For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finite-dimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category ${\mathcal{O}}$ and Koszul duality.

This paper sets up the foundations for derived algebraic geometry, Goerss–Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën–Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and $\text{E}_{\infty }$-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of $\text{E}_{\infty }$-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.

We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers ℚ, we recover the classical results on moments.

We define a motivic conductor for any presheaf with transfers F using the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. If F is a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on $F(L)$ , where L is any henselian discrete valuation field of geometric type over the perfect ground field. We show that if F is a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; if F assigns to X the group of finite characters on the abelianised étale fundamental group of X, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and if F assigns to X the group of integrable rank $1$ connections (in characteristic $0$ ), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type with perfect residue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.

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 a corollary we obtain a new, surprising version of the classical Cramér–Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients.

This second part of the paper strengthens the descent theory described in the first part to rational maps and arbitrary base fields. We obtain in particular a decomposition of any difference field extension into a tower of finite, field-internal and one-based difference field extensions. This is needed in order to obtain the ‘dynamical Northcott’ Theorem 1.11 of Part I in sharp form.

We prove that, under rather general conditions, the 1-cohomology of a von Neumann algebra $M$ with values in a Banach $M$ -bimodule satisfying a combination of smoothness and operatorial conditions vanishes. For instance, we show that, if $M$ acts normally on a Hilbert space ${\mathcal{H}}$ and ${\mathcal{B}}_{0}\subset {\mathcal{B}}({\mathcal{H}})$ is a norm closed $M$ -bimodule such that any $T\in {\mathcal{B}}_{0}$ is smooth (i.e., the left and right multiplications of $T$ by $x\in M$ are continuous from the unit ball of $M$ with the $s^{\ast }$ -topology to ${\mathcal{B}}_{0}$ with its norm), then any derivation of $M$ into ${\mathcal{B}}_{0}$ is inner. The compact operators are smooth over any $M\subset {\mathcal{B}}({\mathcal{H}})$ , but there is a large variety of non-compact smooth elements as well.

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 $X$ and $Y$ are $L_{1}$-preduals with the Daugavet property, in particular, spaces of continuous functions with this property. With the same techniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.

We investigate Tukey functions from the ideal of all closed nowhere-dense subsets of 2ℕ. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of ℕ. We also prove non-existence of various special types of Tukey reductions from the nowhere-dense ideal to analytic P-ideals. In connection with these results, we study families of clopen subsets of 2ℕ with the property that for each nowhere-dense subset of 2ℕ there is a set in not intersecting it. We call such families avoiding.

When the branch character has root number $- 1$, the corresponding anticyclotomic Katz $p$-adic $L$-function vanishes identically. For this case, we determine the $\mu $-invariant of the cyclotomic derivative of the Katz $p$-adic $L$-function. The result proves, as an application, the non-vanishing of the anticyclotomic regulator of a self-dual CM modular form with root number $- 1$. The result also plays a crucial role in the recent work of Hsieh on the Eisenstein ideal approach to a one-sided divisibility of the CM main conjecture.

For a split semisimple Chevalley group scheme G with Lie algebra over an arbitrary base scheme S, we consider the quotient of by the adjoint action of G. We study in detail the structure of over S. Given a maximal torus T with Lie algebra and associated Weyl group W, we show that the Chevalley morphism π : /W → /G is an isomorphism except for the group Sp2n over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient → /G commutes, or does not commute, with base change on S.

In Albano, Bove and Mughetti [J. Funct. Anal. 274(10) (2018), 2725–2753]; Bove and Mughetti [Anal. PDE 10(7) (2017), 1613–1635] it was shown that Treves conjecture for the real analytic hypoellipticity of sums of squares operators does not hold. Models were proposed where the critical points causing a non-analytic regularity might be interpreted as strata. We stress that up to now there is no notion of stratum which could replace the original Treves stratum. In the proposed models such ‘strata’ were non-symplectic analytic submanifolds of the characteristic variety. In this note we modify one of those models in such a way that the critical points are a symplectic submanifold of the characteristic variety while still not being a Treves stratum. We show that the operator is analytic hypoelliptic.

We construct a cohomology theory using quasi-smooth derived schemes as generators and an analog of the bordism relation using derived fiber products as relations. This theory has pull-backs along all morphisms between smooth schemes independent of any characteristic assumptions. We prove that, in characteristic zero, the resulting theory agrees with algebraic cobordism as defined by Levine and Morel. We thus obtain a new set of generators and relations for algebraic cobordism.

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb{Z}$ with finitely generated kernels. The proof uses Bestvina–Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb{H}^{4}$ with fundamental domain the 120-cell or the 24-cell.

In this paper we finish the topological classification of real algebraic surfaces of Kodaira dimension zero and we make a step towards the Enriques classification of real algebraic surfaces, by describing in detail the structure of the moduli space of real hyperelliptic surfaces.

Moreover, we point out the relevance in real geometry of the notion of the orbifold fundamental group of a real variety, and we discuss related questions on real varieties $(X,\sigma)$ whose underlying complex manifold $X$ is a $K(\pi,1)$.

Our first result is that if $(S,\sigma)$ is a real hyperelliptic surface, then the differentiable type of the pair $(S,\sigma)$ is completely determined by the orbifold fundamental group exact sequence.

This result allows us to determine all the possible topological types of $(S,\sigma)$, and to prove that they are exactly 78.

It follows also as a corollary that there are exactly eleven cases for the topological type of the real part of $S$.

Finally, we show that once we fix the topological type of $(S,\sigma)$ corresponding to a real hyperelliptic surface, the corresponding moduli space is irreducible (and connected).

We also give, through a series of tables, explicit analytic representations of the 78 components of the moduli space.

AMS 2000 Mathematics subject classification: Primary 14P99; 14P25; 14J15; 32Q57

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 use of geometric optics and a three-fold wave interaction arising from the cubic non-linearity. Subsequently, the result is generalized to globally hyperbolic Lorentzian manifolds by using Gaussian beams.

We compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

We consider variational properties of some numerical invariants, measuring convergence of local horizontal sections, associated to differential modules on polyannuli over a nonarchimedean field of characteristic 0. This extends prior work in the one-dimensional case of Christol, Dwork, Robba, Young, et al. Our results do not require positive residue characteristic; thus besides their relevance to the study of Swan conductors for isocrystals, they are germane to the formal classification of flat meromorphic connections on complex manifolds.

Let $H$ be a torsion-free compact $p$-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebra $\varLambda_H$ of $H$ has trivial centre and use this result to classify the prime $c$-ideals in the Iwasawa algebra $\varLambda_G$ of $G:=H\times\mathbb{Z}_p$. We also show that a finitely generated torsion $\varLambda_G$-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves.

We describe equations of the universal torsors over del Pezzo surfaces of degrees from 2 to 5 over an algebraically closed field in terms of the equations of the corresponding homogeneous space G/P. We also give a generalization for fields that are not algebraically closed.