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This paper is the second in a series of four papers aiming to describe the (almost integral) Chow ring of \overline {\mathcal {M}}_3, the moduli stack of stable curves of genus 3. In this paper, we introduce the moduli stack \widetilde {\mathcal {H}}_g^r of hyperelliptic A_r-stable curves and generalize the theory of hyperelliptic stable curves to hyperelliptic A_r-stable curves. In particular, we prove that \widetilde {\mathcal {H}}_g^r is a smooth algebraic stack that can be described using cyclic covers of twisted curves of genus 0 and it embeds in \widetilde {\mathcal M}_g^r (the moduli stack of A_r-stable curves) as the closure of the moduli stack of smooth hyperelliptic curves. [ABSTRACT FROM AUTHOR]

Wang and Ye [Adv. Math. 230 (2012), pp. 294–320] prove a Hardy–Moser–Trudinger inequality in dimension two which improves both the classical Moser–Trudinger inequality and the classical Hardy inequality in the unit disc. In this paper, we generalize their result both to the higher dimensional unit ball and to the singular weighted cases as well. More precisely, we prove that \[ \sup _{\substack {u\in W^{1,n}_0(\mathbb {B}^n),\\ \int _{\mathbb {B}^n} |\nabla u|^n dx -\left (\frac {2(n-1)}n\right)^n \int _{\mathbb {B}^n} \frac {|u|^n}{(1-|x|^2)^n} dx \leq 1}}\int _{\mathbb {B}^n} e^{(1-\frac \beta n)\alpha _n |u|^{\frac n{n-1}}} |x|^{-\beta } dx \leq C \] for any \beta \in [0,n), n\geq 3 where \alpha _n = n \omega _{n-1}^{\frac 1{n-1}} and \omega _{n-1} is the surface area of the n-1 dimensional unit sphere. The proof of Wang and Ye is based on the blow-up analysis method which seems not work in the higher dimensions. In this paper, we propose a new approach based on the method of transplantation of Green's functions to prove our inequality. As a consequence, we obtain a singular Moser–Trudinger inequality in the hyperbolic spaces which confirms affirmatively a conjecture by Mancini, Sandeep and Tintarev [Adv. Nonlinear Anal., 2 (2013), pp. 309–324]. [ABSTRACT FROM AUTHOR]

In this paper, we prove several rigidity results for compact initial data sets, in both the boundary and no boundary cases. In particular, under natural energy, boundary, and topological conditions, we obtain a global version of the main result by Galloway and Mendes [Comm. Anal. Geom. 26 (2018), pp. 63–83]. We also obtain some extensions of results by Eichmair, Galloway, and Mendes [Comm. Math. Phys. 386 (2021), pp. 253–268]. A number of examples are given in order to illustrate some of the results presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper we use the Riemann zeta distribution to give a new proof of the Erdös-Kac Central Limit Theorem. That is, if \zeta (s)=\sum _{n\ge 1} \frac {1}{n^s}, s>1, then we consider the random variable X_s with P(X_s=n)=\frac {1}{\zeta (s)n^s}, n\ge 1. In an earlier paper, the first author and Adrien Peltzer derived the analog of the Erdös-Kac Central Limit Theorem (CLT) for the number of distinct prime factors, \omega (X_s), of X_s, as s\searrow 1. In this paper we show, by means of a Tauberian Theorem, how to obtain the Central Limit Theorem of Erdös-Kac for the uniform distribution from the result for the random variable X_s. We also apply the technique to the number of distinct prime divisors of X_s that lie in an arithmetic sequence and a local CLT of the type proved by Dixit and Murty [Hardy-Ramanujan J. 43 (2020), 17–23] as well a version of the CLT for irreducible divisors of a monic polynomial over a finite field. [ABSTRACT FROM AUTHOR]

This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on \mathbb {R}^n. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on \mathbb {R}^n. [ABSTRACT FROM AUTHOR]

In this paper, we study one-dimensional hyperbolic Anderson models (HAM) driven by space-time pure-jump Lévy white noise in a finite-variance setting. Motivated by recent active research on limit theorems for stochastic partial differential equations driven by Gaussian noises, we present the first study in this Lévy setting. In particular, we first establish the spatial ergodicity of the solution and then a quantitative central limit theorem (CLT) for the spatial averages of the solution to HAM in both Wasserstein distance and Kolmogorov distance, with the same rate of convergence. To achieve the first goal (i.e. spatial ergodicity), we exploit some basic properties of the solution and apply a Poincaré inequality in the Poisson setting, which requires delicate moment estimates on the Malliavin derivatives of the solution. Such moment estimates are obtained in a soft manner by observing a natural connection between the Malliavin derivatives of HAM and a HAM with Dirac delta velocity. To achieve the second goal (i.e. CLT), we need two key ingredients: (i) a univariate second-order Poincaré inequality in the Poisson setting that goes back to Last, Peccati, and Schulte (Probab. Theory Related Fields, 2016) and has been recently improved by Trauthwein (arXiv:2212.03782); (ii) aforementioned moment estimates of Malliavin derivatives up to second order. We also establish a corresponding functional CLT by (a) showing the convergence in finite-dimensional distributions and (b) verifying Kolmogorov's tightness criterion. Part (a) is made possible by a linearization trick and the univariate second-order Poincaré inequality, while part (b) follows from a standard moment estimate with an application of Rosenthal's inequality. [ABSTRACT FROM AUTHOR]

It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the q-Morris constant term identity[J. Combin. Theory Ser. A 81 (1998), pp. 69–87]. This conjecture was proved and extended by Károlyi, Nagy, Petrov, and Volkov (KNPV) in 2015 [Adv. Math. 277 (2015), pp. 252–282]. In this paper, we obtain two symmetric function generalizations of the q-Baker–Forrester ex-conjecture. These include: (i) a q-Baker–Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's result. [ABSTRACT FROM AUTHOR]

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, L^{p}-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures. [ABSTRACT FROM AUTHOR]

In this paper, we study, for functionals having a minimax geometry on a constraint, the existence of bounded Palais-Smale sequences carrying Morse index type information. [ABSTRACT FROM AUTHOR]

This paper is to investigate the spectral properties of separable covariance matrices under elliptical populations. The separable covariance matrix model can handle both cross-row and cross-column correlations thus gain more popularity recently. Under the high-dimensional setting where the dimension p and the sample size n tend to infinity proportionally, we find the limit of the empirical spectral distribution and establish the central limit theorems (CLT) for linear spectral statistics of such kinds of sample covariance matrices. Some applications of our established CLT are also given. [ABSTRACT FROM AUTHOR]

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds (M^{n},g) with boundary and with dimension n<8 that was established by McCormick [Proc. Amer. Math. Soc. 146 (2018), pp. 4039–4046]. First, we show that any asymptotically flat static (M^{n},g) which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems. As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension n < 8 under the natural condition of Schwarzschild stability. This generalizes an earlier result of Miao [Classical Quantum Gravity 22 (2005), pp. L53–L59]. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes, e.g. see V. Agostiniani and L. Mazzieri [Comm. Math. Phys. 355 (2017), pp. 261–301], C. Cederbaum and G. J. Galloway [J. Math. Phys. 62 (2021), p. 22], and S. Raulot [Classical Quantum Gravity 38 (2021), p. 22]. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a poset isomorphism between two support \tau-tilting posets. We take several information from combinatorial properties of support \tau-tilting posets and give an analogue of Happel-Unger's reconstruction theorem. [ABSTRACT FROM AUTHOR]

Let G be a reductive group over a field of prime characteristic. An indecomposable tilting module for G whose highest weight lies above the Steinberg weight has a character that is divisible by the Steinberg character. The resulting "Steinberg quotient" carries important information about G-modules, and in previous work we studied patterns in the weight multiplicities of these characters. In this paper we broaden our scope to include quantum Steinberg quotients, and show how the multiplicities in these characters relate to algebraic Steinberg quotients, Weyl characters, and evaluations of Kazhdan-Lusztig polynomials. We give an explicit algorithm for computing minimal characters that possess a key attribute of Steinberg quotients. We provide computations which show that these minimal characters are not always equal to quantum Steinberg quotients, but are close in several nontrivial cases. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE. [ABSTRACT FROM AUTHOR]

Let \mu and \nu be probability measures in the complex plane, and let p and q be independent random polynomials of degree n, whose roots are chosen independently from \mu and \nu, respectively. Under assumptions on the measures \mu and \nu, the limiting distribution for the zeros of the sum p+q was computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021), p. 124719] as n \to \infty. In this paper, we generalize and extend this result to the case where p and q have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of \mu and \nu, scaled by the limiting ratio of the degrees of p and q. Additionally, our approach provides a complete description of the limiting distribution for the zeros of p + q for any pair of measures \mu and \nu, with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment. [ABSTRACT FROM AUTHOR]

A chain complex can be viewed as a representation of a certain self-injective quiver with relations, Q. To define Q, include a vertex q_n and an arrow q_n \xrightarrow {\partial } q_{n-1} for each integer n. The relations are \partial ^2 = 0. Replacing Q by a general self-injective quiver with relations, it turns out that some of the key properties of chain complexes generalise. Indeed, consider the representations of such a Q with values in {}_{A}\mspace {-1mu}\operatorname {Mod} where A is a ring. We showed in earlier work that these representations form the objects of the Q-shaped derived category , \mathcal {D}_{Q}(A), which is triangulated and generalises the classic derived category \mathcal {D}_{}(A). This follows ideas of Iyama and Minamoto. While \mathcal {D}_{Q}(A) has many good properties, it can also diverge dramatically from \mathcal {D}_{}(A). For instance, let Q be the quiver with one vertex q, one loop \partial, and the relation \partial ^2 = 0. By analogy with perfect complexes in the classic derived category, one may expect that a representation with a finitely generated free module placed at q is a compact object of \mathcal {D}_{Q}(A), but we will show that this is, in general, false. The purpose of this paper, then, is to compare and contrast \mathcal {D}_{Q}(A) and \mathcal {D}_{}(A) by investigating several key classes of objects: Perfect and strictly perfect, compact, fibrant, and cofibrant. [ABSTRACT FROM AUTHOR]

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C^\ast-algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C^\ast-algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space. [ABSTRACT FROM AUTHOR]

In this paper we explore the extent to which the algebraic structure of a monoid M determines the topologies on M that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or T_1 topologies; Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids and inverse monoids. If M is a topological monoid such that every homomorphism from M to a second countable topological monoid N is continuous, then we say that M has automatic continuity. We show that many well-known, and extensively studied, monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid \mathbb {N} ^\mathbb {N}; the full binary relation monoid B_{\mathbb {N}}; the partial transformation monoid P_{\mathbb {N}}; the symmetric inverse monoid I_{\mathbb {N}}; the monoid \operatorname {Inj}(\mathbb {N}) consisting of the injective transformations of \mathbb {N}; and the monoid C(2^{\mathbb {N}}) of continuous functions on the Cantor set 2^{\mathbb {N}}. The monoid \mathbb {N} ^\mathbb {N} can be equipped with the product topology, where the natural numbers \mathbb {N} have the discrete topology; this topology is referred to as the pointwise topology. We show that the pointwise topology on \mathbb {N} ^\mathbb {N}, and its analogue on P_{\mathbb {N}}, is the unique Polish semigroup topology on these monoids. The compact-open topology is the unique Polish semigroup topology on C(2 ^\mathbb {N}), and on the monoid C([0, 1] ^\mathbb {N}) of continuous functions on the Hilbert cube [0, 1] ^\mathbb {N}. The symmetric inverse monoid I_{\mathbb {N}} has at least 3 Polish semigroup topologies, but a unique Polish inverse semigroup topology. The full binary relation monoid B_{\mathbb {N}} has no Polish semigroup topologies, nor do the partition monoids. At the other extreme, \operatorname {Inj}(\mathbb {N}) and the monoid \operatorname {Surj}(\mathbb {N}) of all surjective transformations of \mathbb {N} each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on \mathbb {N} ^\mathbb {N}, P_{\mathbb {N}}, and \operatorname {Inj}(\mathbb {N}) coincide with the pointwise topology; and we characterise the Zariski topology on B_{\mathbb {N}}. Along the way we provide many additional results relating to the Markov topology, the small index property for monoids, and topological embeddings of semigroups in \mathbb {N}^{\mathbb {N}} and inverse monoids in I_{\mathbb {N}}. Finally, the techniques developed in this paper to prove the results about monoids are applied to function clones. In particular, we show that: the full function clone has a unique Polish topology; the Horn clone, the polymorphism clones of the Cantor set and the countably infinite atomless Boolean algebra all have automatic continuity with respect to second countable function clone topologies. [ABSTRACT FROM AUTHOR]

In this paper, we construct a compactification of the space of Bridgeland stability conditions on a smooth projective curve, as an analogue of Thurston compactifications in Teichmüller theory. In the case of elliptic curves, we compare our results with the classical one of the torus via homological mirror symmetry and give the Nielsen–Thurston classification of autoequivalences using the compactification. Furthermore, we observe an interesting phenomenon in the case of the projective line. [ABSTRACT FROM AUTHOR]

Given a fixed \alpha \in (0,1), we study the inverse problem of recovering the isometry class of a smooth closed and connected Riemannian manifold (M,g), given the knowledge of a source-to-solution map for the fractional Laplace equation (-\Delta _g)^\alpha u=f on the manifold subject to an arbitrarily small observation region \mathcal O where sources can be placed and solutions can be measured. This can be viewed as a non-local analogue of the well known anisotropic Calderón problem that is concerned with the limiting case \alpha =1. In this paper, we solve the non-local problem under the assumption that the a priori known observation region \mathcal O belongs to some Gevrey class while making no geometric assumptions on the inaccessible region of the manifold, namely M\setminus \mathcal O. Our proof is based on discovering a connection to a variant of Carlson's theorem in complex analysis that reduces the inverse problem to Gel'fand inverse spectral problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a constrained maximization problem related to planar vortex pairs with prescribed impulse. We prove existence, stability and asymptotic behavior for the maximizers, hence obtain a family of stable traveling vortex pairs approaching a pair of point vortices with equal magnitude and opposite signs. As a corollary, we get fine asymptotic estimates for Burton's vortex pairs with large impulse. We also consider a special non-concentrated case and prove a form of stability for the Chaplygin-Lamb dipole. [ABSTRACT FROM AUTHOR]

The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of C_{0}-semigroups \left (\mathcal {V}(t)\right)_{t \geqslant 0} in L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d}) governing conservative linear kinetic equations on the torus with general scattering kernel \boldsymbol {k}(v,v') and degenerate (i.e. not bounded away from zero) collision frequency \sigma (v)=\int _{\mathbb {R}^{d}}\boldsymbol {k}(v',v)\boldsymbol {m}(\mathrm {d}v'), (with \boldsymbol {m}(\mathrm {d}v) being absolutely continuous with respect to the Lebesgue measure). We show in particular that if N_{0} is the maximal integer s \geqslant 0 such that \begin{equation*} \frac {1}{\sigma (\cdot)}\int _{\mathbb {R}^{d}}\boldsymbol {k}(\cdot,v)\sigma ^{-s}(v)\boldsymbol {m}(\mathrm {d}v) \in L^{\infty }(\mathbb {R}^{d}), \end{equation*} then, for initial datum f such that \displaystyle \int _{\mathbb {T}^{d}\times \mathbb {R}^{d}}|f(x,v)|\sigma ^{-N_{0}}(v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v) <\infty it holds \begin{equation*} \left \|\mathcal {V}(t)f-\varrho _{f}\Psi \right \|_{L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})}=\dfrac {{\varepsilon }_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho _{f}≔\int _{\mathbb {R}^{d}}f(x,v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v), \end{equation*} where \Psi is the unique invariant density of \left (\mathcal {V}(t)\right)_{t \geqslant 0} and \lim _{t\to \infty }{\varepsilon }_{f}(t)=0. We in particular provide a new criteria of the existence of invariant density. The proof relies on the explicit computation of the time decay of each term of the Dyson-Phillips expansion of \left (\mathcal {V}(t)\right)_{t \geqslant 0} and on suitable smoothness and integrability properties of the trace on the imaginary axis of Laplace transform of remainders of large order of this Dyson-Phillips expansion. Our construction resorts also on collective compactness arguments and provides various technical results of independent interest. Finally, as a by-product of our analysis, we derive essentially sharp "subgeometric" convergence rate for Markov semigroups associated to general transition kernels. [ABSTRACT FROM AUTHOR]

In this paper we prove that among all convex domains of the plane with two axes of symmetry, the maximizer of the first non-trivial Neumann eigenvalue \mu _1 with perimeter constraint is achieved by the square and the equilateral triangle. Part of the result follows from a new general bound on \mu _1 involving the minimal width over the area. Our main result partially answers to a question addressed in 2009 by R. S. Laugesen, I. Polterovich, and B. A. Siudeja. [ABSTRACT FROM AUTHOR]

The paper explores the symbiotic relation between the Navier-Stokes equations and the associated stochastic cascades. Specifically, we examine how some well-known existence and uniqueness results for the Navier-Stokes equations can inform about the probabilistic features of the associated stochastic cascades, and how some probabilistic features of the stochastic cascades can, in turn, inform about the existence and uniqueness (or the lack thereof) of solutions. Our method of incorporating the stochastic explosion gives a simpler and more natural method to construct the solution compared to the original construction by Le Jan and Sznitman. This new stochastic construction is then used to show the finite-time blowup, non-existence of minimal blowup data, and non-uniqueness of the initial value problem for the Montgomery-Smith equation. We exploit symmetry properties inherent in our construction to give a simple proof of the global well-posedness results for small initial data in scale-critical Fourier-Besov spaces. We also obtain the pointwise convergence of the Picard iteration associated with the Fourier-transformed Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

The exponential random graph model (ERGM) is a central object in the study of clustering properties in social networks as well as canonical ensembles in statistical physics. Despite some breakthrough works in the mathematical understanding of ERGM, most notably by Bhamidi, Bresler, and Sly [Ann. Appl. Probab. 21 (2011), pp. 2146–2170] through the analysis of a natural Heat-bath Glauber dynamics, and by Chatterjee and Diaconis [Ann. Statist. 41 (2013), pp. 2428–2461; Eldan [Geom. Funct. Anal. 28 (2018), pp. 1548–1596; and Eldan and Gross [Ann. Appl. Probab. 28 (2018), pp. 3698–3735] via a large deviation theoretic perspective, several basic questions have remained unanswered owing to the lack of exact solvability unlike the much studied Curie-Weiss model (Ising model on the complete graph). In this paper, we establish a series of new concentration of measure results for the ERGM throughout the entire sub-critical phase , including a Poincaré inequality, Gaussian concentration for Lipschitz functions, and a central limit theorem. In addition, a new proof of a quantitative bound on the W_1-Wasserstein distance to Erdős-Rényi graphs, previously obtained by Reinert and Ross [Ann. Appl. Probab. 29 (2019), pp. 3201–3229], is also presented. The arguments rely on translating temporal mixing properties of Glauber dynamics to static spatial mixing properties of the equilibrium measure and have the potential of being useful in proving similar functional inequalities for other Gibbsian systems beyond the perturbative regime. [ABSTRACT FROM AUTHOR]

In this paper we study Cuntz–Pimsner algebras associated to 퐶*-correspondences over commutative \mathrm {C}^*-algebras from the point of view of the \mathrm {C}^*-algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite dimensional infinite compact metric space X twisted by a vector bundle, the resulting Cuntz–Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these \mathrm {C}^*-algebras by the Elliott invariant. This establishes a dichotomy: when the vector bundle has rank one, the Cuntz–Pimsner algebra has stable rank one. Otherwise, it is purely infinite. For a Cuntz–Pimsner algebra of a minimal homeomorphism of an infinite compact metric space X twisted by a line bundle over X, we introduce orbit-breaking subalgebras. With no assumptions on the dimension of X, we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of X is finite, they are furthermore \mathcal {Z}-stable and hence classified by the Elliott invariant. [ABSTRACT FROM AUTHOR]

In this paper we introduce a noncommutative analogue of the notion of linear system, which we call a helix \underline {\mathcal {L}} ≔(\mathcal {L}_{i})_{i \in \mathbb {Z}} in an abelian category \mathsf {C} over a quadratic \mathbb {Z}-indexed algebra A. We show that, under natural hypotheses, a helix induces a morphism of noncommutative spaces from Proj End(\underline {\mathcal {L}}) to Proj A. We construct examples of helices of vector bundles on elliptic curves generalizing the elliptic helices of line bundles constructed by Bondal-Polishchuk, where A is the quadratic part of B≔End(\underline {\mathcal {L}}). In this case, we identify B as the quotient of the Koszul algebra A by a normal family of regular elements of degree 3, and show that Proj B is a noncommutative elliptic curve in the sense of Polishchuk [J. Geom. Phys. 50 (2004), pp. 162–187]. One interprets this as embedding the noncommutative elliptic curve as a cubic divisor in some noncommutative projective plane, hence generalizing some well-known results of Artin-Tate-Van den Bergh. [ABSTRACT FROM AUTHOR]

Let X \subset \mathbb {P}^r be a linearly normal variety defined by a very ample line bundle L on a projective variety X. Recently it is shown by Kangjin Han, Wanseok Lee, Hyunsuk Moon, and Euisung Park [Compos. Math. 157 (2021), pp. 2001–2025] that there are many cases where (X,L) satisfies property \mathsf {QR} (3) in the sense that the homogeneous ideal I(X,L) of X is generated by quadratic polynomials of rank 3. The locus \Phi _3 (X,L) of rank 3 quadratic equations of X in \mathbb {P}\left (I(X,L)_2 \right) is a projective algebraic set, and property \mathsf {QR} (3) of (X,L) is equivalent to that \Phi _3 (X) is nondegenerate in \mathbb {P}\left (I(X)_2 \right). In this paper we study geometric structures of \Phi _3 (X,L) such as its minimal irreducible decomposition. Let \begin{equation*} \Sigma (X,L) \!=\! \{ (A,B) \mid A,B \!\in \! {Pic}(X),~L \!=\! A^2 \otimes B,~h^0 (X,A) \!\geq \! 2,~h^0 (X,B) \!\geq \! 1 \}. \end{equation*} We first construct a projective subvariety W(A,B) \subset \Phi _3 (X,L) for each (A,B) in \Sigma (X,L). Then we prove that the equality \begin{equation*} \Phi _3 (X,L) ~=~ \bigcup _{(A,B) \in \Sigma (X,L)} W(A,B) \end{equation*} holds when X is locally factorial. Thus this is an irreducible decomposition of \Phi _3 (X,L) when {Pic} (X) is finitely generated and hence \Sigma (X,L) is a finite set. Also we find a condition that the above irreducible decomposition is minimal. For example, it is a minimal irreducible decomposition of \Phi _3 (X,L) if {Pic}(X) is generated by a very ample line bundle. [ABSTRACT FROM AUTHOR]

Let V be a vertex operator algebra, k a positive integer and \sigma a permutation automorphism of the vertex operator algebra V^{\otimes k}. In this paper, we determine the fusion product of any V^{\otimes k}-module with any \sigma-twisted V^{\otimes k}-module. [ABSTRACT FROM AUTHOR]

In this paper, we study 4-dimensional complete hypersurfaces with w-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for 4-dimensional complete hypersurfaces with w-constant mean curvature. As a by-product, we give a new proof of the result of Deng-Gu-Wei [Adv. Math. 314 (2017), pp. 278–305] under the weaker topological condition. [ABSTRACT FROM AUTHOR]

In this paper, we study distributional properties of the sequence of partial quotients in the continued fraction expansion of fractions a/N, where N is fixed and a runs through the set of mod N residue classes which are coprime with N. Our methods cover statistics such as the sum of partial quotients, the maximal partial quotient, the empirical distribution of partial quotients, Dedekind sums, and much more. We prove a sharp concentration inequality for the sum of partial quotients, and sharp tail estimates for the maximal partial quotient and for Dedekind sums, all matching the tail behavior in the limit laws which are known under an extra averaging over the set of possible denominators N. We show that the distribution of partial quotients of reduced fractions with fixed denominator gives a very good fit to the Gauß–Kuzmin distribution. As corollaries we establish the existence of reduced fractions with a small sum of partial quotients resp. a small maximal partial quotient. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of an orthonormal basis in at least one orbit of every generic irreducible representation of a simply connected and connected nilpotent Lie group. Our result has a wide-ranging impact, encompassing all irreducible representations of a nilpotent Lie group that are square-integrable modulo its center. This resolves a fundamental open problem in time-frequency analysis and frame theory, originally posed by Karlheinz Gröchenig. The implications of our findings are significant and far-reaching. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions to the centroaffine Minkowski problem, namely the existence of closed convex hypersurfaces in the Euclidean space \mathbb {R}^{n+1} with prescribed centroaffine curvature. This problem can be described as a variational problem with a functional resembling Kantorovich's dual functional in optimal transportation. By using the Gauss curvature flow, we obtain new conditions for the existence of solutions to the problem. [ABSTRACT FROM AUTHOR]

In this paper we consider which families of finite simple groups G have the property that for each \epsilon > 0 there exists N > 0 such that, if |G| \ge N and S, T are normal subsets of G with at least \epsilon |G| elements each, then every non-trivial element of G is the product of an element of S and an element of T. We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form \mathrm {PSL}_n(q) where q is fixed and n\to \infty. However, in the case S=T and G alternating this holds with an explicit bound on N in terms of \epsilon. Related problems and applications are also discussed. In particular we show that, if w_1, w_2 are non-trivial words, G is a finite simple group of Lie type of bounded rank, and for g \in G, P_{w_1(G),w_2(G)}(g) denotes the probability that g_1g_2 = g where g_i \in w_i(G) are chosen uniformly and independently, then, as |G| \to \infty, the distribution P_{w_1(G),w_2(G)} tends to the uniform distribution on G with respect to the L^{\infty } norm. [ABSTRACT FROM AUTHOR]

In this paper we study \times _0-products of Lannér diagrams. We prove that every \times _0-product of at least four Lannér diagrams with at least one diagram of order \geq 3 is superhyperbolic. As a corollary, we obtain that known classifications exhaust all compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of simplices. We also consider compact hyperbolic Coxeter polytopes whose every Lannér subdiagram has order 2. The second result of this paper slightly improves recent Burcroff's upper bound on the dimension of such polytopes to 12. [ABSTRACT FROM AUTHOR]

This paper is part of a program to establish the existence theory for the conformally invariant Dirac equation \[ D_g\psi =f(x)|\psi |_g^{\frac 2{m-1}}\psi \] on a closed spin manifold (M,g) of dimension m\geq 2 with a fixed spin structure, where f:M\to \mathbb {R} is a given function. The study on such nonlinear equation is motivated by its important applications in Spin Geometry: when m=2, a solution corresponds to an isometric immersion of the universal covering \widetilde M into \mathbb {R}^3 with prescribed mean curvature f; meanwhile, for general dimensions and f\equiv constant, a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant. Comparing with the existing issues, the aim of this paper is to establish multiple existence results in a new geometric context, which have not been considered in the previous literature. Precisely, in order to examine the dependence of solutions of the aforementioned nonlinear Dirac equations on geometrical data, concrete analyses are made for two specific models on the manifold (S^m,g): the geometric potential f is a perturbation from constant with g=g_{S^m} being the canonical round metric; and f\equiv 1 with the metric g being a perturbation of g_{S^m} that is not conformally flat somewhere on S^m. The proof is variational: solutions of these problems are found as critical points of their corresponding energy functionals. The emphasis is that the solutions are always degenerate: they appear as critical manifolds of positive dimension. This is very different from most situations in elliptic PDEs and classical critical point theory. As corollaries of the existence results, multiple distinct embedded spheres in \mathbb {R}^3 with a common mean curvature are constructed, and furthermore, a strict inequality estimate for the Bär-Hijazi-Lott invariant on S^m, m\geq 4, is derived, which is the first result of this kind in the non-locally-conformally-flat setting. [ABSTRACT FROM AUTHOR]

We consider random orthonormal polynomials \begin{equation*} F_{n}(x)=\sum _{i=0}^{n}\xi _{i}p_{i}(x), \end{equation*} where \xi _{0}, ..., \xi _{n} are independent random variables with zero mean, unit variance and uniformly bounded (2+\varepsilon) moments, and (p_n)_{n=0}^{\infty } is the system of orthonormal polynomials with respect to a fixed compactly supported measure on the real line. Under mild technical assumptions satisfied by many classes of classical polynomial systems, we establish universality for the leading asymptotics of the average number of real roots of F_n, both globally and locally. Prior to this paper, these results were known only for random orthonormal polynomials with Gaussian coefficients (see D. D. Lubinsky, I. E. Pritsker, and X. Xie [Proc. Amer. Math. Soc. 144 (2016), pp. 1631–1642]) using the Kac-Rice formula, a method that does not extend to the generality of our paper. [ABSTRACT FROM AUTHOR]

We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. We prove that it is always possible to decompose the domain of such a mapping into pieces on which the mapping "behaves like a projection mapping" along with a "garbage set" that is arbitrarily small in an appropriate sense. Moreover, our control is quantitative, i.e., independent of both the particular mapping and the metric space it maps into. This improves a theorem of Azzam-Schul from the paper "Hard Sard", and answers a question left open in that paper. The proof uses ideas of quantitative differentiation, as well as a detailed study of how to supplement Lipschitz mappings by additional coordinates to form bi-Lipschitz mappings. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear Klein-Gordon equation \begin{align*} \partial _{tt}u-\Delta u+u=|u|^{p-1}u,\qquad t\in \mathbb {R},\ x\in \mathbb {R}^d, \end{align*} with 1 1, and d=1, in which case we prove that the standing wave solution u_\omega is orbitally unstable. [ABSTRACT FROM AUTHOR]

In this paper, we study a special type of compact Hermitian manifolds that are Strominger Kähler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler manifold. Previously, we have shown that any SKL manifold (M^n,g) is always pluriclosed, and when the manifold is compact and g is not Kähler, it cannot admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n=2, the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-Kähler SKL manifolds in dimension 3 and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, given any compact non-Kähler SKL manifold, its Kähler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal Kähler when n\geq 3, and the manifold does not admit any Hermitian symplectic metric. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the two-dimensional Dirac–Klein-Gordon system, which is a basic model in particle physics. We investigate the global behavior of small data solutions to this system in the case of a massive scalar field and a massless Dirac field. More precisely, our main result is twofold: (1) we show sharp time decay for the pointwise estimates of the solutions, which implies the asymptotic stability of this system; (2) we show the linear scattering result of this system which is a fundamental problem when it is viewed as dispersive equations. Our result is valid for general small, high-regular initial data, and in particular, there is no restriction on the support of the initial data. [ABSTRACT FROM AUTHOR]

Let f(z) be in 1+z\mathbb {Q}[[z]] and \mathcal {S} be an infinite set of prime numbers such that, for all p\in \mathcal {S}, we can reduce f(z) modulo p. We let f(z)_{\mid p} denote the reduction of f(z) modulo p. Generally, when f(z) is D-finite, f_{\mid p}(z) is algebraic over \mathbb {F}_p(z). It turns out that if f_{\mid p}(z) is a solution of a polynomial of the form X-A_p(z)X^{p^l}, we can use this type of equations to obtain results of transcendence and algebraic independence over \mathbb {Q}(z). In the present paper, we look for conditions on the differential operators annihilating f(z) to guarantee the existence of these particular equations. Suppose that f(z) is solution of a differential operator \mathcal {H}\in \mathbb {Q}(z)[d/dz] having a strong Frobenius structure for all p\in \mathcal {S} and we also suppose that f(z) annihilates a Fuchsian differential operator \mathcal {D}\in \mathbb {Q}(z)[d/dz] such that zero is a regular singular point of \mathcal {D} and the exponents of \mathcal {D} at zero are equal to zero. Our main result states that, for almost every prime p\in \mathcal {S}, f_{\mid p}(z) is solution of a polynomial of the form X-A_p(z)X^{p^l}, where A_p(z) is a rational function with coefficients in \mathbb {F}_p of height less than or equal to Cp^{2l} with C a positive constant that does not depend on p. We also study the algebraic independence of these power series over \mathbb {Q}(z). [ABSTRACT FROM AUTHOR]

In this paper, we prove the rational coefficient case of the global ACC for foliated threefolds. Specifically, we consider any lc foliated log Calabi-Yau triple (X,\mathcal {F},B) of dimension 3 whose coefficients belong to a set \Gamma of rational numbers satisfying the descending chain condition, and prove that the coefficients of B belong to a finite set depending only on \Gamma. To prove our main result, we introduce the concept of generalized foliated quadruples, which is a mixture of foliated triples and Birkar-Zhang's generalized pairs. With this concept, we establish a canonical bundle formula for foliations in any dimension. As for applications, we extend Shokurov's global index conjecture in the classical MMP to foliated triples and prove this conjecture for threefolds with nonzero boundaries and for surfaces. Additionally, we introduce the theory of rational polytopes for functional divisors on foliations and prove some miscellaneous results. [ABSTRACT FROM AUTHOR]

In this paper, we study the distribution of the cokernel of a general random Hermitian matrix over the ring of integers \mathcal {O} of a quadratic extension K of \mathbb {Q}_p. For each positive integer n, let X_n be a random n \times n Hermitian matrix over \mathcal {O} whose upper triangular entries are independent and their reductions are not too concentrated on certain values. We show that the distribution of the cokernel of X_n always converges to the same distribution which does not depend on the choices of X_n as n \rightarrow \infty and provide an explicit formula for the limiting distribution. This answers Open Problem 3.16 from the ICM 2022 lecture note of Wood [ Probability theory for random groups arising in number theory , 2022] in the case of the ring of integers of a quadratic extension of \mathbb {Q}_p. [ABSTRACT FROM AUTHOR]

We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy [Topology Appl. 14 (1982), pp. 151–161], which was later reposed by A. Miller, T. Orenshtein, and B. Tsaban. Namely, we show that under \mathfrak {p}=\mathfrak {c} there is a \delta-set that is not a \gamma-set. Thus we constructed a subset A of reals such that the space \mathrm {C}_p(A) of all real-valued continuous functions on A is not Fréchet–Urysohn, but possesses the Pytkeev property. Moreover, under \mathbf {CH} we construct a \pi-set that is not a \delta-set solving a problem by M. Sakai. In fact, we construct various examples of \delta-sets that are not \gamma-sets, satisfying finer properties parametrized by ideals on natural numbers. Finally, we distinguish ideal variants of the Fréchet–Urysohn property for many different Borel ideals in the realm of functional spaces. [ABSTRACT FROM AUTHOR]

We establish in this paper the logarithmic Bramson correction for Fisher-KPP equations on the lattice \mathbb {Z}. The level sets of solutions with step-like initial conditions are located at position c_*t-\frac {3}{2\lambda _*}\ln t+O(1) as t\rightarrow +\infty for some explicit positive constants c_* and \lambda _*. This extends a well-known result of Bramson in the continuous setting to the discrete case using only PDE arguments. A by-product of our analysis also gives that the solutions approach the family of logarithmically shifted traveling front solutions with minimal wave speed c_* uniformly on the positive integers, and that the solutions converge along their level sets to the minimal traveling front for large times. [ABSTRACT FROM AUTHOR]

As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in TDA to articulate stability and inference theorems. Intuitively, whereas a strict interleaving between filtered spaces X and Y certifies that X and Y are approximately isomorphic, a homotopy interleaving between X and Y certifies that X and Y are approximately weakly equivalent. The main results of this paper are that homotopy interleavings induce an extended pseudometric d_{HI} on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms. To motivate these axioms, we also observe that d_{HI} (or more generally, any pseudometric satisfying these two axioms and an additional "homology bounding" axiom) can be used to formulate lifts of several fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces. Finally, we consider the problem of establishing a persistent Whitehead theorem in terms of homotopy interleavings. We provide a counterexample to a naive formulation of the result. [ABSTRACT FROM AUTHOR]

Let p be a prime, and let K be a finite extension of \mathbf {Q}_p, with absolute Galois group \mathcal {G}_K. Let \pi be a uniformizer of K and let K_\infty be the Kummer extension obtained by adjoining to K a system of compatible p^n-th roots of \pi, for all n, and let L be the Galois closure of K_\infty. Using these extensions, Caruso has constructed é tale (\phi,\tau)-modules, which classify p-adic Galois representations of K. In this paper, we use locally analytic vectors and theories of families of \phi-modules over Robba rings to prove the overconvergence of (\phi,\tau)-modules in families. As examples, we also compute some explicit families of (\phi,\tau)-modules in some simple cases. [ABSTRACT FROM AUTHOR]

In the present paper we consider elliptic operators L=-div(A\nabla) in a domain bounded by a chord-arc surface \Gamma with small enough constant, and whose coefficients A satisfy a weak form of the Dahlberg-Kenig-Pipher condition of approximation by constant coefficient matrices, with a small enough Carleson norm, and show that the elliptic measure with pole at infinity associated to L is A_\infty-absolutely continuous with respect to the surface measure on \Gamma, with a small A_\infty constant. In other words, we show that for relatively flat uniformly rectifiable sets and for operators with slowly oscillating coefficients the elliptic measure satisfies the A_\infty condition with a small constant and the logarithm of the Poisson kernel has small oscillations. [ABSTRACT FROM AUTHOR]

Let G be a finite subgroup of \mathrm {SU}(4) such that its elements have age at most one. In the first part of this paper, we define K-theoretic stable pair invariants on a crepant resolution of the affine quotient {\mathbb {C}}^4/G, and conjecture a closed formula for their generating series in terms of the root system of G. In the second part, we define degree zero Donaldson-Thomas invariants of Calabi-Yau 4-orbifolds, develop a vertex formalism that computes the invariants in the toric case, and conjecture closed formulae for their generating series for the quotient stacks [{\mathbb {C}}^4/{\mathbb {Z}}_r], [{\mathbb {C}}^4/{\mathbb {Z}}_2\times {\mathbb {Z}}_2]. Combining these two parts, we formulate a crepant resolution correspondence which relates the above two theories. [ABSTRACT FROM AUTHOR]