This paper is concerned with the resolution of an inverse problem related to the recovery of a function $V$ from the source to solution map of the semi-linear equation $(\Box _{g}+V)u+u^{3}=0$ on a globally hyperbolic Lorentzian manifold $({\mathcal{M}},g)$ . We first study the simpler model problem, where $({\mathcal{M}},g)$ is the Minkowski space, and prove the unique recovery of $V$ through the

We examine 2-complexes $X$ with the property that for any compact connected  $Y$ , and immersion $Y\rightarrow X$ , either $\unicode[STIX]{x1D712}(Y)\leqslant 0$ or $\unicode[STIX]{x1D70B}_{1}Y=1$ . The mapping torus of an endomorphism of a free group has this property. Every irreducible 3-manifold with boundary has a spine with this property. We show that the fundamental group of any 2-complex with

Let $p:X\rightarrow Y$ be an algebraic fiber space, and let $L$ be a line bundle on $X$ . In this article, we obtain a curvature formula for the higher direct images of $\unicode[STIX]{x1D6FA}_{X/Y}^{i}\otimes L$ restricted to a suitable Zariski open subset of $X$ . Our results are particularly meaningful if $L$ is semi-negatively curved on $X$ and strictly negative or trivial on smooth fibers of $p$

Let $n$ be either  $2$ or an odd integer greater than  $1$ , and fix a prime  $p>2(n+1)$ . Under standard ‘adequate image’ assumptions, we show that the set of components of $n$ -dimensional $p$ -adic potentially semistable local Galois deformation rings that are seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field is independent of the particular

In this short paper, we combine the representability theorem introduced in [Porta and Yu, Representability theorem in derived analytic geometry, preprint, 2017, arXiv:1704.01683; Porta and Yu, Derived Hom spaces in rigid analytic geometry, preprint, 2018, arXiv:1801.07730] with the theory of derived formal models introduced in [António, $p$ -adic derived formal geometry and derived Raynaud localization

Suppose that $G$ is a simple reductive group over $\mathbf{Q}$ , with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unico

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families

The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$ -varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ are dominant rational maps from an (possibly nonreduced) irreducible scheme

We generalize Uhlenbeck’s generator theorem of ${\mathcal{L}}^{-}\operatorname{U}_{n}$ to the full rational loop group ${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{C}$ and its subgroups ${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{R}$ , ${\mathcal{L}}^{-}\operatorname{U}_{p,q}$ : they are all generated by just simple projective loops. Recall that Terng–Uhlenbeck studied the dressing actions of

The Beauville–Voisin conjecture for a hyperkähler manifold $X$ states that the subring of the Chow ring $A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of  $X$ . We prove a weak version of this conjecture when $X$ is the Hilbert scheme of points on a K3 surface for the subring generated by divisor classes and tautological classes

The base surface $B$ of a Lagrangian fibration of a projective, irreducible symplectic fourfold $X$ is shown to be isomorphic to $\mathbb{P}^{2}$ .

We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks, where the $i$ -morphisms are given by $i$ -fold coisotropic correspondences. Assuming an expected equivalence of different models of higher Morita categories, we prove that all derived Poisson stacks are fully dualizable and so determine framed

In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$ , where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$ , which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra

In this article, we study some Kramers–Fokker–Planck operators with a polynomial potential $V(q)$ of degree greater than two having quadratic limiting behaviour. This work provides an accurate global subelliptic estimate for Kramers–Fokker–Planck operators under some conditions imposed on the potential.

We give a necessary and sufficient condition for the existence of an enhancement of a finite triangulated category. Moreover, we show that enhancements are unique when they exist, up to Morita equivalence.

We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$ , we identify a game-theoretic condition equivalent to $K(X)$

Under an assumption on the existence of $p$ -adic Galois representations, we carry out Taylor–Wiles patching (in the derived category) for the completed homology of the locally symmetric spaces associated with $\operatorname{GL}_{n}$ over a number field. We use our construction, and some new results in non-commutative algebra, to show that standard conjectures on completed homology imply ‘big $R=\

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $v$ above $p$ . Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We describe the $\mathfrak{m}$ -torsion in the $\text{mod}\,p$ cohomology

We completely classify the possible extensions between semistable vector bundles on the Fargues–Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder–Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze’s language of diamonds. This analysis reduces our main

Given a perfect valuation ring $R$ of characteristic $p$ that is complete with respect to a rank- $1$ nondiscrete valuation, we show that the ring $\mathbb{A}_{\inf }$ of Witt vectors of $R$ has infinite Krull dimension.

The symbolic analytic spread of an ideal $I$ is defined in terms of the rate of growth of the minimal number of generators of its symbolic powers. In this article, we find upper bounds for the symbolic analytic spread under certain conditions in terms of other invariants of $I$ . Our methods also work for more general systems of ideals. As applications, we provide bounds for the (local) Kodaira dimension

The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over  $\overline{\mathbb{F}}_{p}$ of parallel weight 1 and level prime to $p$ is unramified above  $p$ . This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight 1. The proof is based on the observation that parallel

Let $q$ be an anisotropic quadratic form defined over a general field $F$ . In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization

We study the germs at the origin of $G$ -representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid

Let $G$ be a semisimple Lie group with associated symmetric space $D$ , and let $\unicode[STIX]{x1D6E4}\subset G$ be a cocompact arithmetic group. Let $\mathscr{L}$ be a lattice inside a $\mathbb{Z}\unicode[STIX]{x1D6E4}$ -module arising from a rational finite-dimensional complex representation of $G$ . Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion

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 $\frac{1}{4}\mathbb{Z}$

We prove a compactness theorem for metrics with bounded integral curvature on a fixed closed surface $\unicode[STIX]{x1D6F4}$ . As a corollary we obtain a new convergence result for sequences of metrics with conical singularities, where an accumulation of singularities is allowed.

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$ . We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$ , that if $X$ has only klt singularities, then so does $H$ .

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We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable sections with explicit bounded degree. To prove these results, we study log del Pezzo surfaces defined over imperfect fields.

We prove a comparison isomorphism between certain moduli spaces of $p$ -divisible groups and strict ${\mathcal{O}}_{K}$ -modules (RZ-spaces). Both moduli problems are of PEL-type (polarization, endomorphism, level structure) and the difficulty lies in relating polarized $p$ -divisible groups and polarized strict ${\mathcal{O}}_{K}$ -modules. We use the theory of relative displays and frames, as developed

The purpose of this note is to establish isomorphisms up to bounded torsion between relative $K_{0}$ -groups and Chow groups with modulus as defined by Binda and Saito.

We consider the Klein–Gordon equation on asymptotically anti-de-Sitter spacetimes subject to Neumann or Robin (or Dirichlet) boundary conditions and prove propagation of singularities along generalized broken bicharacteristics. The result is formulated in terms of conormal regularity relative to a twisted Sobolev space. We use this to show the uniqueness, modulo regularizing terms, of parametrices

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension, we attach mixed Hodge structures with modulus. We combine this construction with

We formulate a $q$ -Schur algebra associated with an arbitrary $W$ -invariant finite set $X_{\text{f}}$ of integral weights for a complex simple Lie algebra with Weyl group $W$ . We establish a $q$ -Schur duality between the $q$ -Schur algebra and Hecke algebra associated with $W$ . We then realize geometrically the $q$ -Schur algebra and duality and construct a canonical basis for the $q$ -Schur algebra

We prove the integral Hodge conjecture for all 3-folds $X$ of Kodaira dimension zero with $H^{0}(X,K_{X})$ not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic

We construct a separately continuous function $e:E\times K\rightarrow \{0,1\}$ on the product of a Baire space $E$ and a compact space $K$ such that no restriction of $e$ to any non-meagre Borel set in $E\times K$ is continuous. The function $e$ has no points of joint continuity, and, hence, it provides a negative solution of Talagrand’s problem in Talagrand [Espaces de Baire et espaces de Namioka

We study the partial Gelfand–Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated with a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase space. By taking advantage of the associated Gevrey regularizing effects, we study the null-controllability of parabolic equations posed on the whole

We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over  $\mathbb{Q}$ . More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over $\mathbb{Q}_{p}(x)$

In this paper, we prove a conjecture of Wei Zhang on comparison of certain local relative characters from which we draw some consequences for the Ichino–Ikeda conjecture for unitary groups.

We prove that the category of (rigidified) Breuil–Kisin–Fargues modules up to isogeny is Tannakian. We then introduce and classify Breuil–Kisin–Fargues modules with complex multiplication mimicking the classical theory for rational Hodge structures. In particular, we compute an avatar of a ‘ $p$ -adic Serre group’.

We prove that the class of reflexive asymptotic- $c_{0}$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic- $c_{0}$ space $Y$ , then $X$ is also reflexive and asymptotic- $c_{0}$ . In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a

Nous construisons un espace adapté à l’étude de l’entropie des applications méromorphes en utilisant des limites projectives. Nous en déduisons un principe variationnel pour ces applications.

We develop a theory of $R$ -module Thom spectra for a commutative symmetric ring spectrum $R$ and we analyze their multiplicative properties. As an interesting source of examples, we show that $R$ -algebra Thom spectra associated to the special unitary groups can be described in terms of quotient constructions on $R$ . We apply the general theory to obtain a description of the $R$ -based topological

Let $\unicode[STIX]{x1D707}$ be the projection on $[0,1]$ of a Gibbs measure on $\unicode[STIX]{x1D6F4}=\{0,1\}^{\mathbb{N}}$ (or more generally a Gibbs capacity) associated with a Hölder potential. The thermodynamic and multifractal properties of $\unicode[STIX]{x1D707}$ are well known to be linked via the multifractal formalism. We study the impact of a random sampling procedure on this structure

Two measurable sets $S,\unicode[STIX]{x1D6EC}\subseteq \mathbb{R}^{d}$ form a Heisenberg uniqueness pair, if every bounded measure $\unicode[STIX]{x1D707}$ with support in $S$ whose Fourier transform vanishes on $\unicode[STIX]{x1D6EC}$ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathbb{R}^{d}$ . As

Estimating averages of Dirichlet convolutions $1\ast \unicode[STIX]{x1D712}$ , for some real Dirichlet character $\unicode[STIX]{x1D712}$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin’s conjecture for Châtelet surfaces. We introduce a far-reaching generalisation

Colmez [Périodes des variétés abéliennes a multiplication complexe, Ann. of Math. (2)138(3) (1993), 625–683; available at http://www.math.jussieu.fr/∼colmez] conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the

In this paper, we construct the first examples of complex surfaces of general type with arbitrarily large geometric genus whose canonical maps induce non-hyperelliptic fibrations of genus $g=4$ , and on the other hand, we prove that there is no complex surface of general type whose canonical map induces a hyperelliptic fibrations of genus $g\geqslant 4$ if the geometric genus is large.

In this paper, we consider the Stokes equations in a perforated domain. When the number of holes increases while their radius tends to 0, it is proven in Desvillettes et al. [J. Stat. Phys. 131 (2008) 941–967], under suitable dilution assumptions, that the solution is well approximated asymptotically by solving a Stokes–Brinkman equation. We provide here quantitative estimates in $L^{p}$ -norms of

We present a construction that enables one to find Banach spaces $X$ whose sets $\operatorname{NA}(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2, Israel

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Let $p$ be an odd prime. For a number field $K$ , we let $K_{\infty }$ be the maximal unramified pro- $p$ extension of  $K$ ; we call the group $\text{Gal}(K_{\infty }/K)$ the $p$ -class tower group of  $K$ . In a previous work, as a non-abelian generalization of the work of Cohen and Lenstra on ideal class groups, we studied how likely it is that a given finite $p$ -group occurs as the $p$ -class

In the joint work with Elmanto, Hoyois, Khan and Sosnilo, we computed infinite $\mathbb{P}^{1}$ -loop spaces of motivic Thom spectra using the technique of framed correspondences. This result allows us to express non-negative $\mathbb{G}_{m}$ -homotopy groups of motivic Thom spectra in terms of geometric generators and relations. Using this explicit description, we show that the unit map of the algebraic

We provide a class of separable II1 factors $M$ whose central sequence algebra is not the ‘tail’ algebra associated with any decreasing sequence of von Neumann subalgebras of $M$ . This settles a question of McDuff [On residual sequences in a II1 factor, J. Lond. Math. Soc. (2) (1971), 273–280].

Let ${\mathcal{F}}_{g}^{\mathbf{N}}$ be the moduli space of polarized Nikulin surfaces $(Y,H)$ of genus $g$ and let ${\mathcal{P}}_{g}^{\mathbf{N}}$ be the moduli of triples $(Y,H,C)$ , with $C\in |H|$ a smooth curve. We study the natural map $\unicode[STIX]{x1D712}_{g}:{\mathcal{P}}_{g}^{\mathbf{N}}\rightarrow {\mathcal{R}}_{g}$ , where ${\mathcal{R}}_{g}$ is the moduli space of Prym curves of genus

We study the Daugavet property in tensor products of Banach spaces. We show that $L_{1}(\unicode[STIX]{x1D707})\widehat{\otimes }_{\unicode[STIX]{x1D700}}L_{1}(\unicode[STIX]{x1D708})$ has the Daugavet property when $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$ are purely non-atomic measures. Also, we show that $X\widehat{\otimes }_{\unicode[STIX]{x1D70B}}Y$ has the Daugavet property provided