Search

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most L on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichmüller geodesic flow. [ABSTRACT FROM AUTHOR]

This year the journal Acta Scientiarum Mathematicarum, which is commonly called Acta Szeged, celebrates its 100th anniversary. In this article, we provide a brief overview of the past and the present of the journal, which is currently one of the 20 oldest and still active mathematical periodicals worldwide. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the editor discusses articles in the issue focusing on American mathematician Dennis Sullivan's significant contributions to mathematics, including arithmeticity in topology, the characteristic variety theorem, his work on holomorphic dynamics, and lectures.

This work is devoted to the mathematical and numerical study of a framework handling a system of coupled variational inequalities. We prove both the existence and the uniqueness of a weak solution to the problem. Then, we introduce a convergent iterative scheme. Using this latter, we decouple the problem into further subproblems and derive their corresponding minimization problems. As a practical application of this class of coupled abstract variational inequalities, we consider a class of problems that model an electroelastic body coming into frictional contact with a rigid electrically conductive foundation. Both electrical and mechanical contacts are of Signorini type. In other words, our model prescribes the mechanical response produced by the foundation and the outflow of the free charges across the contact zone. The last part of this paper is mainly reserved for the numerical resolution of the problem at hand. For this purpose, we have developed an alternating direction method of multipliers and convex dualities to compute and illustrate the solutions. [ABSTRACT FROM AUTHOR]

The matching problem often arises in image processing and involves finding a correspondence between similar objects. In particular, variational matching models optimize suitable energies that evaluate the dissimilarity between the current shape and the relative template. A penalty term often appears in the energy to constrain the regularity of the solution. To perform numerical computation, a discrete version of the energy is defined. Then, the question of consistency between the continuous and discrete solutions arises. This paper proves a Γ-convergence result for the discrete energy to the continuous one. In particular, we highlight some geometric properties that must be guaranteed in the discretization process to ensure the convergence of minimizers. We prove the result in the framework introduced in the 2017 paper of Charlier et al., which studies the matching problem between geometric structures carrying on a signal (fshapes). The matching energy is defined for L2 signals and evaluates the difference between fshapes in terms of the varifold norm. This paper maintains a dual attachment term, but we consider a BV penalty term in place of the original L2 norm. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring. Denote by modR the category of finitely generated R-modules. In the present paper, we introduce the notion of solid subcategories of modR and investigate it. The main result of this paper not only recovers results of Schoutens, Krause and Stevenson, and Takahashi on thick subcategories, but also unifies and extends them to solid subcategories. Moreover, it provides some contributions to the study of the question asking when a thick subcategory is Serre. [ABSTRACT FROM AUTHOR]

In previous papers, we were able to prove that, much like in classical algebraic geometry, it is possible to recover our monoid scheme X from the topos Qcoh(X). This was achieved using topos points and localisations of Qcoh(X). With this philosophy in mind, the aim of this paper is to study the topos of M-sets for non-commutative monoids, especially their points and localisations. We will classify the points and localisations of M-sets for finite monoids in terms of the idempotent elements of M and idempotent ideals of M, respectively. Some of the results obtained in this paper can already be found in previous works, in direct or indirect forms. See the last part of the introduction. [ABSTRACT FROM AUTHOR]

In this paper, we analyze submersions with Poisson fibres. These are submersions whose total space carries a Poisson structure, on which the ambient Poisson structure pulls back, as a Dirac structure, to Poisson structures on each individual fibre. Our "Poisson--Dirac viewpoint" is prompted by natural examples of Poisson submersions with Poisson fibres -- in toric geometry and in Poisson--Lie groups -- whose analysis was not possible using the existing tools in the Poisson literature. The first part of the paper studies the Poisson--Dirac perspective of inducing Poisson structures on submanifolds. This is a rich landscape, in which subtle behaviours abound, as illustrated by a surprising "jumping phenomenon" concerning the complex relation between the induced and the ambient symplectic foliations, which we discovered here. These pathologies, however, are absent from the well-behaved and abundant class of coregular submanifolds, with which we are mostly concerned here. The second part of the paper studies Poisson submersions with Poisson fibres -- the natural Poisson generalization of flat symplectic bundles. These Poisson submersions have coregular Poisson--Dirac fibres, and behave functorially with respect to such submanifolds. We discuss the subtle collective behavior of the Poisson fibres of such Poisson fibrations, and explain their relation to pencils of Poisson structures. The third and final part applies the theory developed to Poisson submersions with Poisson fibres which arise in Lie theory. We also show that such submersions are a convenient setting for the associated bundle construction, and we illustrate this by producing new Poisson structures with a finite number of symplectic leaves. Some of the points in the paper being fairly new, we illustrate the many fine issues that appear with an abundance of (counter-)examples. [ABSTRACT FROM AUTHOR]

This is the third paper in a series of three dealing with sums of squares and hypoellipticity in the infinitely degenerate regime.We establish a C2,d generalization of M. Christ's smooth sum of squares theorem, and then use a bootstrap argument with the sum of squares decomposition for matrix functions, obtained in our second paper of this series, to prove a hypoellipticity theorem that generalizes some cases of the results of Christ, Hoshiro, Koike, Kusuoka and Stroock and Morimoto for sums of squares, and of Fedıi and Kohn for degeneracies not necessarily a sum of squares. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish Green function estimates for a class of purely discontinuous symmetric Markov processes with jump kernels degenerate at the boundary and critical killing potentials. The jump kernel and the killing potential depend on several parameters. We establish sharp two-sided estimates on the Green functions of these processes for all admissible values of the parameters involved. Depending on the regions where the parameters belong, the estimates on the Green functions are different. In fact, the estimates have three different forms. As applications, we prove that the boundary Harnack principle holds in certain region of the parameters and fails in some other region of the parameters. Together with the main results of our previous paper [Potential Anal., online, 2021], we completely determine the region of the parameters where the boundary Harnack principle holds. [ABSTRACT FROM AUTHOR]

In this two-paper series, we prove the invariance of the Gibbs measure for a threedimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a paracontrolled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences. [ABSTRACT FROM AUTHOR]

This paper investigates analytic Besov functions of n variables which act on the generators of n commuting C0-semigroups on a Banach space. The theory for n = 1 has already been published, and the present paper uses a different approach to that case as well as extending to the cases when n ≥ 2. It also clarifies some spectral mapping properties and provides some operator norm estimates. [ABSTRACT FROM AUTHOR]

In deformation-rigidity theory, it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra C[Γ] with Γ a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten Sp class p ∈ [2,∞), then the n-fold tensor power HΓ⊗n for n ≥ p/2 is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups, we give a number of characterizations of having coefficients in Sp for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-Sp property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups, (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by Sinclair for discrete groups admitting a proper cocycle into a p-integrable representation. [ABSTRACT FROM AUTHOR]

In this paper we study the dynamics of an incompressible viscous fluid evolving in an open-top container in two dimensions. The fluid mechanics are dictated by the Navier-Stokes equations. The upper boundary of the fluid is free and evolves within the container. The fluid is acted upon by a uniform gravitational field, and capillary forces are accounted for along the free boundary. The triple-phase interfaces where the fluid, air above the vessel, and solid vessel wall come in contact are called contact points, and the angles formed at the contact point are called contact angles. The model that we consider integrates boundary conditions that allow for full motion of the contact points and angles. Equilibrium configurations consist of quiescent fluid within a domain whose upper boundary is given as the graph of a function minimizing a gravity-capillary energy functional, subject to a fixed mass constraint. The equilibrium contact angles can take on any values between 0 and π depending on the choice of capillary parameters. The main thrust of the paper is the development of a scheme of a priori estimates that show that solutions emanating from data sufficiently close to the equilibrium exist globally in time and decay to equilibrium at an exponential rate. [ABSTRACT FROM AUTHOR]

This paper is devoted to the local null-controllability of the nonlinear KdV equation equipped the Dirichlet boundary conditions using the Neumann boundary control on the right. Rosier proved that this KdV system is small-time locally controllable for all noncritical lengths and that the uncontrollable space of the linearized system is of finite dimension when the length is critical. Concerning critical lengths, Coron and Crépeau showed that the same result holds when the uncontrollable space of the linearized system is of dimension 1; later Cerpa, and then Cerpa and Crépeau, established that the local controllability holds at a finite time for all other critical lengths. In this paper, we prove that, for a class of critical lengths, the nonlinear KdV system is not small-time locally controllable. [ABSTRACT FROM AUTHOR]

For a set E ⊂ Rn that contains the origin, we consider Im(E) - the set of all mth degree Taylor approximations (at the origin) of Cm functions on Rn that vanish on E. This set is a proper ideal in Pm(Rn) - the ring of all mth degree Taylor approximations of Cm functions on Rn. Which ideals in Pm(Rn) arise as Im(E) for some E? In this paper we introduce the notion of a closed ideal in Pm(Rn), and prove that any ideal of the form Im(E) is closed. We do not know whether in general any closed proper ideal is of the form Im(E) for some E, however we prove in a subsequent paper that all closed proper ideals in Pm(Rn) arise as Im(E) when m + n ≤ 5. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason-Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the "quaternionic Geller operators" and the "octonionic Geller operators", which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason-Fourier analysis on symmetric spaces, some precise estimates for the heat and the Bessel-Green-Riesz kernels, and the Kunze-Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy-Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on all rank one symmetric spaces of noncompact type. [ABSTRACT FROM AUTHOR]

In the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three there exists a cone such that the heat kernel (the continuous analog of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles. [ABSTRACT FROM AUTHOR]

We give a characterization of the singular points of the free boundary ∂{u>0} for viscosity solutions of the nonlinear equation F(D²u)=-χ {u>0}, where F is a fully nonlinear elliptic operator and χ is the characteristic function. This equation models the propagation of a nerve impulse along an axon. We analyze the structure of the free boundary ∂{u>0} near the singular points where u and ∇u vanish simultaneously. Our method uses the stratification approach developed in Dipierro and the author's 2018 paper. In particular, when n=2 we show that near a flat singular free boundary point, ∂{u>0} is a union of four C 1 arcs tangential to a pair of crossing lines. [ABSTRACT FROM AUTHOR]

We investigate a phase-field version of the Faber-Krahn theorem based on a phase-field optimization problem introduced by Garcke et al. in their 2023 paper formulated for the principal eigenvalue of the Dirichlet-Laplacian. The shape that is to be optimized is represented by a phase-field function mapping into the interval [0,1]. We show that any minimizer of our problem is a radially symmetric-decreasing phase-field attaining values close to 0 and 1 except for a thin transition layer whose thickness is of order ε>0. Our proof relies on radially symmetric-decreasing rearrangements and corresponding functional inequalities. Moreover, we provide a Γ-convergence result which allows us to recover a variant of the Faber-Krahn theorem for sets of finite perimeter in the sharp interface limit. [ABSTRACT FROM AUTHOR]

In this paper, we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank 2 if and only if each open face in the ideal boundary has dimension at most one. We also introduce the “coarse Hilbert dimension” of a subset of a convex set and use it to characterize when a naive convex co-compact subgroup is word hyperbolic or relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank 2. [ABSTRACT FROM AUTHOR]

Charney and Morris-Wright showed acylindrical hyperbolicity of Artin groups of infinite type associated with graphs that are not joins, by studying clique-cube complexes and the actions on them. In this paper, by developing their study and formulating some additional discussion, we demonstrate that acylindrical hyperbolicity holds for more general Artin groups. Indeed, we are able to treat Artin groups of infinite type associated with graphs that are not cones. [ABSTRACT FROM AUTHOR]

In this paper, we study the action of a countable group Γ on the space of orders on the group. In particular, we are concerned with the invariant probability measures on this space, known as invariant random orders. We show that for any countable group, the space of random invariant orders is rich enough to contain an isomorphic copy of any free ergodic action, and characterize the non-free actions realizable. We prove a Glasner–Weiss dichotomy regarding the simplex of invariant random orders. We also show that the invariant partial order on SL3. Z corresponding to the semigroup generated by the standard unipotents cannot be extended to an invariant random total order. We thus provide the first example for a partial order (deterministic or random) that cannot be randomly extended. [ABSTRACT FROM AUTHOR]

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the p-Laplace equation --Δpu = μ with p ∈ (1, 2). The cases when p ∈ (2 -- 1/n, 2) and p ∈ (3n--2/2n--1, 2 -- 1/n] were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379-418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when p ∈ (1, 3n--2/2n--1)]. Interior and global modulus of continuity estimates of the gradients of solutions are also established. [ABSTRACT FROM AUTHOR]

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. In this paper, we study the asymptotic dimension of metric spaces generated by graphs and their shortest path metric and show their applications to some continuous spaces. The asymptotic dimension of such graph metrics can be seen as a large scale generalisation of weak diameter network decomposition which has been extensively studied in computer science. We prove that every proper minor-closed family of graphs has asymptotic dimension at most 2, which gives optimal answers to a question of Fujiwara and Papasoglu and (in a strong form) to a problem raised by Ostrovskii and Rosenthal on minor excluded groups. For some special minorclosed families, such as the class of graphs embeddable in a surface of bounded Euler genus, we prove a stronger result and apply this to show that complete Riemannian surfaces have Assouad--Nagata dimension at most 2. Furthermore, our techniques allow us to determine the asymptotic dimension of graphs of bounded layered treewidth and graphs with any fixed growth rate, which are graph classes that are defined by purely combinatorial notions and properly contain graph classes with some natural topological and geometric flavours. [ABSTRACT FROM AUTHOR]

In this paper, we construct a smooth vector bundle over the deformation to the normal cone DNC(V,M) through a rescaling of a vector bundle E → V, which generalizes the construction of the spinor rescaled bundle over the tangent groupoid by Nigel Higson and Zelin Yi. We also provide an equivariant version of their construction. As the main application, we recover the Kirillov character formula for the equivariant index of Dirac-type operators. As another application, we get an equivariant generalization of the description of the Witten and the Novikov deformations of the de Rham-Dirac operator using the deformation to the normal cone obtained recently by O. Mohsen. [ABSTRACT FROM AUTHOR]

In this paper, we construct odd finitely summable spectral triples based on length functions of bounded doubling on noncommutative solenoids. Our spectral triples induce a Leibniz Lip-norm on the state spaces of the noncommutative solenoids, giving them the structure of Leibniz quantum compact metric spaces. By applying methods of R. Floricel and A. Ghorbanpour, we also show that our odd spectral triples on noncommutative solenoids can be considered as inductive limits of spectral triples on rotation algebras. In the final section, we prove a noncommutative version of Wiener's lemma and show that our odd spectral triples can be defined to have an associated smooth dense subalgebra which is stable under the holomorphic functional calculus, thus answering a question of B. Long and W. Wu. The construction of the smooth subalgebra also extends to the case of nilpotent discrete groups. [ABSTRACT FROM AUTHOR]

In this paper, we prove a non-commutative version of the Weyl-von Neumann theorem for representations of unital, separable AH algebras into countably decomposable, semifinite, properly infinite, von Neumann factors, where an AH algebra means an approximately homogeneous C*-algebra. We also prove a result for approximate summands of representations of unital, separable AH algebras into finite von Neumann factors. [ABSTRACT FROM AUTHOR]

This paper uses noncommutative resolutions of non-Gorenstein singularities to construct classical deformation spaces by recovering the Artin component of the deformation space of a cyclic surface singularity using only the quiver of the corresponding reconstruction algebra. The relations of the reconstruction algebra are then deformed, and the deformed relations together with variation of the GIT quotient achieve the simultaneous resolution. This extends the work of Brieskorn, Kronheimer, Grothendieck, Cassens-Slodowy, and Crawley-Boevey-Holland into the setting of singularities C²/H with H ≤ GL(2, C) and furthermore gives a prediction for what is true more generally. [ABSTRACT FROM AUTHOR]

In this paper, we generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces X so that any bi-Lipschitz embedding of a subset of the real line into X extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset Y of X has small Assouad dimension, then it is a uniform domain. Finally, we prove a quantitative approximation of continua in X by bi-Lipschitz curves. [ABSTRACT FROM AUTHOR]

Consider a finite connected 2-complex X endowed with a piecewise Riemannian metric, and whose fundamental group is freely indecomposable, of rank at least 3, and in which every 2-generated subgroup is free. In this paper, we show that we can always find a connected graph Γ⊂X such that π1​ Γ≃F2 ​ ↪π1 X (in short, a 2-incompressible graph) whose length satisfies the following curvature-free inequality: ℓ(Γ)≤4 √2Area(X). This generalizes a previous inequality proved by Gromov for closed Riemannian surfaces with negative Euler characteristic. As a consequence, we obtain that the volume entropy of such 2-complexes with unit area is always bounded away from zero. [ABSTRACT FROM AUTHOR]

In his 1957 paper, John Milnor introduced link invariants which measure the homotopy class of the longitudes of a link relative to the lower central series of the link group. Consequently, these invariants determine the lower central series quotients of the link group. This work has driven decades of research with profound influence. One of Milnor's original problems remained unsolved: to extract similar invariants from the transfinite lower central series of the link group. We reformulate and extend Milnor's invariants in the broader setting of 3-manifolds, with his original invariants as special cases. We present a solution to Milnor's problem for general 3-manifold groups, developing a theory of transfinite invariants and realizing nontrivial values. [ABSTRACT FROM AUTHOR]

The embedding of a torus into an inner form of PGL2 defines an adelic toric period. A general version of Duke's theorem states that this period equidistributes as the discriminant of the splitting field tends to infinity. In this paper we consider a torus embedded diagonally into two distinct inner forms of PGL2. Assuming the Generalized Riemann Hypothesis (and some additional technical assumptions), we show simultaneous equidistribution as the discriminant tends to infinity, with an effective logarithmic rate. Our proof is based on a probabilistic approach to estimating fractional moments of L-functions twisted by extended class group characters. [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time behavior of a nonlinear porous elasticity system. The system is subject to a viscoporous damping and a nonlinear source term which is locally Lipschitz and depends only on the volume fraction. The dynamical system associated with the solutions of the model is gradient, and under the hypothesis of equal speeds of propagation for the waves, we prove that it is also quasi-stable, which allows us to show the existence of a global attractor for the system, which is the main result of the paper. [ABSTRACT FROM AUTHOR]

This paper seeks conditions that ensure that the attractor of a graph directed iterated function system (GD-IFS) cannot be realised as the attractor of a standard iterated function system (IFS). For a strongly connected directed graph, it is known that, if all directed circuits go through a particular vertex, then for any GD-IFS of similarities on R based on the graph and satisfying the convex open set condition (COSC), its attractor associated with that vertex is also the attractor of a (COSC) standard IFS. In this paper we show the following complementary result. If there exists a directed circuit which does not go through a certain vertex, then there exists a GD-IFS based on the graph such that the attractor associated with that vertex is not the attractor of any standard IFS of similarities. Indeed, we give algebraic conditions for such GD-IFS attractors not to be attractors of standard IFSs, and thus show that 'almost-all' COSC GD-IFSs based on the graph have attractors associated with this vertex that are not the attractors of any COSC standard IFS. [ABSTRACT FROM AUTHOR]

The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group, we mean any probability measure ν on the collection O of linear orders of type Z on G, invariant under the natural action of G on such orders. Multiorders exist on any countable amenable group (and only on such groups) and every multiorder has the Følner property, meaning that almost surely the order intervals starting at the unit form a Følner sequence. Every free measure-preserving G-action (X,μ,G) has a multiorder (O,ν,G) as a factor and has the same orbits as the Z-action (X,μ,S), where S is the successor map determined by the multiorder factor. Moreover, the sub-sigma-algebra ΣO ​ associated with the multiorder factor is invariant under S, which makes the corresponding Z-action (O,ν,S) a factor of (X,μ,S). We prove that the entropy of any G-process generated by a finite partition of X, conditional with respect to ΣO ​, is preserved by the orbit equivalence with (X,μ,S). Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to h(μ,T,P)=H(μ,P∣P-) known for Z-actions. The above fact is then applied to prove a variant of a result by Rudolph and Weiss (2000). The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a sub-sigma-algebra Σ, as soon as the "orbit change" is measurable with respect to Σ. In our variant, we replace the measurability assumption by a simpler one: Σ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion, we provide a characterization of the Pinsker sigma-algebra of any G-process in terms of an appropriately defined remote past arising from a multiorder. The paper has an appendix in which we present an explicit construction of a particularly regular (uniformly Følner) multiorder based on an ordered dynamical tiling system of G. [ABSTRACT FROM AUTHOR]

In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of d-regular graphs for all d, as n→∞. The key to our results is a new approach to estimating ratios between point probabilities in the space of degree sequences of the random graph, including analysis of fixed points of the associated operators. [ABSTRACT FROM AUTHOR]

In this paper we prove that if G is a finite group, then the k-th term of the lower central series is nilpotent if and only if for every γk-values x, y ∈ G with coprime orders, either π(o(x)o(y)) ⊆ π(o)x y)) or o(x)o(y)) ≤ o(x y). We obtain an analogous version for the derived series of finite solvable groups, but replacing γk-values by δk-values. We will also discuss the existence of normal Sylow subgroups in the derived subgroup in terms of the order of the product of certain elements. [ABSTRACT FROM AUTHOR]

Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and a universal equivariant quantization with respect to a reductive group acting on it by C*-equivariant Poisson automorphisms. We go on to study these definitions in the context of nilpotent Slodowy slices. First, we give a complete description of the cases in which the finite W -algebra is a universal filtered quantization of the slice, building on the work of Lehn--Namikawa--Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices. The subregular slices in non-simply laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type, we prove that the finite W -algebra is a universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowy's theorem. Finally, we apply this result to give a presentation of the subregular finiteW -algebra of type B as a quotient of a shifted Yangian. [ABSTRACT FROM AUTHOR]

In this paper we discuss the notion of research data for the field of mathematics and report on the status quo of research-data management and planning. A number of decentralized approaches are presented and compared to needs and challenges faced in three use cases from different mathematical subdisciplines. We highlight the importance of tailoring research-data management plans to mathematicians’ research processes and discuss their usage all along the data life cycle. [ABSTRACT FROM AUTHOR]

In this paper and the companion work [J. Funct. Anal. 281 (2021)], we prove that the Schrödinger map flows from Rd with d ≥ 2 to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. The main difficulty compared with the constant sectional curvature case is that the gauged equation now is not self-contained due to the curvature part. Our main idea is to use a novel bootstrap-iteration scheme to reduce the gauged equation to an approximate constant curvature system in finite times of iteration. This paper together with the companion work solves the open problem raised by Tataru. [ABSTRACT FROM AUTHOR]

In this paper we investigate a model for quantum gravity on finite noncommutative spaces using the theory of blobbed topological recursion. The model is based on a particular class of random finite real spectral triples (A,H,D,γ,J), called random matrix geometries of type (1,0), with a fixed fermion space (A,H,γ,J), and a distribution of the form e-S(D)dD over the moduli space of Dirac operators. The action functional S(D) is considered to be a sum of terms of the form ∏si=1Tr(Dni) for arbitrary s⩾1. The Schwinger-Dyson equations satisfied by the connected correlators Wn of the corresponding multi-trace formal 1-Hermitian matrix model are derived by a differential geometric approach. It is shown that the coefficients Wg,n of the large N expansion of Wn's enumerate discrete surfaces, called stuffed maps, whose building blocks are of particular topologies. The spectral curve (Σ,ω0,1,ω0,2) of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential ω0,2 in terms of the formal parameters of the model. [ABSTRACT FROM AUTHOR]

In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schrödinger operator (1 + Δ)log corresponding to the logarithmic symbol log(1 + |ξ|²) which is a singular integral operator given by (I+ Δ)log u(x) = cNP.V. u(x)-u(y) /∫ℝN u(x)-u(y)/|x-y|n k(|x-y|dy, where cN = π - N/2, k(r) =21-N/2 r N/2 K N/2 (r)and Kv is the modified Bessel function of the second kind with index v. The proof hinges on a direct method of moving planes for the logarithmic Schrödinger operator. [ABSTRACT FROM AUTHOR]

In this paper, we study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Lefflertype function to study oscillatory-type integrals appearing in the analysis of time-fractional partial differential equations. Several generalisations of the first and second van der Corput lemmas are proved. Optimal estimates on decay orders for particular cases of the Mittag-Leffler functions are also obtained. As an application of the above results, the generalised Riemann–Lebesgue lemma and the Cauchy problem for the time-fractional evolution equation are considered. [ABSTRACT FROM AUTHOR]

The mysterious phenomenon of revivals in linear dispersive periodic equations was discovered first experimentally in optics in the 19th century, then rediscovered several times by theoretical and experimental investigations. While the term has been used systematically and consistently by many authors, there is no consensus on a rigorous definition. In this paper, we describe revivals modulo a regularity condition in a large class of Schrödinger’s equations with complex bounded potentials. As we show, at rational times, the solution is given explicitly by finite linear combinations of translations and dilations of the initial datum, plus an additional continuous term. [ABSTRACT FROM AUTHOR]

A celebrated theorem of Serrin asserts that one overdetermined condition on the boundary is enough to obtain radial symmetry in the so-called one-phase overdetermined torsion problem. It is also known that imposing just one overdetermined condition on the boundary is not enough to obtain radial symmetry in the corresponding multi-phase overdetermined problem. In this paper we show that, in order to obtain radial symmetry in the two-phase overdetermined torsion problem, two overdetermined conditions are needed. Moreover, it is noteworthy that this pattern does not extend to multi-phase problems with three or more layers, for which we show the existence of nonradial configurations satisfying countably infinitely many overdetermined conditions on the outer boundary. [ABSTRACT FROM AUTHOR]

In this paper, we consider the semiclassical 2D magnetic Schrödinger operator in the case where the magnetic field vanishes along a smooth closed curve. Assuming that this curve has an axis of symmetry, we prove that semiclassical tunneling occurs. The main result is an expression of the splitting of the first two eigenvalues and an explicit tunneling formula. [ABSTRACT FROM AUTHOR]