Advances in Mathematics最新文献

We show that the total number of non-torsion integral points on the elliptic curves $E_D:y^2=x^3-D^{2}x$, where D ranges over positive squarefree integers less than N, is $O\left(N{(\log N)}^{-\frac{1}{4}+ϵ}\right)$. The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the 2-Selmer groups of the curves in this family.

We establish a close connection between acceleration and dynamical degree for one-frequency quasi-periodic compact cocycles, by showing that two vectors derived separately from each coincide. Based on this, we provide a dynamical classification of one-frequency quasi-periodic $\mathrm{SO}(3,R)$-cocycles.

In this paper we solve in the positive the question of whether any finite set of integers, containing 0, is the mapping degree set between two oriented closed connected manifolds of the same dimension. We extend this question to the rational setting, where an affirmative answer is also given.

In the topological dynamical system $(X,T)$, a point x simultaneously approximates a point y if there exists a sequence $n_1$, $n_2$, …of natural numbers for which $T^{n_i}x$, $T^{2n_i}x$, …, $T^{k{n}_i}x$ all tend to y. In 1978, Furstenberg and Weiss showed that every system possesses a point which simultaneously approximates itself (a multiply recurrent point) and deduced refinements of van der Waerden's theorem on arithmetic progressions. In this paper, we study the denseness of the set of points that are simultaneously approximated by a given point. We show that in a minimal nilsystem, all points simultaneously approximate a δ-dense set of points under a necessarily restricted set of powers of T. We tie this theorem to the multiplicative combinatorial properties of return-time sets, showing that all nil-Bohr sets and typical return-time sets in a minimal system are multiplicatively thick in a coset of a multiplicative subsemigroup of the natural numbers. This yields an inhomogeneous multiple recurrence result that generalizes Furstenberg and Weiss' theorem and leads to new enhancements of van der Waerden's theorem. This work relies crucially on continuity in the prolongation relation (the closure of the orbit-closure relation) developed by Auslander, Akin, and Glasner; the theory of rational points and polynomials on nilmanifolds developed by Leibman, Green, and Tao; and the machinery of topological characteristic factors developed recently by Glasner, Huang, Shao, Weiss, and Ye.

For w in the symmetric group, we provide an exact formula for the smallest positive power $q^{h\left(w\right)}$ appearing in the Kazhdan–Lusztig polynomial $P_{e,w}\left(q\right)$. We also provide a tight upper bound on $h\left(w\right)$ in simply-laced types, resolving a conjecture of Billey–Postnikov from 2002.

We define a notion of a connectivity structure on an ∞-category, analogous to a t-structure but applicable in unstable contexts—such as spaces, or algebras over an operad. This allows us to generalize notions of n-skeleta, minimal skeleta, and cellular approximation from the category of spaces. For modules over an Eilenberg–Mac Lane spectrum, these are closely related to the notion of projective amplitude.

We apply these to ring spectra, where they can be detected via the cotangent complex and higher Hochschild homology with coefficients. We show that the spectra $Y\left(n\right)$ of chromatic homotopy theory are minimal skeleta for $H{F}_2$ in the category of associative ring spectra. Similarly, Ravenel's spectra $T\left(n\right)$ are shown to be minimal skeleta for BP in the same way, which proves that these admit canonical associative algebra structures.

Given an elliptic curve E over a perfect defectless henselian valued field $(F,\mathrm{val})$ with perfect residue field ${\text{k}}_F$ and valuation ring $O_F$, there exists an integral separated smooth group scheme $E$ over $O_F$ with $E{\times }_{\text{Spec }{O}_F}\text{Spec }F\cong E$. If $\mathrm{char}\left({\text{k}}_F\right)\ne 2,3$ then one can be found over $O_{F^{alg}}$ such that the definable group $E\left(O\right)$ is the maximal generically stable subgroup of E. We also give some partial results on general Abelian varieties over F.

The construction of $E$ is by means of generating a birational group law over $O_F$ by the aid of a generically stable generic type of a definable subgroup of E.

We study a neighborhood of generic singularities formed by mean curvature flow (MCF). For various possibilities when the singularities are modeled on $S^3\times R$, we provide a detailed description for a small, but fixed, neighborhood of singularity, including proving that a small neighborhood is mean convex, and the singularity is isolated. For the remaining possibilities, we conjecture that an entire neighborhood of the singularity becomes singular at the time of blowup, and present evidence to support it. A key technique is that, when looking for a dominating direction for the rescaled MCF, we need a normal form transformation, as a result, the rescaled MCF is parametrized over some chosen curved cylinder, instead of a standard straight one.

A contact form on the tight 3-sphere $(S^3,{\xi }_0)$ is called weakly convex if the Conley-Zehnder index of every Reeb orbit is at least 2. In this article, we study Reeb flows of weakly convex contact forms on $(S^3,{\xi }_0)$ admitting a prescribed finite set of index-2 Reeb orbits, which are all hyperbolic and mutually unlinked. We present conditions so that these index-2 orbits are binding orbits of a genus zero transverse foliation whose additional binding orbits have index 3. In addition, we show in the real-analytic case that the topological entropy of the Reeb flow is positive if the branches of the stable/unstable manifolds of the index-2 orbits are mutually non-coincident.