Abstract In this paper, we describe a new approach to the problem of classification of transitive Anosov flows on 3‐manifolds up to orbital equivalence. To every transitive Anosov flow on is associated a bifoliated plane endowed with an action of . Thanks to a theorem of Barbot, the previous action characterizes up to orbital equivalence. The goal of this paper is to classify the above class of actions arising from transitive Anosov flows using Markovian families , introduced here as a group‐action analogue of Markov partitions. More specifically, we prove that every transitive Anosov flow admits infinitely many Markovian families; given a Markovian family of , say , the number of orbits of rectangles of and their pattern of intersection can be encoded by a finite combinatorial object, called a geometric type, which describes completely up to Dehn–Goodman–Fried surgeries on a finite set of periodic orbits of ; equipping a geometric type of with additional combinatorial data, referred to as cycles , produces a finite combinatorial invariant that completely characterizes up to orbital equivalence.
Abstract Let denote the Chow variety of effective algebraic ‐cycles of degree in complex projective space . In this paper, we compute the rational Lawson homology groups for . Additionally, we prove that the rational Lawson homology groups of a natural completion of the Chow monoid of algebraic ‐cycles in projective spaces are isomorphic to the corresponding rational singular homology groups. We also establish the stability of Lawson homology groups of Chow varieties under natural embeddings and algebraic suspension maps within a specified range.
Abstract We study homological invariants of the Steinberg algebra of an ample groupoid over a commutative ring . For any ample Hausdorff groupoid , we find that is a direct summand of ; using this and the Dennis trace we obtain a map . We study this map when is the (twisted) Exel–Pardo groupoid associated to a self‐similar action of a group on a graph, and compute and in terms of the homology of , and the ‐theory of in terms of that of .
Abstract Let and with . We consider homogeneous Diophantine equations of degree in variables and whether they have solutions in the primes. In particular, we show that a certain local–global principle holds for almost all such equations, following on from previous work of Holdridge. We do this by adapting the methods of Browning, Le Boudec and Sawin, with the main input coming from some results on counting points with prime coordinates in lattices.
Abstract The aim of this paper is to clarify the asymptotic behavior of global classical solutions with small initial data for the Cauchy problem of incompressible neo‐Hookean elastodynamics. To this end, we first provide a new and streamlined proof of the global existence result, employing only general derivatives and spatial rotation operators as commuting vector fields. Building upon this new proof, we then show that the global solution scatters in the energy sense; namely, it converges to a solution of the homogeneous linear wave equations as time tends to infinity. We also prove the following rigidity result: if the scattering data vanish, then the global solution vanishes identically. Finally, we show that for compactly supported initial data, the global solution is uniquely determined by the scattering data; in other words, the inverse scattering property holds.
Abstract A real sequence is called ‐generating if there exists a function whose translates span the space . While the ‐generating sets were completely characterized for and , the case remains not well understood. In this case, both the size and the arithmetic structure of the set play an important role. In this paper, (i) we show that a ‐generating set of positive real numbers can be very sparse, namely, the ratios may tend to 1 arbitrarily slowly; (ii) we prove that every “almost integer” sequence , that is, satisfying , , is ‐generating; and (iii) we construct ‐generating sets such that the successive differences attain only two different positive values. The constructions are, in a sense, sharp: it is well known that cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.
Abstract The classic paper by Littlewood and Paley [J. Lond. Math. Soc. (1), 6 (1931), 230–233] marked the birth of Littlewood–Paley theory. We discuss this paper and its impact from a historical perspective, include an outline of the results in the paper and their subsequent significance in relation to developments over the last century, and set them into the context of the current state of the art in harmonic analysis and beyond.
Abstract We study the emergent weak flocking behaviors of the generalized kinetic Justh–Krishnaprasad (GKJK) model in a spatially extended setting. For this, we provide a sufficient framework to ensure the emergence of weak flocking behaviors. Weak flocking behaviors refer to the situation in which the second moment for heading angle fluctuation around the heading angle average tends to zero asymptotically, whereas the second moment for spatial fluctuations around the center of mass remains bounded uniformly in time. On an unbounded spatial domain, the nonlocal alignment kernel may lose a uniform positive lower bound, so standard energy‐type argument breaks down as it is. To overcome this difficulty, we impose a suitable tail– decay ansatz on the one‐particle distribution function and introduce the method of an effective time‐varying domain whose complement carries vanishing mass asymptotically. Our results show that polynomial spatial decay of the initial distribution yields at least polynomial convergence to weak flocking, while exponential decay implies exponential convergence. In particular, we also demonstrate that even if the heading angle diameter remains constant, weak flocking behavior can still emerge in the GKJK model. This is the new phenomenon that was not observed in the literature.
Abstract Finite arc‐transitive bicirculants with valency 3,4, or 5 have been classified in several previous papers. In 2022, Devillers, Giudici, and Jin gave a reductive characterization of finite arc‐transitive bicirculants, and using this, in 2023 Jin obtained a classification of finite 2‐arc‐transitive bicirculants. In this paper, we give a complete classification of finite vertex‐transitive and locally primitive bicirculants.
Abstract Consider ‐analytic mapping‐germs, . They can be equivalent (by coordinate changes) ‐analytically, but not ‐analytically. However, if the transformation of ‐equivalence is modulo higher order terms, then it implies the ‐equivalence. On the other hand, starting from ‐analytic map‐germs , and taking any field extension , one has: if , then . These (quite useful) properties seem to be not well known. We prove slightly stronger properties in a more general form: for , where are (formal/analytic/‐Nash) germs of spaces, with arbitrary singularities, over a base ring for the classical groups of (right/left–right/contact) equivalence of singularity theory; for faithfully flat extensions of rings . In particular, for arbitrary extensions of fields. The case “ is a ring” is important for the study of deformations/unfoldings. For example, it implies the statement for field extensions: if a family of ‐maps is ‐trivial, then it is also ‐trivial. Similar statements for germs of spaces (“isomorphism over vs. isomorphism over ”) follow by the standard reduction “Two maps are contact equivalent if and only if their zero sets are ambient isomorphic.” This study involves the contact equivalence of maps with singular targets, which seems to be not well established. We write down the relevant part of this theory.
Abstract We survey ideas surrounding the study of the number of integers that can be represented as the sum of three positive cubes. We focus on the early contribution of Davenport using elementary techniques, and the subsequent developments due to Vaughan, which introduced Fourier analysis and mirrored many of the important developments of the Hardy–Littlewood circle method.
Abstract In this paper, we study the K‐stability of del Pezzo surfaces with a unique quotient singularity whose minimal resolution has exceptional locus , where , , and for .
Abstract For a (not necessarily smooth) bounded domain of , and a Carathéodory vector‐valued function , we study the compactness of the inverse of the Leray–Lions operator , , . Denoting by the Sobolev exponent, in the main result it is proved the compactness of if , provided . Also, for , the operators and are compact for every . In contrast, it is also established that and are not compact. The compactness of the set of solutions for the more general operator is also studied. The proofs of the main results are based on an appropriate adaptation of Stampacchia's technique, which provides a more elementary approach than the usual regularity theory up to the boundary, that cannot be employed for nonsmooth domains. As an application, we improve previous results on the existence and multiplicity of solutions for a class of problems involving the ‐Laplacian operator under a local Landesman–Lazer condition for arbitrary bounded domains.
Abstract We are concerned with the fluid limit to KdV equations for the one‐dimensional Vlasov–Poisson–Landau system that describes the dynamics of ions in plasma with the electron density determined by the self‐consistent electric potential through the so‐called Boltzmann relation. Formally, it is well known that as the Knudsen number the Vlasov–Poisson–Landau system in the compressible scaling converges to the Euler–Poisson equations which further under the Gardner–Morikawa transformation converge to the KdV equations as the parameter . Our goal of this paper is to construct smooth solutions of the correspondingly rescaled Vlasov–Poisson–Landau system over an arbitrary finite time interval that can converge uniformly to smooth solutions of the KdV equations as and simultaneously under an extra condition . Moreover, the explicit rate of convergence in is also obtained. The proof is established by an appropriately chosen scaling and an intricate weighted energy method through the macro–micro decomposition around local Maxwellians. We design a ‐‐dependent high‐order energy functional to capture the singularity of such fluid limit problem.
Abstract Let be a sequence of independent copies of a random vector in . We revisit the question to determine the asymptotic shape of the random polytope where . We show that for any there exists a constant such that the following holds true: if is a Borel probability measure on then, for all we have that with probability greater than , where is the convex set of all points with half‐space depth greater than or equal to . Our approach does not require any additional assumptions about the measure and hence it generalizes and/or improves a sequence of previous results. Moreover, for the class of strongly regular measures we compare the family to other natural families of convex bodies associated with , such as the ‐centroid bodies of or the level sets of the Cramér transform of , and use this information in order to estimate the size of a random .
We review the paper `Models for the Eremenko–Lyubich class' by Chris Bishop, which appeared in the Journal of the London Mathematical Society in 2015, the first of two LMS papers for which Bishop received the LMS Senior Berwick Prize in 2024. This is one of a series of papers where Bishop introduced his new technique of quasiconformal folding, which transformed the range of approaches to constructing examples of transcendental entire functions leading to the resolution of many open problems, in particular in transcendental dynamics.
We prove the precise inversion of adjunction formula for finite linear group quotients of complete intersection varieties defined by semi-invariant equations. As an application, we prove the semi-continuity of minimal log discrepancies for them. These results extend the results in our first paper, where we prove the same results for complete intersection varieties defined by “invariant equations".
Abstract In 2002 Zilber, motivated by some model‐theoretical study of the formal theory of exponentiation, proposed a conjecture on ‘atypical’ intersections of subvarieties in tori. This conjecture, raised also independently by Bombieri, Masser and Zannier, and its generalization in the more general setting of mixed Shimura varieties due to Pink, led to the formulation of the so called problems of unlikely intersections , which have been intensively studied in the last three decades during which many important results in Diophantine Geometry have been proved. In this paper we will describe the conjecture about unlikely intersections in tori and in abelian varieties in different formulations, showing its connection with Schanuel's conjecture and we will review the state of the art on the subject.
Abstract In 1975, Bollobás, Erdős, and Szemerédi asked for the smallest such that an tripartite graph with minimum degree must contain , conjecturing that for . We prove that which confirms their conjecture and is best possible assuming the widely believed conjecture that the Zarankiewicz number satisfies . Our proof uses a density increment argument. We also construct an infinite family of extremal graphs that are pairwise far apart (requiring the change of edges to get between any two).
Abstract We study projections in the bidual of a ‐algebra that are null with respect to a subalgebra , that is, projections satisfying for every annihilating . In the separable case, ‐null projections are precisely the peak projections in the bidual of at which the subalgebra interpolates the entire ‐algebra . These are analogs of null sets in classical function theory, on which several profound results rely. This motivates the development of a noncommutative variant, which we use to find appropriate “quantized” versions of some of these classical facts. Through a delicate generalization of a theorem of Varopoulos, we show that, roughly speaking, sufficiently regular interpolation projections are null precisely when their atomic parts are. As an application, we give alternative proofs and sharpenings of some recent peak interpolation results of Davidson and Hartz for algebras on Hilbert function spaces, also illuminating thereby how earlier noncommutative peak‐interpolation theory may be applied. In another direction, given a convex subset of the state space of , we characterize when the associated Riesz projection is null. This is then applied to various important topics in noncommutative function theory, such as the F.& M. Riesz property, the existence of Lebesgue decompositions, the description of Henkin functionals, and Arveson's noncommutative Hardy spaces (maximal subdiagonal algebras).
