Suppose that $(X,\unicode[STIX]{x1D714})$ is a compact Kähler manifold. In the present work we propose a construction for weak geodesic rays in the space of Kähler potentials that is tied together with properties of the class ${\mathcal{E}}(X,\unicode[STIX]{x1D714})$. As an application of our construction, we prove a characterization of ${\mathcal{E}}(X,\unicode[STIX]{x1D714})$ in terms of envelopes.

We show that any $n$-dimensional Fano manifold $X$ with $\unicode[STIX]{x1D6FC}(X)=n/(n+1)$ and $n\geqslant 2$ is K-stable, where $\unicode[STIX]{x1D6FC}(X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any such $X$ admits Kähler–Einstein metrics and the holomorphic automorphism group $\operatorname{Aut}(X)$ of $X$ is finite.

In a scale of Fock spaces with radial weights ϕ we study the existence of Riesz bases of (normalized) reproducing kernels. We prove that these spaces possess such bases if and only if ϕ(x) grows at most like (log x)2.

We give a topological criterion for the minimality of the strong unstable (or stable) foliation of robustly transitive partially hyperbolic diffeomorphisms.

As a consequence we prove that, for $3$-manifolds, there is an open and dense subset of robustly transitive diffeomorphisms (far from homoclinic tangencies) such that either the strong stable or the strong unstable foliation is robustly minimal.

We also give a topological condition (existence of a central periodic compact leaf) guaranteeing (for an open and dense subset) the simultaneous minimality of the two strong foliations.

AMS 2000 Mathematics subject classification: Primary 37D25; 37C70; 37C20; 37C29

We develop an explicit descent theory in the context of Whitehead groups of non-commutative Iwasawa algebras. We apply this theory to describe the precise connection between main conjectures of non-commutative Iwasawa theory (in the spirit of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture and proving special cases of the equivariant Tamagawa number conjecture.

We reduce the openness conjecture of Demailly and Kollár on the singularities of plurisubharmonic functions to a purely algebraic statement.

We prove an explicit formula for periods of certain automorphic forms on SO5 × SO4 along the diagonal subgroup SO4 in terms of L-values. Our formula also involves a quantity from the theory of endoscopy, as predicted by the refined Gross–Prasad conjecture.

We study several types of curves and higher-dimensional objects inside the moduli spaces of curves, insisting on their arithmetic properties in the perspective of Grothendieck–Teichmüller theory. On the way we explicitly identify those curves which were originally associated by W. Veech with certain rational polygonal billiards.

AMS 2000 Mathematics subject classification: Primary 14D22; 32G15; 14H30

The notion of a prolongation of an algebraic variety is developed in an abstract setting that generalizes the difference and (Hasse) differential contexts. An interpolating map that compares these prolongation spaces with algebraic jet spaces is introduced and studied.

An exact control theorem is proven for nearly ordinary $p$-adic automorphic forms on symplectic and unitary groups over totally real fields if the algebraic group is split at $p$. In particular, a given nearly ordinary holomorphic Hecke eigenform can be lifted to a family of holomorphic Hecke eigenforms indexed by weights of the standard maximal split torus of the group. Their $q$-expansion coefficients are Iwasawa functions on the Iwasawa algebra of ${\mathbb{Z}}_p$-points of the split torus. The method is applicable to any reductive algebraic groups yielding Shimura varieties of PEL type under mild assumptions on the existence of integral toroidal compactification of the variety. Even in the Hilbert modular case, the result contains something new: freeness of the universal nearly ordinary Hecke algebra over the Iwasawa algebra, which eluded my scrutiny when I studied general theory of the $p$-adic Hecke algebra in the 1980s.

AMS 2000 Mathematics subject classification: Primary 11F03; 11F30; 11F33; 11F41; 11F60; 11G15; 11G18

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic. The main goal of this paper is to introduce a Riemann–Hilbert problem whose solution gives the solution of our initial–boundary-value problem. This is a preliminary step to obtain uniform long-time asymptotics for the solution of this equation using both the stationary phase and the Deift–Zhou methods.

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac–Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ($C_{2}$-cofinite) $W$-algebras that are not coming from admissible representations of affine Kac–Moody algebras.

Let $X$ be a smooth complex projective manifold of dimension $n$ equipped with an ample line bundle $L$ and a rank $k$ holomorphic vector bundle $E$. We assume that $1\leqslant k\leqslant n$, that $X$, $E$ and $L$ are defined over the reals and denote by $\mathbb{R}X$ the real locus of $X$. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in $\mathbb{R}X$ of holomorphic real sections of $E\otimes L^{d}$, where $d$ is a large enough integer. Moreover, given any closed connected codimension $k$ submanifold ${\it\Sigma}$ of $\mathbb{R}^{n}$ with trivial normal bundle, we prove that a real section of $E\otimes L^{d}$ has a positive probability, independent of $d$, of containing around $\sqrt{d}^{n}$ connected components diffeomorphic to ${\it\Sigma}$ in its vanishing locus.

Let K be an arbitrary number field, and let ρ : Gal(/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal(/ℚ) when n > 2.

We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups $\text{GL}(n)$ and $\text{SL}(n)$ we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur’s trace formula obtained in [Finis, Lapid, and Müller, Ann. of Math. (2) 174(1) (2011), 173–195; Finis and Lapid, Ann. of Math. (2) 174(1) (2011), 197–223], which allows us to show that for $\text{GL}(n)$ and $\text{SL}(n)$ the contribution of the continuous spectrum is negligible in the limit.

The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point for a small amount of disorder. The question has been mathematically settled in most cases in the last few years, giving in particular a rigorous validation of the Harris criterion on disorder relevance. However, the marginal case, where the return probability exponent is equal to $1/2$, that is, where the interarrival law of the renewal process is given by $\text{K}(n)=n^{-3/2}\unicode[STIX]{x1D719}(n)$ where $\unicode[STIX]{x1D719}$ is a slowly varying function, has been left partially open. In this paper, we give a complete answer to the question by proving a simple necessary and sufficient criterion on the return probability for disorder relevance, which confirms earlier predictions from the literature. Moreover, we also provide sharp asymptotics on the critical point shift: in the case of the pinning of a one-dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptotics

$$\begin{eqnarray}\lim _{\unicode[STIX]{x1D6FD}\rightarrow 0}\unicode[STIX]{x1D6FD}^{2}\log h_{c}(\unicode[STIX]{x1D6FD})=-\frac{\unicode[STIX]{x1D70B}}{2}.\end{eqnarray}$$

We deal with aspects of direct and inverse problems in parameterized Picard–Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) $G$ is a PPV Galois group over these fields if and only if $G$ contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs $G$, including unipotent groups, $G$ is such a group if and only if it has differential type $0$. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois group of a parameterized linear differential equation if its Galois group has differential type $0$.

The S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental group of a product of two complete varieties is a product of their S-fundamental groups as conjectured by Mehta and the author. We also compute the abelian part of the S-fundamental group scheme and the S-fundamental group scheme of an abelian variety or a variety with trivial étale fundamental group.

Given a compact Lie group $G$, in this paper we establish $L^{p}$-bounds for pseudo-differential operators in $L^{p}(G)$. The criteria here are given in terms of the concept of matrix symbols defined on the noncommutative analogue of the phase space $G\times \widehat{G}$, where $\widehat{G}$ is the unitary dual of $G$. We obtain two different types of $L^{p}$ bounds: first for finite regularity symbols and second for smooth symbols. The conditions for smooth symbols are formulated using $\mathscr{S}_{\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF}}^{m}(G)$ classes which are a suitable extension of the well-known $(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D6FF})$ ones on the Euclidean space. The results herein extend classical $L^{p}$ bounds established by C. Fefferman on $\mathbb{R}^{n}$. While Fefferman’s results have immediate consequences on general manifolds for $\unicode[STIX]{x1D70C}>\max \{\unicode[STIX]{x1D6FF},1-\unicode[STIX]{x1D6FF}\}$, our results do not require the condition $\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$. Moreover, one of our results also does not require $\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$. Examples are given for the case of $\text{SU}(2)\cong \mathbb{S}^{3}$ and vector fields/sub-Laplacian operators when operators in the classes $\mathscr{S}_{0,0}^{m}$ and $\mathscr{S}_{\frac{1}{2},0}^{m}$ naturally appear, and where conditions $\unicode[STIX]{x1D70C}>\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D70C}>1-\unicode[STIX]{x1D6FF}$ fail, respectively.

let $\pi$ be a cuspidal automorphic representation of $\mathrm{gl}_n(\mathbb{a}_{\mathbb{q}})$ with non-vanishing cohomology. under a certain local non-vanishing assumption we prove the rationality of the values of the automorphic $l$-function attached to $\pi$ at critical points. conjecturally, any motivic $l$-function coincides with an $l$-function attached to an automorphic representation on $\mathrm{gl}_n$, hence, our result corresponds to a conjecture of deligne on critical values of motivic $l$-functions.