New Class Of Examples Research Articles

Limit theorems for multifractal products of random fields

This paper investigates asymptotic properties of multifractal products of random fields and multidimensional multifractal measures. The obtained limit theorems provide sufficient conditions for the convergence of cumulative fields in the spaces Lq. New results on the rate of convergence of cumulative fields are presented. Simple unified conditions for the limit theorems and the calculation of the Rényi function are given. They are less restrictive than those in the known one-dimensional results. The developed methodology is also applied to multidimensional multifractal measures. Finally, an application to a new class of examples of geometric φ-sub-Gaussian random fields is presented. In this case, the general assumptions have a simple form and can be expressed in terms of covariance functions only.

On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most$3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric$3$-folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

The equivalence between rational G-sheaves and rational G-Mackey functors for profinite G

AbstractFor G a profinite group, we construct an equivalence between rational G-Mackey functors and a certain full subcategory of G-sheaves over the space of closed subgroups of G called Weyl-G-sheaves. This subcategory consists of those sheaves whose stalk over a subgroup K is K-fixed.This extends the classification of rational G-Mackey functors for finite G of Thévenaz and Webb, and Greenlees and May to a new class of examples. Moreover, this equivalence is instrumental in the classification of rational G-spectra for profinite G, as given in the second author’s thesis.

Flops of any length, Gopakumar-Vafa invariants and 5d Higgs branches

The conifold is a basic example of a noncompact Calabi-Yau threefold that admits a simple flop, and in M-theory, gives rise to a 5d hypermultiplet at low energies, realized by an M2-brane wrapped on the vanishing sphere. We develop a novel gauge-theoretic method to construct new classes of examples that generalize the simple flop to so-called length ℓ = 1, . , 6. The method allows us to naturally read off the Gopakumar-Vafa invariants. Although they share similar properties to the beloved conifold, these threefolds are expected to admit M2-bound states of higher degree ℓ. We demonstrate this through our computations of the GV invariants. Furthermore we characterize the associated Higgs branches by computing their dimensions and flavor groups. With our techniques we extract more refined data such as the charges of the hypers under the flavor group.

Bourgeois contact structures: Tightness, fillability and applications

Given a contact structure on a manifold V together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on V times {mathbb {T}}^2. We prove that all such structures are universally tight in dimension 5, independent of whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact 5-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for {mathbb {S}}^1-invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the n-torus has a unique symplectically aspherical strong filling up to diffeomorphism.

Spectrum of weak model sets with Borel windows

AbstractConsider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the articlePeriods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

Examples around the strong Viterbo conjecture

A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on $$S^1$$ -invariant convex domains. We introduce a new class of examples called “monotone toric domains”, which are not necessarily convex, and which include all dynamically convex toric domains in four dimensions. We prove that for monotone toric domains in four dimensions, all normalized symplectic capacities agree. For monotone toric domains in arbitrary dimension, we prove that the Gromov width agrees with the first equivariant capacity. We also study a family of examples of non-monotone toric domains and determine when the conclusion of the strong Viterbo conjecture holds for these examples. Along the way, we compute the cylindrical capacity of a large class of “weakly convex toric domains” in four dimensions.

Hochschild cohomology of twisted tensor products

The tensor product $$R\otimes S$$ of two algebras can have its multiplication deformed by a bicharacter to yield a twisted tensor product $$R\otimes ^t S$$ . We completely describe the Hochschild cohomology of $$R\otimes ^t S$$ in terms of the Hochschild cohomology of the components R and S, including the full Gerstenhaber algebra structure. This description generalizes a result of Bergh and Oppermann. A number of interesting classes of noncommutative algebras arise as bicharacter twisted tensor products, sometimes in non-obvious ways. The main result thereby allows us to significantly simplify various calculations in the literature, and to compute Hochschild cohomology in several new classes of examples. In particular, we fully compute the Hochschild cohomology of quantum complete intersection algebras, with any number of indeterminates. One new tool which goes into the main theorem is orbit Hochschild cohomology, which can be defined for algebras with a group action, and which satisfies twisted versions of the usual Gerstenhaber algebra axioms.

Cohomology of finite tensor categories: Duality and Drinfeld centers

We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes.

BMO spaces of $\sigma $-finite von Neumann algebras and Fourier–Schur multipliers on ${\rm SU}_q(2)$

We consider semi-group BMO spaces associated with an arbitrary $\sigma$-finite von Neumann algebra $(\mathcal{M}, \varphi)$. We prove that the associated row and column BMO spaces always admit a predual, extending results from the finite case. Consequently, we can prove that the semi-group BMO spaces considered are Banach spaces and they interpolate with $L_p$ as in the commutative situation, namely $[\mathrm{BMO}(\mathcal{M}), L_p^\circ(\mathcal{M})]_{1/q} \approx L_{pq}^\circ(\mathcal{M})$. We then study a new class of examples. We introduce the notion of Fourier-Schur multiplier on a compact quantum group and show that such multipliers naturally exist for $SU_q(2)$.

Separation of variables, Lax-integrable systems and [formula omitted]-valued classical [formula omitted]-matrices

In the present paper we consider a problem of separation of variables for Lax-integrable hamiltonian system generated by general non-skew-symmetric gl(2)⊗gl(2)-valued classical r-matrix r(u,v) with spectral parameters. We specify the general separability condition of Dubrovin and Skrypnyk (2018) on the components of the r-matrix and consider two new classes of examples of classical non-skew-symmetric r-matrices r(u,v) for which the separability condition is satisfied and the whole construction produces a complete set of canonical coordinates. For each of these r-matrices we consider in details the corresponding classical Gaudin-type models with N spins. In the case N=2 we explicitly find coordinates and momenta of separation, reconstruction formulae and Abel–Jacobi quadratures. Some potential physical applications of the considered models are discussed.

Multifractal processes: Definition, properties and new examples

We investigate stochastic processes possessing scale invariance properties which we refer to as multifractal processes. The examples of such processes known so far do not go much beyond the original cascade construction of Mandelbrot. We provide a new definition of the multifractal process by generalizing the definition of the self-similar process. We establish general properties of these processes and show how existing examples fit into our setting. Finally, we define a new class of examples inspired by the idea of Lamperti transformation. Namely, for any pair of infinitely divisible distribution and a stationary process one can construct a multifractal process.

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

We consider the analogue for regular maps from affine varieties to suitable algebraic manifolds of Oka theory for holomorphic maps from Stein spaces to suitable complex manifolds. The goal is to understand when the obstructions to approximation or interpolation are purely topological. We propose a definition of an algebraic Oka property, which is stronger than the analytic Oka property. We review the known examples of algebraic manifolds satisfying the algebraic Oka property and add a new class of examples: smooth nondegenerate toric varieties. On the other hand, we show that the algebraic analogues of three of the central properties of analytic Oka theory fail for all compact manifolds and manifolds with a rational curve; in particular, for projective manifolds.

A general construction of n-angulated categories using periodic injective resolutions

Let C be an additive category equipped with an automorphism Σ. We show how to obtain n-angulations of (C,Σ) using some particular periodic injective resolutions. We give necessary and sufficient conditions on (C,Σ) admitting an n-angulation. Then we apply these characterizations to explain the standard construction of n-angulated categories and the n-angulated categories arising from some local rings. Moreover, we obtain a class of new examples of n-angulated categories from quasi-periodic selfinjective algebras.

New classes of examples satisfying the three matrix analog of Gerstenhaber's theorem

In 1961, Gerstenhaber proved the following theorem: if k is a field and X and Y are commuting d×d matrices with entries in k, then the unital k-algebra generated by these matrices has dimension at most d. The analog of this statement for four or more commuting matrices is false. The three matrix version remains open. We use commutative–algebraic techniques to prove that the three matrix analog of Gerstenhaber's theorem is true for some new classes of examples.In particular, we translate this three commuting matrix statement into an equivalent statement about certain maps between modules, and prove that this commutative–algebraic reformulation is true in special cases. We end with ideas for an inductive approach intended to handle the three matrix analog of Gerstenhaber's theorem more generally.

On the similarity problem for locally compact quantum groups

A well-known theorem of Day and Dixmier states that any uniformly bounded representation of an amenable locally compact group G on a Hilbert space is similar to a unitary representation. Within the category of locally compact quantum groups, the conjectural analogue of the Day–Dixmier theorem is that every completely bounded Hilbert space representation of the convolution algebra of an amenable locally compact quantum group should be similar to a ⁎-representation. We prove that this conjecture is false for a large class of non-Kac type compact quantum groups, including all q-deformations of compact simply connected semisimple Lie groups. On the other hand, within the Kac framework, we prove that the Day–Dixmier theorem does indeed hold for several new classes of examples, including amenable discrete quantum groups of Kac-type.

Term algebras of elementarily equivalent atom structures

Generalizing results of J\'onsson and Tarski, Maddux introduced the notion of a pair-dense relation algebra and proved that every pair-dense relation algebra is representable. The notion of a pair below the identity element is readily definable within the equational framework of relation algebras. The notion of a triple, a quadruple, or more generally, an element of size (or measure) n>2 is not definable within this framework, and therefore it seems at first glance that Maddux's theorem cannot be generalized. It turns out, however, that a very far-reaching generalization of Maddux's result is possible if one is willing to go outside of the equational framework of relation algebras, and work instead within the framework of the first-order theory. In the present paper, we define the notion of an atom below the identity element in a relation algebra having measure n for an arbitrary cardinal number n>0, and we define a relation algebra to be measurable if it's identity element is the sum of atoms each of which has some (finite or infinite) measure. The main purpose of the present paper is to construct a large class of new examples of group relation algebras using systems of groups and corresponding systems of quotient isomorphisms (instead of the classic example of using a single group and forming its complex algebra), and to prove that each of these algebras is an example of a measurable set relation algebra. In a subsequent paper, the class of examples will be greatly expanded by adding a third ingredient to the mix, namely systems of "shifting" cosets. The expanded class of examples---called coset relation algebras---will be large enough to prove a representation theorem saying that every atomic, measurable relation algebra is essentially isomorphic to a coset relation algebra.

Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces

Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication d-tuple Mz on a reproducing kernel Hilbert space H of E-valued holomorphic functions on Ω, where E is a separable Hilbert space and Ω is a bounded domain in Cd admitting bounded approximation by polynomials. In case E is a finite dimensional cyclic subspace for Mz, under some natural conditions on the B(E)-valued kernel associated with H, the commutant of Mz is shown to be the algebra HB(E)∞(Ω) of bounded holomorphic B(E)-valued functions on Ω, provided Mz satisfies the matrix-valued von Neumann's inequality. This generalizes a classical result of Shields and Wallen (the case of dim⁡E=1 and d=1). As an application, we determine the commutant of a Bergman shift on a leafless, locally finite, rooted directed tree T of finite branching index. As the second main result of this paper, we show that a multiplication d-tuple Mz on H satisfying the von Neumann's inequality is reflexive. This provides several new classes of examples as well as recovers special cases of various known results in one and several variables. We also exhibit a family of tri-diagonal B(C2)-valued kernels for which the associated multiplication operators Mz are non-hyponormal reflexive operators with commutants equal to HB(C2)∞(D).

Relation algebras and groups

Generalizing results of Jonsson and Tarski, Maddux introduced the notion of a pair-dense relation algebra and proved that every pair-dense relation algebra is representable. The notion of a pair below the identity element is readily definable within the equational framework of relation algebras. The notion of a triple, a quadruple, or more generally, an element of size (or measure) $$n>2$$ is not definable within this framework, and therefore it seems at first glance that Maddux’s theorem cannot be generalized. It turns out, however, that a very far-reaching generalization of Maddux’s result is possible if one is willing to go outside of the equational framework of relation algebras, and work instead within the framework of the first-order theory. Moreover, this generalization sheds a great deal of light not only on Maddux’s theorem, but on the earlier results of Jonsson and Tarski. In the present paper, we define the notion of an atom below the identity element in a relation algebra having measure n for an arbitrary cardinal number $$n>0$$ , and we define a relation algebra to be measurable if it’s identity element is the sum of atoms each of which has some (finite or infinite) measure. The main purpose of the present paper is to construct a large class of new examples of group relation algebras using systems of groups and corresponding systems of quotient isomorphisms (instead of the classic example of using a single group and forming its complex algebra), and to prove that each of these algebras is an example of a measurable set relation algebra. In a subsequent paper, the class of examples will be greatly expanded by adding a third ingredient to the mix, namely systems of “shifting” cosets. The expanded class of examples—called coset relation algebras—will be large enough to prove a representation theorem saying that every atomic, measurable relation algebra is essentially isomorphic to a coset relation algebra.

Molino theory for matchbox manifolds

A matchbox manifold is a foliated space with totally disconnected transversals, and an equicontinuous matchbox manifold is the generalization of Riemannian foliations for smooth manifolds in this context. In this paper, we develop the Molino theory for all equicontinuous matchbox manifolds. Our work extends the Molino theory developed in the work of \'Alvarez L\'opez and Moreira Galicia which required the hypothesis that the holonomy actions for these spaces satisfy the strong quasi-analyticity condition. The methods of this paper are based on the authors' previous works on the structure of weak solenoids, and provide many new properties of the Molino theory for the case of totally disconnected transversals, and examples to illustrate these properties. In particular, we show that the Molino space need not be uniquely well-defined, unless the global holonomy dynamical system is tame, a notion defined in this work. We show that examples in the literature for the theory of weak solenoids provide examples for which the strong quasi-analytic condition fails. Of particular interest is a new class of examples of equicontinuous minimal Cantor actions by finitely generated groups, whose construction relies on a result of Lubotzky. These examples have non-trivial Molino sequences, and other interesting properties.