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This paper is devoted to the study of nonlinear stability of steady incompressible Euler flows in two dimensions. We prove that a steady Euler flow is nonlinearly stable in L^p norm of the vorticity if its stream function is a semistable solution of some semilinear elliptic problem with nondecreasing nonlinearity. The idea of the proof is to show that such a flow has strict local maximum energy among flows whose vorticities are rearrangements of a given function, with the help of an improved version of Wolansky and Ghil's stability theorem. The result can be regarded as an extension of Arnol'd's second stability theorem. [ABSTRACT FROM AUTHOR]

In this paper we use the method of geometric flow on the problem of nonlinear spline interpolations for non-closed curves in n-dimensional Euclidean spaces. The method applies theory of fourth-order parabolic PDEs to each piece of the curve between two successive knot points at which certain dynamic boundary conditions are imposed. We show the existence of global solutions of the elastic flow in suitable Hölder spaces. In the asymptotic limit, as time approaches infinity, solutions subconverge to a stationary solution of the problem. The method of geometric flows provides a new approach for the problem of nonlinear spline interpolations. [ABSTRACT FROM AUTHOR]

In this paper we give a complete characterisation of linear repetitivity for cut and project schemes with convex polytopal windows satisfying a weak homogeneity condition. This answers a question of Lagarias and Pleasants from the 90s for a natural class of cut and project schemes which is large enough to cover almost all such polytopal schemes which are of interest in the literature. We show that a cut and project scheme in this class has linear repetitivity exactly when it has the lowest possible patch complexity and satisfies a Diophantine condition. Finding the correct Diophantine condition is a major part of the work. To this end we develop a theory, initiated by Forrest, Hunton and Kellendonk, of decomposing polytopal cut and project schemes to factors. We also demonstrate our main theorem on a wide variety of examples, covering all classical examples of canonical cut and project schemes, such as Penrose and Ammann–Beenker tilings. [ABSTRACT FROM AUTHOR]

In this paper we study sharp pointwise inequalities for maximal operators. In particular, we strengthen DeVore's inequality for the moduli of smoothness and a logarithmic variant of Bennett–DeVore–Sharpley's inequality for rearrangements. As a consequence, we improve the classical Stein–Zygmund embedding deriving \dot {B}^{d/p}_\infty L_{p,\infty }(\mathbb {R}^d) \hookrightarrow \text {BMO}(\mathbb {R}^d) for 1 < p < \infty. Moreover, these results are also applied to establish new Fefferman–Stein inequalities, Calderón–Scott type inequalities, and extrapolation estimates. Our approach is based on the limiting interpolation techniques. [ABSTRACT FROM AUTHOR]

Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which P can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more specifically considering various families of low-dimensional polytopes. First, we prove that for a fixed dimension d, the extension complexity of a random d-dimensional polytope (obtained as the convex hull of random points in a ball or on a sphere) is typically on the order of the square root of its number of vertices. Second, we prove that any cyclic n-vertex polygon (whose vertices lie on a circle) has extension complexity at most 24\sqrt n. This bound is tight up to the constant factor 24. Finally, we show that there exists an n^{o(1)}-dimensional polytope with at most n vertices and extension complexity n^{1-o(1)}. Our theorems are proved with a range of different techniques, which we hope will be of further interest. [ABSTRACT FROM AUTHOR]

For a primitive Dirichlet character \chi modulo q, we define M(\chi)=\max _{t } |\sum _{n \leq t} \chi (n)|. In this paper, we study this quantity for characters of a fixed odd order g\geq 3. Our main result provides a further improvement of the classical Pólya-Vinogradov inequality in this case. More specifically, we show that for any such character \chi we have \begin{equation*} M(\chi)\ll _{\varepsilon } \sqrt {q}(\log q)^{1-\delta _g}(\log \log q)^{-1/4+\varepsilon }, \end{equation*} where \delta _g ≔1-\frac {g}{\pi }\sin (\pi /g). This improves upon the works of Granville and Soundararajan [J. Amer. Math. Soc. 20 (2007), pp. 357–384] and of Goldmakher [Algebra Number Theory 6 (2012), pp. 123–163]. Furthermore, assuming the Generalized Riemann Hypothesis (GRH) we prove that \begin{equation*} M(\chi) \ll \sqrt {q} \left (\log _2 q\right)^{1-\delta _g} \left (\log _3 q\right)^{-\frac {1}{4}}\left (\log _4 q\right)^{O(1)}, \end{equation*} where \log _j is the j-th iterated logarithm. We also show unconditionally that this bound is best possible (up to a power of \log _4 q). One of the key ingredients in the proof of the upper bounds is a new Halász-type inequality for logarithmic mean values of completely multiplicative functions, which might be of independent interest. [ABSTRACT FROM AUTHOR]

Recent progress in the study of the contact process (see Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly [Ann. Probab. 49 (2021), pp. 244–286]) has verified that the extinction-survival threshold \lambda _1 on a Galton-Watson tree is strictly positive if and only if the offspring distribution \xi has an exponential tail. In this paper, we derive the first-order asymptotics of \lambda _1 for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if \xi is appropriately concentrated around its mean, we demonstrate that \lambda _1(\xi) \sim 1/\mathbb {E} \xi as \mathbb {E}\xi \rightarrow \infty, which matches with the known asymptotics on d-regular trees. The same results for the short-long survival threshold on the Erdős-Rényi and other random graphs are shown as well. [ABSTRACT FROM AUTHOR]

One of the main open problems in the theory of automorphic products is to classify reflective modular forms. Scheithauer [Invent. Math. 164 (2006), pp. 641-678] classified strongly reflective modular forms of singular weight on lattices of prime level. In this paper we classify symmetric reflective modular forms on lattices of prime level. This yields a full classification of lattices of prime level which have reflective modular forms. We also present some applications. [ABSTRACT FROM AUTHOR]

In this paper, we study quasi post-critically finite degenerations for rational maps. We construct limits for such degenerations as geometrically finite rational maps on a finite tree of Riemann spheres. We prove the boundedness for such degenerations of hyperbolic rational maps with Sierpinski carpet Julia set and give criteria for the convergence for quasi-Blaschke products QBd, making progress towards the analogues of Thurston's compactness theorem for acylindrical 3-manifold and the double limit theorem for quasi-Fuchsian groups in complex dynamics. In the appendix, we apply such convergence results to show the existence of certain polynomial matings. [ABSTRACT FROM AUTHOR]

In this paper, over imaginary quadratic fields, we consider the family of L-functions L (s, ƒ) for an orthonormal basis of spherical Hecke–Maass forms f with Archimedean parameter tƒ. We establish asymptotic formulae for the twisted first and second moments of the central values L (1/2, ƒ), which can be applied to prove that at least 33 % of L (1/2, ƒ) with ƒ ≤ T are non-vanishing as T → ∞. Our main tools are the spherical Kuznetsov trace formula and the Voronoï summation formula over imaginary quadratic fields. [ABSTRACT FROM AUTHOR]

A spherical conical metric g on a surface Σ is a metric of constant curvature 1 with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds 2π. The eigenfunctions of the Friedrichs Laplacian Δg with eigenvalue λ = 2 play a special role in this problem, as they represent local obstructions to deformations of the metric g in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eigenfunctions. In the first part we establish a new criterion for the existence of 2-eigenfunctions, given in terms of a certain meromorphic data on Σ. As an application we give a description of all 2-eigenfunctions for metrics on the sphere with at most three conical singularities. The second part is an algebraic construction of metrics with large number of 2-eigenfunctions via the deformation of multivalued harmonic maps. We provide new explicit examples of metrics with many 2-eigenfunctions via both approaches, and describe the general algorithm to find metrics with arbitrarily large number of 2-eigenfunctions. [ABSTRACT FROM AUTHOR]

In this paper we prove that the Cauchy problem of the Muskat equation is wellposed locally in time for any initial data in ⋅ C1(Rd) ∩ L2(Rd). [ABSTRACT FROM AUTHOR]

In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general Rd-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates either in the central limit theorem for the Wasserstein distance, or in the strong approximation. A part of this paper is devoted to circumvent this issue. We then exhibit near optimal rates both in the central limit theorem in terms of the Wasserstein distance and in the almost sure invariance principle for Rd-valued martingales with stationary increments having moments of order p ∈ (2, 3] (the case of sequences of reversed martingale differences is also considered). Note also that, as an application of our results for general Rd-valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups (like for instance GLd(R)). [ABSTRACT FROM AUTHOR]

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on R+ or R with values in a general Riemannian manifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper of Röckner and the second, third, and fourth authors [SIAM J. Math. Anal. 52 (2020), pp. 2237-2274] on finite volume case. Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution of the Ricci curvature is strictly positive and the non-ergodicity of the process if the sectional curvature is negative. [ABSTRACT FROM AUTHOR]

In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing so, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding ''finite'' case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calderón-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm classes. The same occurs with the multilinear Calderón-Zygmund operators, the bilinear Hilbert transform, and the corresponding commutators with BMO functions. Extrapolation along with the already established weighted norm inequalities easily give scalar and vector-valued inequalities with multilinear weights and these include the end-point cases. [ABSTRACT FROM AUTHOR]

A Cohen-Macaulay local ring R satisfies trivial vanishing if ToriR(M,N) = 0 for all large i implies that M or N has finite projective dimension. If R satisfies trivial vanishing, then we also have that ExiR(M,N) = 0 for all large i implies that M has finite projective dimension or N has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with results of Gasharov and Peeva, provide sufficient conditions for R to satisfy trivial vanishing; we provide sharpened conditions when R is generalized Golod. Our methods allow us to settle the Auslander-Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes-Huneke, and Jorgensen-Leuschke. [ABSTRACT FROM AUTHOR]

We correct a mistake that appeared in the first section of the original article, which appeared in Tran. Amer. Math. Soc. 370 (2018), 99-130. Namely, Corollary 1.16 was false as stated and was subsequently used in later proofs in the paper. In this note it is argued that all the relevant statements after Corollary 1.16 can be saved with at most minor modifications. In particular, all the main results of the original paper remain valid as stated, but some intermediate claims are slightly modified or proved more directly without Corollary 1.16. [ABSTRACT FROM AUTHOR]

In a companion paper [arXiv:1903.00833], we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the above-mentioned paper. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time. [ABSTRACT FROM AUTHOR]

The problem of toroidalization is to construct a toroidal lifting of a dominant morphism \varphi :X\to Y of algebraic varieties by blowing up in the target and domain. This paper contains a solution to this problem when \varphi is locally toroidal. [ABSTRACT FROM AUTHOR]

This paper investigates the combinatorial \alpha-curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial \alpha-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization theorem for classical combinatorial curvature [J. Differential Geom. 109 (2018), pp. 431–466]. We further introduce combinatorial \alpha-Yamabe flow and combinatorial \alpha-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial \alpha-curvatures. To handle the potential singularities along the combinatorial curvature flows, we do surgery along the flows by edge flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu [J. Differential Geom. 109 (2018), pp. 431–466], we prove the longtime existence and convergence of combinatorial \alpha-Yamabe flow and combinatorial \alpha-Calabi flow with surgery, which provide effective algorithms for finding piecewise hyperbolic metrics with prescribed combinatorial \alpha-curvatures. [ABSTRACT FROM AUTHOR]

This paper deals with non-Archimedean representations of punctured surface groups in \operatorname {PGL}_3, associated actions on (not necessarily discrete) Euclidean buildings of type A_2, and degenerations of convex real projective structures on surfaces. The main result is that, under good conditions on Fock-Goncharov generalized shear parameters, non-Archimedean representations acting on the Euclidean building preserve a cocompact weakly convex subspace, which is part flat surface and part tree. In particular the eigenvalue and length(s) spectra are given by an explicit finite A_2-complex. We use this result to describe degenerations of convex real projective structures on surfaces for an open cone of parameters. The main tool is a geometric interpretation of Fock-Goncharov parameters in A_2-buildings. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a special function on the Drinfeld period domain \Omega ^{r} for r\geq 2 which gives the false Eisenstein series of Gekeler when r=2. We also study its functional equation and relation with quasi-periodic functions of a Drinfeld module as well as transcendence of its values at CM points. [ABSTRACT FROM AUTHOR]

Let p\equiv 8\mod 9 be a prime. In this paper we give a sufficient condition such that at least one of p and p^2 is the sum of two rational cubes. This is the first general result on the 8 case of the so-called Sylvester conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we present a general construction of 3-transposition groups as automorphism groups of vertex operator algebras. Applying to the moonshine vertex operator algebra, we establish the Conway-Miyamoto correspondences between Fischer 3-transposition groups \mathrm {Fi}_{23} and \mathrm {Fi}_{22} and c=25/28 and c=11/12 Virasoro vectors of subalgebras of the moonshine vertex operator algebra. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to introduce the so-called ideal approximation theory into higher homological algebra. To this end, we introduce some important notions from approximation theory into the theory of n-exact categories and prove some results. In particular, the higher version of notions such as ideal cotorsion pairs, phantom ideals, Salce's Lemma and Wakamatsu's Lemma for ideals are introduced and studied. Our results motivate the definitions and show that n-exact categories are the appropriate context for the study of higher ideal approximation theory. [ABSTRACT FROM AUTHOR]

A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 136 (1969), pp. 269–286] says that given an integer n \geqslant 1, if the entrywise application of a smooth function f : (0,\infty) \to \mathbb {R} preserves the set of n \times n positive semidefinite matrices with positive entries, then f and its first n-1 derivatives are non-negative on (0,\infty). In a recent joint work with Belton–Guillot–Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg–Rudin characterization of dimension-free positivity preservers [Duke Math. J. 26 (1959), pp. 617–622; Duke Math. J. 9 (1942), pp. 96–108]. In recent works with Belton–Guillot–Putinar [Adv. Math. 298 (2016), pp. 325–368] and with Tao [Amer. J. Math. 143 (2021), pp. 1863-1929] we used local, real-analytic versions at the origin of the Horn–Loewner condition, and discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a Master Theorem (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first n nonzero Taylor coefficients at individual points rather than on all of (0,\infty). A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials arise naturally from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy's (1841) determinant (whose matrix entries are geometric series 1 / (1 - u_j v_k)), as well as of a determinant of Frobenius [J. Reine Angew. Math. 93 (1882), pp. 53–68] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings. [ABSTRACT FROM AUTHOR]

In this paper we give a version of Bergman's diamond lemma which applies to certain monoidal categories presented by generators and relations. In particular, it applies to: the Coxeter presentation of the symmetric groups, the quiver Hecke algebras of Khovanov-Lauda-Rouquier, the Webster tensor product algebras, and various generalizations of these. We also give an extension of Manin-Schechtmann theory to non-reduced expressions. [ABSTRACT FROM AUTHOR]

We prove that it is relatively consistent with \mathsf {ZFC} that every maximal ideal is Katětov above of all F_\sigma ideals. In particular, we prove that it is consistent that there is no Hausdorff ultrafilter. The main theorem answers questions from Mauro Di Nasso and Marco Forti [Proc. Amer. Math. Soc. 134 (2006), pp. 1809–1818], Jana Flašková [WDS'05 proceedings of contributed papers: part I - mathematics and computer sciences, 2005; Comment. Math. Univ. Carolin. 47 (2006), pp. 617–621; 10th Asian logic conference, World Sci. Publ., Hackensack, NJ, 2010], Osvaldo Guzmán and Michael Hrušák [Topology Appl. 259 (2019), pp. 242–250], and Mauro Di Nasso and Marco Forti [Logic and its applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 2005], and gives a different model for a question from Michael Benedikt [J. Symb. Log. 63 (1998), pp. 638–662], which was originally solved by S. Shelah [Logic colloquium '95 (Haifa), lecture notes logic, Springer, Berlin, 1998]. [ABSTRACT FROM AUTHOR]

Neretin in [Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), pp. 95–108] constructed an analogue of the Hua measures on the infinite p-adic matrices \mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right). Bufetov and Qiu in [Compos. Math. 153 (2017), pp. 2482–2533] classified the ergodic measures on \mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right) that are invariant under the natural action of \mathrm {GL}(\infty,\mathbb {Z}_p)\times \mathrm {GL}(\infty,\mathbb {Z}_p). In this paper we solve the problem of ergodic decomposition for the p-adic Hua measures introduced by Neretin. We prove that the probability measure governing the ergodic decomposition has an explicit expression which identifies it with a Hall-Littlewood measure on partitions. Our arguments involve certain Markov chains. [ABSTRACT FROM AUTHOR]

Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation \mathcal {A}: \mathbb {R}[t] \to \mathbb {R}[t] defined by \mathcal {A}(t^n) = A_n(t), where A_n(t) denotes the n-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator \mathcal {A}, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials. [ABSTRACT FROM AUTHOR]

An important result with many applications in the theory of finite groups is the following: Let S \not =1 be a finite p-group for some prime p. Then S contains a characteristic subgroup W(S) \not = 1 with the property that W(S) is normal in every finite group G of characteristic p with S \in Syl_p(G) that does not possess a section isomorphic to the semi-direct product of SL_{2}(p) with its natural module. For odd primes, this was first established by Glauberman with his celebrated ZJ-Theorem. The case p=2 proved more elusive and was only established much later by the second author. Unlike the ZJ-Theorem, it was not possible to give an explicit description of the subgroup W(S) in terms of the internal structure of S. In this paper we introduce the notion of "almost quadratic action" and show that in each finite p-group S there exists a unique maximal elementary abelian characteristic subgroup W(S) which contains \Omega _1(Z(S)) and does not allow non-trivial almost quadratic action from S. We show that W(S) is normal in all finite groups G of characteristic p with S \in Syl_p(G) provided that G does not possess a section isomorphic to the semi-direct product of SL_{2}(p) with its natural module. This provides a unified approach for all primes p and gives a concrete description of the subgroup W(S) in terms of the internal structure of S. [ABSTRACT FROM AUTHOR]

In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine approximation to hold almost surely for stochastically self-similar and self-affine random iterated function systems. [ABSTRACT FROM AUTHOR]

We discuss various questions of the following kind: for a continuous map X \to Y from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The d-width measures how well a space can be approximated by a d-dimensional complex. The results of this paper include the following. Any piecewise linear map f: [0,1]^{m+2} \to Y^m from the unit euclidean (m+2)-cube to an m-polyhedron must have a fiber of 1-width at least \frac {1}{2\beta m +m^2 + m + 1}, where \beta = \sup _{y\in Y} rkH_1(f^{-1}(y)) measures the topological complexity of the map. There exists a piecewise smooth map X^{3m+1} \to \mathbb {R}^m, with X a riemannian (3m+1)-manifold of large 3m-width, and with all fibers being topological (2m+1)-balls of arbitrarily small (m+1)-width. [ABSTRACT FROM AUTHOR]

We investigate criteria for circle packing (CP) types of disk triangulation graphs embedded into simply connected domains in C. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem involving boundary turns, we show that a disk triangulation graph is CP parabolic if ∑n=1∞ 1/∑j=0n-1 (kj + 6) = ∞, where kn is the degree excess sequence defined by kn = ∑v∈Bn (deg v − 6) for combinatorial balls Bn of radius n and centered at a fixed vertex. It is also shown that the simple random walk on a disk triangulation graph is recurrent if ∑n=1∞ 1/∑j=0n-1 (kj + 6) + ∑j=0n (kj + 6) = ∞. These criteria are sharp, and generalize a conjecture by He and Schramm in their paper from 1995, which was later proved by Repp in 2001. We also give several criteria for CP hyperbolicity, one of which generalizes a theorem of He and Schramm, and present a necessary and sufficient condition for CP types of layered circle packings generalizing and confirming a criterion given by Siders in 1998. [ABSTRACT FROM AUTHOR]

This paper builds model-theoretic tools to detect changes in complexity among the simple theories. We develop a generalization of dividing, called shearing, which depends on a so-called context c. This leads to defining c-superstability, a syntactical notion, which includes supersimplicity as a special case. The main result is a separation theorem showing that for any countable context c and any two theories T1, T2 such that T1 is c-superstable and T2 is c-unsuperstable, and for arbitrarily large μ, it is possible to build models of any theory interpreting both T1 and T2 whose restriction to τ (T1) is μ-saturated and whose restriction to τ (T2) is not ℵ1-saturated. (This suggests "c-superstable" is really a dividing line.) The proof uses generalized Ehrenfeucht-Mostowski models, and along the way, we clarify the use of these techniques to realize certain types while omitting others. In some sense, shearing allows us to study the interaction of complexity coming from the usual notion of dividing in simple theories and the more combinatorial complexity detected by the general definition. This work is inspired by our recent progress on Keisler's order, but does not use ultrafilters, rather aiming to build up the internal model theory of these classes. [ABSTRACT FROM AUTHOR]

In this paper, we construct a pair of two regular Cantor sets in the Euclidean space of dimension at least two which exhibits C1-stable intersection, and a hyperbolic basic set which exhibits C2-robust homoclinic tangency of the largest codimension for any manifold of dimension at least four. The former implies that an analog of Moreira's theorem on Cantor sets in the real line does not hold in higher dimension. The latter solves a question posed by Barrientos and Raibekas. [ABSTRACT FROM AUTHOR]

We established in our recent work [Adv. Math. 319 (2017), pp. 567-598] the sharp Hardy-Adams inequalities for the bi-Laplacian-Beltrami operator (−ΔH2 on the hyperbolic space B4 in dimension four and for n/2th (integer) power of the Laplacian operator (−ΔH)n/2 on B2 of any even dimension n > 4 in [Adv. Math., 2018]. The proofs of these inequalities rely on the special structures of the integer (n/2th) power and the even dimension n. These are the borderline cases of the sharp higher order Hardy-Sobolev-Maz'ya inequalities by the authors [Amer. J. Math., 2019]. Thus, it remains open if such sharp Hardy-Adams inequalities hold for hyperbolic spaces Bn of any odd dimension n and any fractional power α of the Laplacian (−ΔH)α. The purpose of this paper is to settle the remaining cases completely. One of the main purposes of this paper is to establish sharp Adams type inequalities on Sobolev spaces Wα,n/α(Bn) of any positive fractional order \alpha 4 and n is even. Fourier analysis on hyperbolic spaces play an important role in establishing the Adams and Hardy-Adams inequalities of any fractional order derivatives on hyperbolic spaces of any dimension. We also establish the fractional order Sobolev embedding theorems on hyperbolic spaces which are of their own independent interest. [ABSTRACT FROM AUTHOR]

In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [Compos. Math. 154 (2018), pp. 2403-2425]. [ABSTRACT FROM AUTHOR]

Let L be a fixed branch – that is, an irreducible germ of curve – on a normal surface singularity X. If A, B are two other branches, define uL(A,B) := (L·A)(L·B)/A·B, where A · B denotes the intersection number of A and B. Cal lX arborescent if all the dual graphs of its good resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of Płoski by proving that whenever X is arborescent, the function uL is an ultrametric on the set of branches on X different from L. In the present paper we prove that, conversely, if uL is an ultrametric, then X is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on X, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which uL is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing L to be an arbitrary semivaluation on X and by defining uL on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if X is arborescent, and without any restriction on X we exhibit special subspaces of the space of semivaluations in restriction to which uL is still an ultrametric. [ABSTRACT FROM AUTHOR]

In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and we prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove a kind of stability of the isomorphism classes of the nerves of good coverings in the noncollapsing case. In the proof, we need a version of Perelman’s fibration theorem, which is also proved in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider Chow-Luo's combinatorial Ricci flow in the inversive distance circle packing setting. Although a solution to the flow may develop singularities in finite time, we can always extend the solution so as it exists for all time and converges exponentially fast to a unique packing with prescribed cone angles. We also give partial results on the range of all attainable cone angles, which generalize the classical Andreev-Thurston theorem. This paper opens a program about the study of the deformations of discrete metrics and discrete curvatures. [ABSTRACT FROM AUTHOR]

We derive a Ray–Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both of these cases. The Ray–Knight type theorem is applied to study integral functionals of the local time process of the skew Brownian motion. In particular, we determine the distribution of the maximum of the local time process up to a fixed time, which can be seen as the main new result of the paper. [ABSTRACT FROM AUTHOR]

We introduce and develop a theory of limits for sequences of sparse graphs based on Lp graphons, which generalizes both the existing L∞ theory of dense graph limits and its extension by Bollobás and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the Lp theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper. [ABSTRACT FROM AUTHOR]

The main inspiration for this paper is a paper by Elek where he introduces combinatorial cost for graph sequences. We show that having cost equal to 1 and hyperfiniteness are coarse invariants. We also show that “cost−1” for box spaces behaves multiplicatively when taking subgroups. We show that graph sequences coming from Farber sequences of a group have property A if and only if the group is amenable. The same is true for hyperfiniteness. This generalises a theorem by Elek. Furthermore we optimise this result when Farber sequences are replaced by sofic approximations. In doing so we introduce a new concept: property almost-A. [ABSTRACT FROM AUTHOR]

In analogy with the study of representations of GL2n(F) distinguished by Sp2n(F), where F is a local field, we study representations of U2n(F) distinguished by Sp2n(F) in this paper. (Only quasisplit unitary groups are considered in this paper since they are the only ones which contain Sp2n(F).) We prove that there are no cuspidal representations of U2n(F) distinguished by Sp2n(F) for F a nonarchimedean local field. We also prove the corresponding global theorem that there are no cuspidal automorphic representations of U2n(Ak) with nonzero period integral on Sp2n(k)\ Sp2n(Ak) for k any number field or a function field. We completely classify representations of quasisplit unitary groups in four variables over local and global fields with nontrivial symplectic periods using methods of theta correspondence. We propose a conjectural answer for the classification of all representations of a quasisplit unitary group distinguished by Sp2n(F). [ABSTRACT FROM AUTHOR]

For fixed r\geq 3 and n divisible by r, let \boldsymbol {\mathcal {H}}=\boldsymbol {\mathcal {H}}^r_{n,M} be the random M-edge r-graph on V=\{1,\ldots,n\}; that is, \boldsymbol {\mathcal {H}} is chosen uniformly from the M-subsets of \mathcal {K}≔{{V}\choose {r}} (≔\{\text {r-subsets ofV}\}). Shamir's Problem (circa 1980) asks, roughly, \[ \text {\emph {for what \boldsymbol {M=M(n)} is }} \boldsymbol {\mathcal {H}} \text {\emph { likely to contain a perfect matching}} \] (that is, n/r disjoint r-sets)? In 2008 Johansson, Vu and the author showed that this is true for M>C_rn\log n. More recently the author proved the asymptotically correct version of that result: for fixed C> 1/r and M> Cn\log n, \[ \mathbb {P}(\boldsymbol {\mathcal {H}} ~\text {\emph {contains a perfect matching}})\rightarrow 1 \,\,\, \text {\emph {as} n\rightarrow \infty }. \] The present work completes a proof, begun in that recent paper, of the definitive "hitting time" statement: Theorem. If A_1, \ldots ~ is a uniform permutation of \mathcal {K}, \boldsymbol {\mathcal {H}}_t=\{A_1,\ldots,A_t\}, and \[ T=\min \{t:A_1\cup \cdots \cup A_t=V\}, \] then \mathbb {P}(\boldsymbol {\mathcal {H}}_T ~\text {contains a perfect matching})\rightarrow 1 \,\,\, \text {\emph {as}n→∞}. [ABSTRACT FROM AUTHOR]

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares (n^2)_{n\geq 0} and primes (p_n)_{n\geq 1} exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lemańczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function. [ABSTRACT FROM AUTHOR]

A graph X is defined inductively to be (a_0,\dots,a_{n-1})-regular if X is a_0-regular and for every vertex v of X, the sphere of radius 1 around v is an (a_1,\dots,a_{n-1})-regular graph. Such a graph X is said to be highly regular (HR) of level n if a_{n-1}\neq 0. Chapman, Linial and Peled [Combinatorica 40 (2020), pp. 473–509] studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally", and asked about the existence of HR-graphs of level 3. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can be used to construct such graphs. Given a Coxeter system (W,S) and a subset M of S, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope \mathcal {P}_{W,M}, which form an infinite family of expander graphs when (W,S) is indefinite and \mathcal {P}_{W,M} has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of (W,S). The expansion stems from applying superapproximation to the congruence subgroups of the linear group W. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled. [ABSTRACT FROM AUTHOR]

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler–Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that this automorphic data is generically rigid (in the sense of Zhiwei Yun) and identify the resulting Hecke eigenvalue with hypergeometric sheaves. The definition of hypergeometric automorphic data in the tame case involves the mirabolic subgroup, while in the wild case, semistable (but not necessarily stable) vectors coming from principal gradings intervene. [ABSTRACT FROM AUTHOR]

In this paper we introduce the local Nori fundamental group scheme of a reduced scheme or algebraic stack over a perfect field k. We give particular attention to the case of fields: to any field extension K/k we attach a pro-local group scheme over k. We show how this group has many analogies, but also some crucial differences, with the absolute Galois group. We propose two conjectures, analogous to the classical Neukirch-Uchida Theorem and Abhyankar Conjecture, providing some evidence in their favor. Finally we show that the local fundamental group of a normal variety is a quotient of the local fundamental group of an open, of its generic point (as it happens for the étale fundamental group) and even of any smooth neighborhood. [ABSTRACT FROM AUTHOR]