We construct embedded periodic minimal surfaces with planar ends. We use the Weierstrass representation and solve the period problem using the implicit function theorem at a singular point.

AMS 2000 Mathematics subject classification: Primary 53A10

Ellipticity of operators on a manifold with edges can be treated in the framework of a calculus of $2\times2$-block matrix operators with trace and potential operators on the edges. The picture is similar to the pseudodifferential analysis of boundary-value problems. The extra conditions satisfy an analogue of the Shapiro–Lopatinskij condition, provided a topological obstruction for the elliptic edge-degenerate operator in the upper left corner vanishes; this is an analogue of a condition of Atiyah and Bott in boundary-value problems. In general, however, we need global projection data, similarly to global boundary conditions, known for Dirac operators or other geometric operators. The present paper develops a new calculus with global projection data for operators on manifolds with edges. In particular, we show the Fredholm property in a suitable scale of spaces and construct parametrices within the calculus.

It is well known that a finitely generated group ${\rm\Gamma}$ has Kazhdan’s property (T) if and only if the Laplacian element ${\rm\Delta}$ in $\mathbb{R}[{\rm\Gamma}]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in $\mathbb{R}[{\rm\Gamma}]$. Namely, ${\rm\Gamma}$ has property (T) if and only if there exist a constant ${\it\kappa}>0$ and a finite sequence ${\it\xi}_{1},\ldots ,{\it\xi}_{n}$ in $\mathbb{R}[{\rm\Gamma}]$ such that ${\rm\Delta}^{2}-{\it\kappa}{\rm\Delta}=\sum _{i}{\it\xi}_{i}^{\ast }{\it\xi}_{i}$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in $\mathbb{R}[{\rm\Gamma}]$, possibly with the assistance of computers.

Many quantum field theories in one, two and four dimensions possess remarkable limits in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyse the corresponding models as full quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kähler manifolds the space of states exhibits holomorphic factorization. We conclude that in dimensions two and four our theories are logarithmic conformal field theories.

In Part I we describe the class of models under study and present our results in the case of one-dimensional (quantum mechanical) models, which is quite representative and at the same time simple enough to analyse explicitly. Part II will be devoted to supersymmetric two-dimensional sigma models and four-dimensional Yang–Mills theory. In Part III we will discuss non-supersymmetric models.

Nous présentons une expression explicite pour la hauteur normalisée d’une variété torique projective définie sur $\overline{\mathbb{Q}}$. Cette expression se décompose comme somme de contributions locales, chaque terme étant l’intégrale d’une certaine fonction concave et affine par morceaux, définie sur le polytope $Q_{\mathcal{A}}$ classiquement associé à l’action du tore. Plus généralement, nous obtenons une expression explicite pour la multihauteur normalisée d’un tore relative à plusieurs plongements monomiaux.

L’ensemble de fonctions introduit se comporte comme un analogue arithmétique du polytope $Q_{\mathcal{A}}$. En plus des formules pour les hauteur et multihauteurs, nous montrons que cet objet se comporte de manière naturelle par rapport à plusieurs constructions standard : décomposition en orbites, formation de joints, produits de Segre et plongements de Veronese.

La démonstration suit une démarche indirecte : à la place de la définition de la hauteur normalisée, on s’appuie sur le calcul d’une fonction de Hilbert arithmétique appropriée.

We present an explicit expression for the normalized height of a projective toric variety defined over $\overline{\mathbb{Q}}$. This expression decomposes as a sum of local contributions, each term being the integral of a certain function, concave and piecewise linear-affine, defined on the polytope $Q_{\mathcal{A}}$ classically associated with the torus action. More generally, we obtain an explicit expression for the normalized multiheight of a torus with respect to several monomial embeddings.

The set of functions introduced behaves as an arithmetic analogue of the polytope $Q_{\mathcal{A}}$. Besides the formulae for the height and multiheight, we show that this object behaves in a natural way with respect to several standard constructions: decomposition into orbits, joins, Segre products and Veronese embeddings.

The proof follows an indirect path: instead of the definition of the normalized height, we rely on the computation of an appropriate arithmetic Hilbert function.

We investigate the special fibres of Siegel modular varieties with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz–Rapoport (KR) stratification; one would like to understand how these stratifications are related to each other. We give a simple description of all KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne–Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties.

we prove in this work the trace and trace lifting theorems for the sobolev spaces associated with a system of hörmander’s vectors fields of order 2. the case of non-degenerate characteristic points for left invariant vector fields on the heisenberg group is also studied.

We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in Hernandez and Leclerc (Duke Math. J.154 (2010), 265–341; Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statitics, Volume 40, pp. 175–193 (2013)), in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting) to $\mathfrak{sl}_{n+1}[t]$-stable prime Demazure modules in level-two integrable highest-weight representations of the classical affine Lie algebra. It was proved in Chari et al. (arXiv:1408.4090) that a stable Demazure module is isomorphic to the fusion product of stable prime Demazure modules. Our next result proves that such a fusion product is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine $\mathfrak{sl}_{n+1}$.

We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set $K$ is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects $K$ at exactly one point.

In this paper, we construct explicitly endoscopic representations of $\widetilde{\mathrm{Sp}}_{2n}$, the metaplectic cover of a symplectic group of rank $n$. We do this in the automorphic case, and also in the local case, over a $p$-adic field.

AMS 2000 Mathematics subject classification: Primary 11F70

We provide asymptotics for the range $R_{n}$ of a random walk on the $d$-dimensional lattice indexed by a random tree with $n$ vertices. Using Kingman’s subadditive ergodic theorem, we prove under general assumptions that $n^{-1}R_{n}$ converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension $4$, and in the case of a symmetric random walk with exponential moments, we prove that $R_{n}$ grows like $n/\!\log n$. We apply our results to asymptotics for the range of a branching random walk when the initial size of the population tends to infinity.

Let $(X,{\rm\Delta})$ be a projective klt (standing for Kawamata log terminal) three-dimensional pair defined over an algebraically closed field $k$ with $\text{char}(k)>5$. Let $L$ be a nef (numerically eventually free) and big line bundle on $X$ such that $L-K_{X}-{\rm\Delta}$ is big and nef. We show that $L$ is indeed semi-ample.

This paper gives a complete description of the spherical unitary dual of split classical real and p-adic groups. The proof makes heavy use of the affine graded Hecke algebra.

Subconvexity bounds on the critical line are proved for general Epstein zeta-functions of $k$-ary quadratic forms. This is related to sup-norm bounds for unitary Eisenstein series on $\text{GL}(k)$ associated with the maximal parabolic of type $(k-1,1)$, and the exact sup-norm exponent is determined to be $(k-2)/8$ for $k\geqslant 4$. In particular, if $k$ is odd, this exponent is not in $\frac{1}{4}\mathbb{Z}$, which is relevant in the context of Sarnak’s purity conjecture and shows that it can in general not directly be generalized to Eisenstein series.

Let $R$ be a large field of characteristic $p$. We classify the supersingular simple modules of the pro-$p$-Iwahori Hecke $R$-algebra ${\mathcal{H}}$ of a general reductive $p$-adic group $G$. We show that the functor of pro-$p$-Iwahori invariants behaves well when restricted to the representations compactly induced from an irreducible smooth $R$-representation $\unicode[STIX]{x1D70C}$ of a special parahoric subgroup $K$ of $G$. We give an almost-isomorphism between the center of ${\mathcal{H}}$ and the center of the spherical Hecke algebra ${\mathcal{H}}(G,K,\unicode[STIX]{x1D70C})$, and a Satake-type isomorphism for ${\mathcal{H}}(G,K,\unicode[STIX]{x1D70C})$. This generalizes results obtained by Ollivier for $G$ split and $K$ hyperspecial to $G$ general and $K$ special.

Nous construisons une $KK$-théorie pour les algèbres de Banach équivariante par l’action d’un groupoïde et nous montrons la conjecture de Baum–Connes à coefficients commutatifs pour les groupes hyperboliques et pour les groupoïdes de Poincaré des feuilletages à base compacte qui peuvent être munis d’une métrique riemannienne longitudinale à courbure sectionnelle strictement négative.

We construct a $KK$-theory for Banach algebras, equivariant with respect to the action of a groupoid. We prove the Baum–Connes conjecture with commutative coefficients for hyperbolic groups and for the Poincaré groupoids of foliations with a compact base and a longitudinal Riemannian metrics with negative sectional curvature.

We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen’s connected Seiberg–Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction terms $\bar{d}$ and $\text{}\underline{d}$ for certain families of three-manifolds.

We prove that if $M$ is a $\text{JBW}^{\ast }$-triple and not a Cartan factor of rank two, then $M$ satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another real Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$.

We consider a transitive uniformly quasi-conformal Anosov diffeomorphism $f$ of a compact manifold $\mathcal{M}$. We prove that if the stable and unstable distributions have dimensions greater than two, then $f$ is $C^\infty$ conjugate to an affine Anosov automorphism of a finite factor of a torus. If the dimensions are at least two, the same conclusion holds under the additional assumption that $\mathcal{M}$ is an infranilmanifold. We also describe necessary and sufficient conditions for smoothness of conjugacy between such a diffeomorphism and a small perturbation.

AMS 2000 Mathematics subject classification: Primary 37C; 37D