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Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of |$p$| -adic and |$\textrm{mod}\; p$| modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: (1) the analytic continuation at unramified primes |$p$| to the whole Shimura variety of the |$\textrm{mod}\; p$| reduction of |$p$| -adic Maass–Shimura operators a priori defined only over the |$\mu $| -ordinary locus, and (2) the construction of new |$\textrm{mod}\; p$| theta operators that do not arise as the |$\textrm{mod}\; p$| reduction of Maass–Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree. [ABSTRACT FROM AUTHOR]

Let |${\mathcal{M}}_{\operatorname{Dol}}(X,G)$| denote the hyperkähler moduli space of |$G$| -Higgs bundles over a smooth projective curve |$X$|⁠. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of |${\mathcal{M}}_{\operatorname{Dol}}(X,G)$|⁠. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space |$\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$|⁠. Following Gaiotto's suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack , a derived analytic stack with corresponding moduli space |$\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$|⁠ , defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence. [ABSTRACT FROM AUTHOR]

In this paper, we give a new geometric condition in terms of measured asymptotic expanders to ensure rigidity of Roe algebras. Consequently, we obtain the rigidity for all bounded geometry spaces that coarsely embed into some |$L^p$| -space for |$p\in [1,\infty)$|⁠. Moreover, we also verify rigidity for the box spaces constructed by Arzhantseva–Tessera and Delabie–Khukhro even though they do not coarsely embed into any |$L^p$| -space. The key step in our proof of rigidity is showing that a block-rank-one (ghost) projection on a sparse space |$X$| belongs to the Roe algebra |$C^{\ast }(X)$| if and only if |$X$| consists of (ghostly) measured asymptotic expanders. As a by-product, we also deduce that ghostly measured asymptotic expanders are new sources of counterexamples to the coarse Baum–Connes conjecture. [ABSTRACT FROM AUTHOR]

This paper is devoted to coregular submanifolds in Poisson geometry. We show that their local Poisson saturation is an embedded Poisson submanifold, and we give a normal form for this Poisson submanifold around the coregular submanifold. This result recovers the normal form around Poisson transversals, and it yields Poisson versions of some normal form/rigidity results around constant rank submanifolds in symplectic geometry. As an application, we prove a uniqueness result concerning coisotropic embeddings of Dirac manifolds in Poisson manifolds. We also show how our results generalize to the setting of coregular submanifolds in Dirac geometry. [ABSTRACT FROM AUTHOR]

In this paper, we define a Donaldson type functional whose Euler–Lagrange equations are a system of differential equations, which corresponds to Hitchin's self-duality equations for a suitable choice of Higgs bundle on closed Riemann surfaces. The main challenge of this functional is its lack of regularity and lack of compactness when defined in its natural domain of definition. Though a standard variational approach cannot directly be applied, we provide the appropriate analytical tools that make Donaldson functional treatable by a variational viewpoint. We prove that this functional admits a unique critical point corresponding to its global minimum. As an immediate consequence, we find that this system of self-duality equations admits a unique solution. Among the applications in geometry of this fact, we obtain a parametrization of closed constant mean curvature immersions in hyperbolic manifolds (possibly incomplete), and their moduli spaces. [ABSTRACT FROM AUTHOR]

We fix Lemma 6.2 in our recent paper [ 1 ] and correct the constants derived from it, appearing in Theorems 1.1, 1.3, and 1.4 and Corollary 6.5. [ABSTRACT FROM AUTHOR]

Faithfully flat descent of pseudo-coherent complexes in rigid geometry was proved by Mathew. In this paper, we generalize the result of Mathew to solid quasi-coherent complexes on rigid analytic varieties, which have been introduced by Clausen and Scholze by means of condensed mathematics. [ABSTRACT FROM AUTHOR]

In sub-Riemannian geometry there exist, in general, no known explicit representations of the heat kernels, and these functions fail to have any symmetry whatsoever. In particular, they are not a function of the control distance, nor they are for instance spherically symmetric in any of the layers of the Lie algebra. Despite these unfavourable aspects, in this paper we establish a new heat semigroup characterisation of the Sobolev and |$BV$| spaces in a Carnot group by means of an integral decoupling property of the heat kernel. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Keller's deformed Calabi–Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an |$n$| -dimensional smooth variety |$Y$|⁠ , we describe the derived category of the total space of an |$\omega _{Y}$| -torsor as a certain deformed |$(n+1)$| -Calabi–Yau completion of the derived category of |$Y$|⁠. As an application, we investigate the geometry of the derived moduli stack of compactly supported coherent sheaves on a local curve, that is, a Calabi–Yau threefold of the form |$\textrm{Tot}_{C}(N)$|⁠ , where |$C$| is a smooth projective curve and |$N$| is a rank two vector bundle on |$C$|⁠. We show that the derived moduli stack is equivalent to the derived critical locus of a function on a certain smooth moduli space. This result will be used by the first author and Naoki Koseki in their joint work on Higgs bundles and Gopakumar–Vafa invariants. [ABSTRACT FROM AUTHOR]

In this paper, we derive topological and number theoretical consequences of the rigidity of elliptic genera, which are special modular forms associated to compact almost complex manifolds. On the geometry side, we prove that rigidity implies relations between the Betti numbers and the index of a compact symplectic manifold admitting a Hamiltonian action of a circle with isolated fixed points. We investigate the case of maximal index and toric actions. On the number theoretical side, we prove that from each compact almost complex manifold of index greater than one, that can be endowed with the action of a circle with isolated fixed points, one can derive non-trivial relations among Eisenstein series. We give explicit formulas coming from the standard action on |${{\mathbb{C}}} P^n$|⁠. [ABSTRACT FROM AUTHOR]

The first purpose of this paper is to solve completely the finite field cone restriction conjecture in four dimensions with |$-1$| non-square. The second is to introduce a new approach to study incidence problems via restriction theory. More precisely, using the cone restriction estimates, we will prove sharp point-sphere incidence bounds associated with complex-valued functions for sphere sets of small size. Our incidence bounds with a specific function improve significantly, a result given by Cilleruelo, Iosevich, Lund, Roche-Newton, and Rudnev. [ABSTRACT FROM AUTHOR]

Powering the adjacency matrix of an expander graph results in a better expander of higher degree. In this paper we seek an analogue operation for high-dimensional (HD) expanders. We show that the naive approach to powering does not preserve HD expansion and define a new power operation, using geodesic walks on quotients of Bruhat–Tits buildings. Applying this operation results in HD expanders of higher degrees. The crux of the proof is a combinatorial study of flags of free modules over finite local rings. Their geometry describes links in the power complex, and showing that they are excellent expanders implies HD expansion for the power complex by Garland's local-to-global technique. As an application, we use our power operation to obtain new efficient double samplers. [ABSTRACT FROM AUTHOR]

For a general cubic four-fold |$X\subset {\mathbb {P}}^5$| with Fano scheme of lines |$F$|⁠ , we prove a number of properties of the universal family of lines |$I\to F$| and various subloci. We first describe the moduli and ramification theory of the genus four fibration |$p:I\to X$| and explore its relation to a birational model of |$F$| in |$I$|⁠. The main part of the paper is devoted to describing the locus |$V\subset F$| of triple lines, i.e. the fixed locus of the Voisin map |$\phi :F\dashrightarrow F$|⁠ , in particular proving it is an irreducible projective singular surface of class |$21\textrm {c}_2({\mathcal {U}}_F)$| and detailing its intersection with the locus |$S$| of second type lines. A consequence of the analysis of the singularities of |$V$| is a geometric proof of the fact that if |$X$| is very general, then the number of singular (necessarily 1-nodal) rational curves in |$F$| of primitive class is 3780. [ABSTRACT FROM AUTHOR]

In this paper, we study the geometry of the total space |$Y$| of a cotangent bundle to a Kähler manifold |$N$| where |$N$| is obtained as a Kähler reduction from |$\mathbb{C}^n$|⁠. Using the hyperkähler reduction, we construct a hyperkähler metric on |$Y$| and prove that it coincides with the canonical Feix–Kaledin metric. This metric is in general non-complete. We show that the metric completion |$\widetilde Y$| of the space |$Y$| is equipped with a structure of a stratified hyperkähler space (in the sense of [ 31 ]). We give a necessary condition for the Feix–Kaledin metric to be complete using an observation of R. Bielawski. Pick a complex structure |$J$| on |$\widetilde Y$| induced from the quaternions. Suppose that |$J\ne \pm I$| where |$I$| is the complex structure whose restriction to |$Y = T^{*}N$| is induced by the complex structure on |$N$|⁠. We prove that the space |$\widetilde{Y}_J$| admits an algebraic structure and is an affine variety. [ABSTRACT FROM AUTHOR]

We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The |$l$| -dimensional variety |$Y_{(n)}^{(l)}$|⁠ , which is the quotient of the product of a certain curve |$C_{(n)}$| by itself |$l$| times by a group |$G\simeq \left ({\mathbb{Z}}/n{\mathbb{Z}}\right)^{l-1}$| of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If |$n=3^c$| Schreieder constructed an explicit smooth birational model of it, and Flapan proved that the Kodaira dimension of this smooth model is 1, if |$c>1$|⁠ ; if |$l=2$| it is a modular elliptic surface; if |$l=3$| it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on |$n$| and |$l$| we prove that |$Y_{(n)}^{(l)}$| admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if |$l=2$|⁠ , its minimal resolution is a modular elliptic surface, obtained by a base change of order |$n$| on a specific extremal rational elliptic surface; if |$l\geq 3$| it has a birational model that admits a fibration in K3 surfaces and a fibration in |$(l-1)$| -dimensional varieties of Kodaira dimension at most 0. [ABSTRACT FROM AUTHOR]

The o-minimal structure generated by the restricted Pfaffian functions, known as restricted sub-Pfaffian sets, admits a natural measure of complexity in terms of a format |${{\mathcal{F}}}$|⁠ , recording information like the number of variables and quantifiers involved in the definition of the set, and a degree |$D$|⁠ , recording the degrees of the equations involved. Khovanskii and later Gabrielov and Vorobjov have established many effective estimates for the geometric complexity of sub-Pfaffian sets in terms of these parameters. It is often important in applications that these estimates are polynomial in |$D$|⁠. Despite much research done in this area, it is still not known whether cell decomposition, the foundational operation of o-minimal geometry, preserves polynomial dependence on |$D$|⁠. We slightly modify the usual notions of format and degree and prove that with these revised notions, this does in fact hold. As one consequence, we also obtain the first polynomial (in |$D$|⁠) upper bounds for the sum of Betti numbers of sets defined using quantified formulas in the restricted sub-Pfaffian structure. [ABSTRACT FROM AUTHOR]

We construct wonderful compactifications of the spaces of linear maps and symmetric linear maps of a given rank as blowups of secant varieties of Segre and Veronese varieties. Furthermore, we investigate their birational geometry and their relations with some spaces of degree two stable maps. [ABSTRACT FROM AUTHOR]

We define a GL -variety to be an (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used to study asymptotic properties of invariants like strength and tensor rank and played a key role in two recent proofs of Stillman's conjecture. We initiate a systematic study of |$\textbf {GL}$| -varieties and establish a number of foundational results about them. For example, we prove a version of Chevalley's theorem on constructible sets in this setting. [ABSTRACT FROM AUTHOR]

We introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. We focus on genus |$1$| and prove combinatorial classification results for fine compactified Jacobians in the case of a single nodal curve and in the case of the universal family |$\overline {{\mathcal {C}}}_{1,n} / \overline {{\mathcal {M}}}_{1,n}$| over the moduli space of stable pointed curves. We show that if the fine compactified Jacobian of a nodal curve of genus |$1$| can be extended to a smoothing of the curve, then it can be described as the moduli space of stable sheaves with respect to some polarisation. In the universal case we construct new examples of genus |$1$| fine compactified universal Jacobians. Then we give a formula for the Hodge and Betti numbers of each genus |$1$| fine compactified universal Jacobian |$\overline {{\mathcal {J}}}^{d}_{1,n}$| and prove that their even cohomology is algebraic. [ABSTRACT FROM AUTHOR]

Let |$PW_S^1$| be the space of integrable functions on |${\mathbb {R}}$| whose Fourier transform vanishes outside |$S$|⁠ , where |$S = [-\sigma ,-\rho ]\cup [\rho ,\sigma ]$|⁠ , |$0<\rho <\sigma $|⁠. In the case |$\rho>\sigma /2$|⁠ , we present a complete description of the set of extreme and the set of exposed points of the unit ball of |$PW_S^1$|. The structure of these sets becomes more complicated when |$\rho <\sigma /2$|⁠. [ABSTRACT FROM AUTHOR]

We introduce and study a one parameter deformation of the polynuclear growth (PNG) in (1+1)-dimensions, which we call the |$t$| -PNG model. It is defined by requiring that, when two expanding islands merge, with probability |$t$| they sprout another island on top of the merging location. At |$t=0$|⁠ , this becomes the standard (non-deformed) PNG model that, in the droplet geometry, can be reformulated through longest increasing subsequences of uniformly random permutations or through an algorithm known as patience sorting. In terms of the latter, the |$t$| -PNG model allows errors to occur in the sorting algorithm with probability |$t$|⁠. We prove that the |$t$| -PNG model exhibits one-point Tracy–Widom Gaussian Unitary Ensemble asymptotics at large times for any fixed |$t\in [0,1)$|⁠ , and one-point convergence to the narrow wedge solution of the Kardar–Parisi–Zhang equation as |$t$| tends to |$1$|⁠. We further construct distributions for an external source that are likely to induce Baik–Ben Arous–Péché-type phase transitions. The proofs are based on solvable stochastic vertex models and their connection to the determinantal point processes arising from Schur measures on partitions. [ABSTRACT FROM AUTHOR]

For a group |$G$| let |$\gamma ^{\max }_G(n)$| denote the maximum number of length- |$n$| words over an arbitrary |$n$| -letter subset of |$G$|⁠. If |$G$| is finitely generated and residually finite, then either |$\gamma ^{\max }_G(n)$| is exponentially bounded, if and only if |$G$| is virtually abelian, or |$\gamma ^{\max }_G(n)\geq ~\left (e^{-\frac {1}{4}}+o(1)\right)n^n$|⁠ ; the latter bound cannot be improved, namely, it is attained for the Heisenberg group. For higher-step free nilpotent groups we have |$(1-o(1))n^n\leq \gamma ^{\max }_G(n)\lneq n^n$|⁠. As a key ingredient in the proof, we calculate the number of pair histograms of functions |$f\colon [n]\rightarrow [n]$| and the probability that a random function |$f\colon [n]\rightarrow [n]$| can be uniquely determined by its pair histogram. We analyze group laws of the Heisenberg group by means of closed paths attached to words in the free group and their projected oriented polygons. [ABSTRACT FROM AUTHOR]

Deligne [ 9 ] showed that every K3 surface over an algebraically closed field of positive characteristic admits a lift to characteristic 0. We show the same is true for a twisted K3 surface. To do this, we study the versal deformation spaces of twisted K3 surfaces, which are particularly interesting when the characteristic divides the order of the Brauer class. We also give an algebraic construction of certain moduli spaces of twisted K3 surfaces over |${\operatorname {Spec}}\ \textbf {Z}$| and apply our deformation theory to study their geometry. As an application of our results, we show that every derived equivalence between twisted K3 surfaces in positive characteristic is orientation preserving. [ABSTRACT FROM AUTHOR]

Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat–Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviour of Groebner bases under substitution over unique factorisation rings. Finally, we outline applications to the adic Simpson correspondence, when the respective projective variety is a curve. [ABSTRACT FROM AUTHOR]

In this article, we give the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with klt singularities. Building on recent works of Spicer, Cascini—Spicer and Spicer—Svaldi, we then describe the birational geometry of rank two foliations with canonical singularities and canonical class of numerical dimension zero on complex projective three-folds. [ABSTRACT FROM AUTHOR]

We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works by D. Guan and the 1st author. Any such manifold |$Q$| of dimension |$2n-2$| is obtained as a finite degree |$n^2$| cover of some non-Kähler manifold |$W_F$|⁠ , which we call the base of |$Q$|⁠. We show that the algebraic reduction of |$Q$| and its base is the projective space of dimension |$n-1$|⁠. Besides, we give a partial classification of submanifolds in |$Q$|⁠ , describe the degeneracy locus of its algebraic reduction and prove that the automorphism group of |$Q$| satisfies the Jordan property. [ABSTRACT FROM AUTHOR]

We study the quaternionic Calabi–Yau problem in HyperKähler manifolds with torsion geometry, introduced by Alesker and Verbitsky in [ 5 ], on eight-dimensional two-step nilmanifolds |$M$| with an Abelian hypercomplex structure. We show that on these manifolds the quaternionic Monge–Ampère equation can always be solved for any data that are invariant under the action of a three-torus. [ABSTRACT FROM AUTHOR]

Let |$Z$| be a unimodular real spherical space. We develop a theory of constant terms for tempered functions on |$Z$|⁠ , which parallels the work of Harish-Chandra. The constant terms |$f_I$| of an eigenfunction |$f$| are parametrized by subsets |$I$| of the set |$S$| of spherical roots that determine the fine geometry of |$Z$| at infinity. Constant terms are transitive i.e. |$(f_J)_I=f_I$| for |$I\subset J$|⁠ , and our main result is a quantitative bound of the difference |$f-f_I$|⁠ , which is uniform in the parameter of the eigenfunction. [ABSTRACT FROM AUTHOR]

We study the geometry of smooth non-homogeneous horospherical varieties of Picard rank one. These have been classified by Pasquier and include the well-known odd symplectic Grassmannians. We focus our study on quantum cohomology, with a view towards Dubrovin's conjecture. We start with describing the cohomology groups of smooth horospherical varieties of Picard rank one. We show a Chevalley formula for these and establish that many Gromov–Witten invariants are enumerative. This enables us to prove that in many cases the quantum cohomology is semisimple. We give a presentation of the quantum cohomology ring for odd symplectic Grassmannians. In the last sections, we turn to derived categories of coherent sheaves. We first discuss a general construction of exceptional bundles on horospherical varieties. We work out in detail the case of the horospherical variety associated to the exceptional group |$G_2$| and construct a full rectangular Lefschetz exceptional collection in the derived category. [ABSTRACT FROM AUTHOR]

We show quantitative stability results for the geometric "cells" arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma–Trudinger–Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory. [ABSTRACT FROM AUTHOR]

Tropical geometry and the theory of Newton–Okounkov bodies are two methods that produce toric degenerations of an irreducible complex projective variety. Kaveh and Manon showed that the two are related. We give geometric maps between the Newton–Okounkov bodies corresponding to two adjacent maximal-dimensional prime cones in the tropicalization of |$X$|⁠. Under a technical condition, we produce a natural "algebraic wall-crossing" map on the underlying value semigroups (of the corresponding valuations). In the case of the tropical Grassmannian |$Gr(2,m)$|⁠ , we prove that the algebraic wall-crossing map is the restriction of a geometric map. In an appendix by Nathan Ilten, he explains how the geometric wall-crossing phenomenon can also be derived from the perspective of complexity-one |$T$| -varieties; Ilten also explains the connection to the "combinatorial mutations" studied by Akhtar–Coates–Galkin–Kasprzyk. [ABSTRACT FROM AUTHOR]

Dedicated to the memory of Kazumasa Kuwada. Let |$(X,\mathfrak{d},{\mathfrak{m}})$| be an |$\textrm{RCD}^*(K,N)$| space for some |$K\in{\mathbb{R}}$|⁠ , |$N\in [1,\infty)$|⁠ , and let |$H$| be the self-adjoint Laplacian induced by the underlying Cheeger form. Given |$\alpha \in [0,1]$|⁠ , we introduce the |$\alpha$| -Kato class of potentials on |$(X,\mathfrak{d},{\mathfrak{m}})$|⁠ , and given a potential |$V:X\to{\mathbb{R}}$| in this class, we denote with |$H_V$| the natural self-adjoint realization of the Schrödinger operator |$H+V$| in |$L^2(X,{\mathfrak{m}})$|⁠. We use Brownian coupling methods and perturbation theory to prove that for all |$t>0$|⁠ , there exists an explicitly given constant |$A(V,K,\alpha ,t)<\infty$|⁠ , such that for all |$\Psi \in L^{\infty }(X,{\mathfrak{m}})$|⁠ , |$x,y\in X$| one has $$\begin{align*}\big|e^{-tH_V}\Psi(x)-e^{-tH_V}\Psi(y)\big|\leq A(V,K,\alpha,t) \|\Psi\|_{L^{\infty}}\mathfrak{d}(x,y)^{\alpha}.\end{align*}$$ In particular, all |$L^{\infty }$| -eigenfunctions of |$H_V$| are globally |$\alpha$| -Hölder continuous. This result applies to multi-particle Schrödinger semigroups and, by the explicitness of the Hölder constants, sheds some light into the geometry of such operators. [ABSTRACT FROM AUTHOR]

For a field of characteristic |$\ne 2$|⁠ , we study vector spaces that are graded by the weight lattice of a root system and are endowed with linear operators in each simple root direction. We show that these data extend to a weight lattice graded semisimple representation of the corresponding Lie algebra, if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge–Riemann forms in complex geometry. In the 2nd part of the article, we replace the field by the |$p$| -adic integers (with |$p\ne 2$|⁠) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected |$p$| -adic Chevalley group. [ABSTRACT FROM AUTHOR]

Quantum analogues of sets are defined by two simple assumptions, allowing enumeration, reminiscent of the Gram–Schmidt orthogonalization process. It is shown that any symmetric quantum set is a classical set of irreducible components, and that each irreducible component of size |$>3$| is representable by an orthomodular space over a skew field with involution. For finite or sufficiently large irreducible components, invariance of quantum cardinality is proved. Topological quantum sets are introduced as quantum analogues of topological spaces; irreducible ones of size |$>3$| are shown to be representable by Hilbert spaces over |${\mathbb{R}}$|⁠ , |${\mathbb{C}}$|⁠ , or |${\mathbb{H}}$|⁠. Symmetric quantum sets are characterized as a class of |$L$| -algebras with an intrinsic geometry, and they are shown to be equivalent to Piron's quantum formalism. Equivalences between symmetric quantum sets and several other structures are established. To any symmetric quantum set, a group with a right invariant lattice structure is associated as a complete invariant. A simple and self-contained proof of Solèr's theorem is included, which is used to prove that sufficiently large irreducible symmetric quantum sets come from a classical Hilbert space. [ABSTRACT FROM AUTHOR]

We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices. Our arguments combine group theory, ramification theory, and the geometry of numbers. [ABSTRACT FROM AUTHOR]

We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated with a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic curves of genus 4, in particular, such curves lying on a singular quadric, which arise from del Pezzo surfaces of degree 1. Indeed, we obtain a complete level 2 structure of the curves by studying their theta characteristic divisors via exceptional divisors of the del Pezzo surfaces as the structure is required for the method. [ABSTRACT FROM AUTHOR]

For the class of Hardy spaces and standard weighted Bergman spaces of the unit disk, we prove that the spectrum of a generalized Cesàro operator |$T_g$| is unchanged if the symbol |$g$| is perturbed to |$g+h$| by an analytic function |$h$| inducing a quasi-nilpotent operator |$T_h$|⁠ , that is, spectrum of |$T_h$| equals |$\{0\}$|⁠. We also show that any |$T_g$| operator that can be approximated in the operator norm by an operator |$T_h$| with bounded symbol |$h$| is quasi-nilpotent. In the converse direction, we establish an equivalent condition for the function |$g \in \textbf{BMOA}$| to be in the |$\textbf{BMOA}$| norm closure of |$H^{\infty }$|⁠. This condition turns out to be equivalent to quasi-nilpotency of the operator |$T_g$| on the Hardy spaces. This raises the question whether similar statement is true in the context of Bergman spaces and the Bloch space. Furthermore, we provide some general geometric properties of the spectrum of |$T_{g}$| operators. [ABSTRACT FROM AUTHOR]