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When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages

The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$ and $\pi^*$ induced in cohomology by the rational quotient map $\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms., Comment: 28 pages. Version 2: this is the published version of the paper. The last section of the previous version (v1) was erased (the results are only stated) and it is now contained in arXiv:2209.10141

In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large., v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section 5); Some minor corrections in the other sections. v3: some minor corrections in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has been modified; The paper is split into two parts, the complete intersection case will be discussed in a different paper

We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty $-categories) for additive $\infty $-categories, (2) define the notion of nilpotent extensions for suitable $\infty $-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas–Goodwillie–McCarthy theorem for stable $\infty $-categories that are not monogenically generated (such as the stable $\infty $-category of Voevodsky’s motives or the stable $\infty $-category of perfect complexes on some algebraic stacks). The key input in our paper is Bondarko’s notion of weight structures, which provides a “ring-with-many-objects” analog of a connective $\mathbb{E}_1$-ring spectrum. As applications, we prove cdh descent results for truncating invariants of stacks extending the work by Hoyois–Krishna for homotopy $K$-theory and establish new cases of Blanc’s lattice conjecture.

Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions., 40 pages; updates based on referee comments; expanded proof of Proposition 3.3

In our previous paper, viewing $D^b(K(l))$ as a noncommutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of noncommutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The noncommutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition, however, defines a larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on the examples and extend our studies beyond counting. We enrich our invariants with the following structures: the inclusion of subcategories makes them partially ordered sets and considering semi-orthogonal pairs of subcategories as edges amounts to directed graphs. It turns out that the problem for counting $D^b(A_k)$ in $D^b(A_n)$ has a geometric combinatorial parallel - counting of maps between polygons. Estimating the numbers counting noncommutative curves in $D^b({\mathbb P}^2)$ modulo the group of autoequivalences, we prove finiteness and that the exact determining of these numbers leads to a solution of Markov problem. Via homological mirror symmetry, this gives a new approach to this problem. Regarding the structure of a partially ordered set mentioned above, we initiate intersection theory of noncommutative curves focusing on the case of noncommutative genus zero. The above-mentioned structure of a directed graph (and related simplicial complex) is a categorical analogue of the classical curve complex, introduced by Harvey and Harrer. The paper contains pictures of the graphs in many examples and also presents an approach to Markov conjecture via counting of subgraphs in a graph associated with $D^b({{\mathbb{P}}}^2)$. Some of the results proved here were announced in a previous work.

The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations., 20 pages, 1 figure. The condition of finite generation in Thm 1.9 was replaced by countability. The proof of Thm 1.9 has been simplified. The notation used in 5 has been modified. Several minor corrections across the paper

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $\psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $\gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $\gamma$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi g}}\cdot\frac{1}{4^g}$ times less frequent., Comment: The current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042

Let E∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E∗ G-isomorphism, there exists at most one Hughes-free division E∗G-ring. However, the existence of a Hughes-free division E∗ G-ring DE∗G for an arbitrary locally indicable group G is still an open question. Nevertheless, DE∗G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether DE∗G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists DE[G] and it is universal. In Appendix we give a description of DE[G] when G is a RFRS group, This paper is partially supported by the Spanish Ministry of Science and Innovation through the grant MTM2017-82690-P and the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S4). I would like to thank Dawid Kielak and an anonymous referee for useful suggestions and comments

Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group multiple Dirichlet series associated to metaplectic (n-fold) covers of algebraic groups. In subsequent joint work with Puskas, they extended this action to a "metaplectic" representation of the equal parameter affine Hecke algebra, which allowed them to obtain explicit formulas for the p-parts of these Dirichlet series. They have also verified by a computer check the remarkable fact that their formulas continue to define a group action for general (unspecialized) parameters. In the first part of paper we give a conceptual explanation of this fact, by giving a uniform and elementary construction of the "metaplectic" representation for generic Hecke algebras as a suitable quotient of a parabolically induced affine Hecke algebra module, from which the associated Chinta-Gunnells Weyl group action follows through localization. In the second part of the paper we extend the metaplectic representation to the double affine Hecke algebra, which provides a generalization of Cherednik's basic representation. This allows us to introduce a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials. In this paper, we provide the details of the construction of metaplectic polynomials in type A; the general case will be handled in the sequel to this paper., 39 pages. Version 2 is a significant revision. Added second part introducing a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials and metaplectic Iwahori-Whittaker functions. Title has been changed and the introduction has been expanded

This paper considers the problem of establishing L p -improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension n of the ambient space and the dimension k of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang [28] and extended by Zorin-Kranich [29] , and on recent results for geometric nonconcentration inequalities [11] . As an initial application of this new approach, this paper establishes sharp restricted strong type L p -improving inequalities for certain model quadratic submanifolds in the range k n ≤ 2 k .

We present a covering conjecture that we expect to be true below superstrong cardinals. We then show that the conjecture is true in hod mice. This work is a continuation of the work that started in Covering with Universally Baire Functions Advances in Mathematics, and the main conjecture of the current paper is a revision of the UB Covering Conjecture of the aforementioned paper.

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry., Comment: 25 pages

This paper presents a curvature-free version of the Log ( 2 k − 1 ) \text {Log}(2k-1) Theorem of Anderson, Canary, Culler, and Shalen [J. Differential Geometry 44 (1996), pp. 738–782]. It generalizes a result by Hou [J. Differential Geometry 57 (2001), no. 1, pp. 173–193] and its proof is rather straightforward once we know the work by Lim [Trans. Amer. Math. Soc. 360 (2008), no. 10, pp. 5089–5100] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the Collar Lemma in all dimensions.

Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studied in Matsui-Nam-Takahashi-Tri-Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R. In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), and explore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrations of subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi and Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria., Comment: 17 pages, to appear in IMRN

In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are developed, such as the behaviour of the entropy when transformations on the left variables or on the right variables are performed. In addition, the microstate bi-free entropy is demonstrated to be additive over bi-free collections provided additional regularity assumptions are included and is computed for all bi-free central limit distributions. Moreover, an orbital version of bi-free entropy is examined which provides a tighter upper bound for the subadditivity of microstate bi-free entropy and provides an alternate characterization of bi-freeness in certain settings., Comment: (new version contains corrections, orbital bi-free entropy, and a new characterization of bi-freenness)

This paper focuses on the spherical fixed point set of a circle action on an ALE space endowed with Kronheimer's hyperkaehler metric. The induced metric on the sphere is described by using the algebraic geometry of rational curves on algebraic surfaces, in particular the lines on a cubic., Dedicated to the memory of Michael Atiyah

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12] . In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of l-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A − D − E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic p greater than 2, called OG6 varieties. We show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces., Comment: Final version. To appear in I.M.R.N

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

The aim of this paper is to describe the structure of the finitely generated subgroups of a family of branch groups, which includes the first Grigorchuk group and the Gupta-Sidki 3-group. This description is made via the notion of block subgroup. We then use this to show that all groups in the above family are subgroup separable (LERF). These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups., 22 pages, 2 figures; V3: Final version, to appear in Izvestiya: Mathematics

For $$(m,p,q)\in (1,\infty )\times {\mathbb {R}}\times {\mathbb {R}}$$ , this paper establishes a sharp Liouville principle for the weak solutions to the quasilinear elliptic inequality of second order $$\Delta _m u+u^p|\nabla u|^q\le 0$$ on the geodesically complete noncompact Riemannian manifolds, which is new even for the Euclidean spaces.

We set out some general criteria to prove the K-property, refining the assumptions used in an earlier paper for the flow case, and introducing the analogous discrete-time result. We also introduce one-sided $\lambda $ -decompositions, as well as multiple techniques for checking the pressure gap required to show the K-property. We apply our results to the family of Mañé diffeomorphisms and the Katok map. Our argument builds on the orbit decomposition theory of Climenhaga and Thompson.

In this paper we generalize an equation studied by Mossino and Temam, to the fully nonlinear case. This equation arises in plasma physics as an approximation to Grad equations, which were introduced by Harold Grad, to model the behavior of plasma confined in a toroidal vessel. We prove existence of a $W^{2,p}$-viscosity solution and regularity up to $C^{1,\alpha}(\overline{\Omega})$ for any $\alpha

The first part of this paper is a generalization of the Feix-Kaledin theorem on the existence of a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of a Kahler manifold. We show that the problem of constructing a hyperkahler structure on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix-Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kahler type has a hyperkahler structure on a neighbourhood of its identity section. More generally, we reduce the existence of a hyperkahler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem., 20 pages. To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal)

We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.

When studying a graded module $M$ over the Cox ring of a smooth projective toric variety $X$, there are two standard types of resolutions commonly used to glean information: free resolutions of $M$ and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen--Macaulay property, though tools for assessing which modules are virtually Cohen--Macaulay have only recently started to be developed. In this paper, we continue this research program in two related ways. The first is that, when $X$ is a product of projective spaces, we produce a large new class of virtually Cohen--Macaulay Stanley--Reisner rings, which we show to be virtually Cohen--Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety $X$, we develop homological tools for assessing the virtual Cohen--Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen--Macaulay properties., Accepted to Transactions of the London Mathematical Society

In this paper, we consider the m-Hessian equation S m [ D 2 u ] = b ( x ) f ( u ) > 0 in Ω, subject to the singular boundary condition u = ∞ on ∂ Ω. We give estimates of the asymptotic behavior of such solutions near ∂ Ω when the nonlinear term f satisfies a new structure condition.

In this paper we prove a strengthening of the generic vanishing result in characteristic $p>0$ given in [HP16]. As a consequence of this result, we show that irreducible $\Theta$ divisors are strongly F-regular and we prove a related result for pluri-theta divisors., Comment: Comments are more than welcome

A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.

The problem of whether the cohomological support map of a finite dimensional Hopf algebra has the tensor product property has attracted a lot of attention following the earlier developments on representations of finite group schemes. Many authors have focussed on concrete situations where positive and negative results have been obtained by direct arguments. In this paper we demonstrate that it is natural to study questions involving the tensor product property in the broader setting of a monoidal triangulated category. We give an intrinsic characterization by proving that the tensor product property for the universal support datum is equivalent to complete primeness of the categorical spectrum. From these results one obtains information for other support data, including the cohomological one. Two theorems are proved giving compete primeness and non-complete primeness in certain general settings. As an illustration of the methods, we give a proof of a recent conjecture of Negron and Pevtsova on the tensor product property for the cohomological support maps for the small quantum Borel algebras for all complex simple Lie algebras., 20 pages

The elementary Reinhardt domain associated to multi-index k = (k 1, …, k n ) ∈ ℤ n is defined by ℋ ( k ) : = { z ∈ D n : z k is defined and | z k | < 1 } . $$\mathcal{H}(\mathbf{k}):=\{z\in\mathbb{D}^n: z^{\mathbf{k}}\ \text{is defined and}\ |z^{\mathbf{k}}| In this paper, we study the mapping properties of the associated Bergman projection on L p spaces and L p Sobolev spaces of order ≥ 1.

The theory of dessins d’enfants on compact Riemann surfaces, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. In this paper, we show how this theory is naturally extended to non-compact orientable surfaces and, in particular, we observe that the Loch Ness monster (LNM; the surface of infinite genus with exactly one end) admits infinitely many regular dessins d’enfants (either chiral or reflexive). In addition, we study different holomorphic structures on the LNM, which come from homology covers of compact Riemann surfaces, and infinite hyperelliptic and infinite superelliptic curves.

In this paper we prove a local exponential synchronization for Markovian random iterations of homeomorphisms of the circle $S^{1}$ , providing a new result on stochastic circle dynamics even for $C^1$ -diffeomorphisms. This result is obtained by combining an invariance principle for stationary random iterations of homeomorphisms of the circle with a Krylov–Bogolyubov-type result for homogeneous Markov chains.

Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action $\rho: G \to \operatorname{Aff}(H)$ is simply transitive by looking only at the induced morphism $\varphi: \mathfrak{g} \to \operatorname{aff}(\mathfrak{h})$ between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group $G$ acts simply transitive on a given nilpotent Lie group $H$, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension $4$., Comment: 22 pages, 8 tables. Comments are welcome

Over the past two decades several different approaches to defining a geometry over$${{\mathbb F}_1}$$F1have been proposed. In this paper, relying on Toën and Vaquié’s formalism (J.K-Theory 3: 437–500, 2009), we investigate a new category$${\mathsf {Sch}}_{\widetilde{{\mathsf B}}}$$SchB~of schemes admitting a Zariski cover by affine schemes relative to the category of blueprints introduced by Lorscheid (Adv. Math. 229: 1804–1846, 2012). A blueprint, which may be thought of as a pair consisting of a monoidMand a relation on the semiring$$M\otimes _{{{\mathbb F}_1}} {\mathbb N}$$M⊗F1N, is a monoid object in a certain symmetric monoidal category$${\mathsf B}$$B, which is shown to be complete, cocomplete, and closed. We prove that every$${\widetilde{{\mathsf B}}}$$B~-scheme$$\Sigma $$Σcan be associated, through adjunctions, with both a classical scheme$$\Sigma _{\mathbb Z}$$ΣZand a scheme$$\underline{\Sigma }$$Σ̲over$${{\mathbb F}_1}$$F1in the sense of Deitmar (in van der Geer G., Moonen B., Schoof R. (eds.) Progress in mathematics 239, Birkhäuser, Boston, 87–100, 2005), together with a natural transformation$$\Lambda :\Sigma _{\mathbb Z}\rightarrow \underline{\Sigma }\otimes _{{{\mathbb F}_1}}{\mathbb Z}$$Λ:ΣZ→Σ̲⊗F1Z. Furthermore, as an application, we show that the category of “$${{\mathbb F}_1}$$F1-schemes” defined by Connes and Consani in (Compos. Math. 146: 1383–1415, 2010) can be naturally merged with that of$${\widetilde{{\mathsf B}}}$$B~-schemes to obtain a larger category, whose objects we call “$${{\mathbb F}_1}$$F1-schemes with relations”.

In this paper, we introduce generalizations of Quaternacci sequences (Quaternaccis), called Split Quaternacci sequences, which arose on a base of split quaternion algebras. Explicit and recurrent formulae for Split Quaternacci sequences are given, as well as generating functions. Also, matrices related to Split Quaternaccis sequences are investigated. Moreover, new identities connecting Horadam sequences with other known sequences are generated. Analogous identities for Horadam quaternions and split Horadam quaternions are proved.

Let $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$ , which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ . In this paper, we consider the $\mathrm {Gal}(N/K)$ -equivariant local $\varepsilon $ -conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$ . For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$ . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval $[\mathcal{U},\mathcal{T}]$ in the lattice of torsion classes, we associate a subcategory $\mathcal{T} \cap \mathcal{U}^\perp$ called the heart. We show that every ICE-closed subcategory can be realized as a heart of some interval of torsion classes, and give a lattice-theoretic characterization of intervals whose hearts are ICE-closed. In particular, we prove that ICE-closed subcategories are precisely torsion classes in some wide subcategories. For an artin algebra, we introduce the notion of wide $\tau$-tilting modules as a generalization of support $\tau$-tilting modules. Then we establish a bijection between wide $\tau$-tilting modules and doubly functorially finite ICE-closed subcategories, which extends Adachi--Iyama--Reiten's bijection on torsion classes. For the hereditary case, we discuss the Hasse quiver of the poset of ICE-closed subcategories by introducing a mutation of rigid modules., Comment: 31 pages, final version, to appear in Math. Z. Section 4.4 rewritten using sincere intervals

In this paper we obtain some improved $L^p$ -Hardy and $L^p$ -Rellich inequalities on bounded domains of Riemannian manifolds. For Cartan–Hadamard manifolds we prove the inequalities with sharp constants and with weights being hyperbolic functions of the Riemannian distance.

Let $$M^{(i)}$$ , $$i=1,2,\ldots , n$$ , be the boundaries of unbounded domains $$\Omega ^{(i)}$$ of finite type in $$\mathbb {C}^2$$ , and let $$\Box _b^{(i)}$$ be the Kohn Laplacian on $$M^{(i)}$$ . In this paper, we study multivariable spectral multipliers $$m(\Box _b^{(1)},\ldots , \Box _b^{(n)})$$ acting on the Shilov boundary $${\widetilde{M}}=M^{(1)} \times \cdots \times M^{(n)}$$ of the product domain $$\Omega ^{(1)}\times \cdots \times \Omega ^{(n)}$$ . We show that if a function $$m(\lambda _1, \ldots ,\lambda _n)$$ satisfies a Marcinkiewicz-type smoothness condition defined using Sobolev norms, then the spectral multiplier operator $$m(\Box _b^{(1)}, \ldots , \Box _b^{(n)})$$ is a product Calderon–Zygmund operator of Journe type.

In [25], higher genus Gromov–Witten invariants of the stack of r-th roots of a smooth projective variety X along a smooth divisor D are shown to be polynomials in r. In this paper we study the degrees and coefficients of these polynomials.

We study the behavior of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence. By working in the "relative setting" (splitting one node at a time), we demonstrate that there are many non-hereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show the above is true when replacing irreducible components by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras., 26 pages. Updated references, typos, metadata. Actually final version

Let $${{\mathcal {A}}}$$ be a Banach algebra and let $$\varphi $$ be a non-zero character on $${{\mathcal {A}}}$$ . Suppose that $${{\mathcal {A}}}_M$$ is the closure of the faithful Banach algebra $${{\mathcal {A}}}$$ in the multiplier norm. In this paper, topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}_M^*$$ are defined and studied. Under some conditions on $${{\mathcal {A}}}$$ , we will show that the set of topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}^*$$ and on $${{\mathcal {A}}}_M^*$$ have the same cardinality. The main applications are concerned with the quantum group algebra $$L^1({\mathbb {G}})$$ of a locally compact quantum group $${\mathbb {G}}$$ . In particular, we obtain some characterizations of compactness of $${\mathbb {G}}$$ in terms of the existence of a non-zero (weakly) compact left or right multiplier on $$L^1_M({\mathbb {G}})$$ or on its bidual in some senses.

Let $$(r_n)_{n=1}^\infty $$ be a non-decreasing sequence of radii in $$(0, \infty )$$ , and let $$(\theta _n)_{n=1}^\infty $$ be a sequence of independent random arguments uniformly distributed in $$[0, 2\pi )$$ . In this paper, we establish a new sufficient condition on the sequence $$(r_n)_{n=1}^\infty $$ under which $$(r_ne^{i\theta _n})_{n=1}^\infty $$ is almost surely a zero set for Fock spaces. The condition is in terms of the sum of two characteristics involving the counting function. The sharpness of this condition is discussed and examples are presented to illustrate it.

In this paper, we extend compactness theorems of Cheeger, Gromov, Taylor, and Sprouse to the Bakry–Emery Ricci tensor and generalized quasi-Einstein tensors. Our results generalize previous results obtained by Yun and Wan.

This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.

In this paper, we apply our previous estimates in Chen and Cheng [On the constant scalar curvature Kähler metrics (I): a priori estimates, Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness ofKK-energy in terms ofL1L^1geodesic distanced1d_1in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of theKK-energy in(E1,d1)(\mathcal {E}^1, d_1)are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilizedL1L^1geodesic ray where theKK-energy is non-increasing, which is a weak version of a conjecture by Donaldson. The continuity path proposed by Xiuxiong Chen [Ann. Math. Qué. 42 (2018), pp. 69–189] is instrumental in the above proofs.

In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of aC0C^0bound for the Kähler potential.

For a given odd prime number p, in this paper we construct a minimal generating set for the mod-p cohomology of the Steinberg summand of a family of Thom spectra over the classifying space of an elementary p-abelian group. This resolves the remaining cases for odd prime numbers of a problem studied previously by M. Inoue (Contemporary Mathematics, vol. 293, pp. 125–139, 2002) and (J. Lond. Math. Soc. 75: 317–329, 2007) and by the author (J. Algebra 381: 164–175, 2013).