. Suppose further that the Følner sequence $\{{{F}_{n}}\}$ indexing the towers of $\text{ }\!\!\chi\!\!\text{ }$ satisfies a “bounded intersection property”: there is a constant $p$ such that each $\{{{F}_{n}}\}$ can intersect no more than $p$ disjoint translates of $\{{{F}_{n}}\}$. Then $\text{ }\!\!\chi\!\!\text{ }$ is mixing of all orders. When $G\,=\,\mathbf{Z}$, this extends the results of Kalikow and Ryzhikov to a large class of “funny” rank one transformations. We follow Ryzhikov’s joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of $k$ copies of $\text{ }\!\!\chi\!\!\text{ }$ is necessarily product measure. This method generalizes Ryzhikov’s technique.

Let $F$ be a divisor on the blow-up $X$ of ${{\mathbf{P}}^{2}}$ at $r$ general points ${{p}_{1}},...,{{p}_{r}}$ and let $L$ be the total transform of a line on ${{\mathbf{P}}^{2}}$. An approach is presented for reducing the computation of the dimension of the cokernel of the natural map ${{\mu }_{F}}:\Gamma ({{\mathcal{O}}_{_{X}}}(F))\otimes \Gamma ({{\mathcal{O}}_{_{X}}}(L))\to \Gamma ({{\mathcal{O}}_{_{X}}}(F)\otimes {{\mathcal{O}}_{_{X}}}(L))$ to the case that $F$ is ample. As an application, a formula for the dimension of the cokernel of ${{\mu }_{_{F}}}$ is obtained when $r\,=\,7$, completely solving the problem of determining the modules in minimal free resolutions of fat point subschemes ${{m}_{1}}\,{{p}_{1}}\,+\,\cdot \cdot \cdot \,+\,{{m}_{7}}\,{{p}_{7}}\,\subset \,{{\mathbf{P}}^{2}}$. All results hold for an arbitrary algebraically closed ground field $k$.

In this paper, we consider the (essential) spectrum of the discrete Laplacian of an infinite graph. We introduce a new quantity for an infinite graph, in terms of which we give new lower bound estimates of the (essential) spectrum and give also upper bound estimates when the infinite graph is bipartite. We give sharp estimates of the (essential) spectrum for several examples of infinite graphs.

We show that if $m,n\ge 0,\lambda >1$, and $R$ is a rational function with numerator, denominator of degree $\le m,n$, respectively, then there exists a set $S\subset [0,1]$ of linear measure $\ge \,\frac{1}{4}\,\exp \left( -\frac{13}{\log \,\text{ }\!\!\lambda\!\!\text{ }} \right)$ such that for $r\in S$,

1

$$_{|z|=r\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,|z|=r}^{\max |R(z)|/min|R(z)|\le {{\lambda }^{m+n}}.}$$

Here, one may not replace $\frac{1}{4}\exp (-\frac{13}{\log \lambda })$ by $\exp (-\frac{2-\varepsilon }{\log \lambda })$, for any $\varepsilon >0$. As our motivating application, we prove a convergence result for diagonal Padé approximants for functions meromorphic in the unit ball.

A unified formulation is given to various generalizations of the classical numerical range including the $c$-numerical range, congruence numerical range, $q$-numerical range and von Neumann range. Attention is given to those cases having connections with classical simple real Lie algebras. Convexity and inclusion relation involving those generalized numerical ranges are investigated. The underlying geometry is emphasized.

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.