Mostrando recursos 1 - 20 de 63

  1. Groups quasi-isometric to right-angled Artin groups

    Huang, Jingyin; Kleiner, Bruce
    We characterize groups quasi-isometric to a right-angled Artin group (RAAG) $G$ with finite outer automorphism group. In particular, all such groups admit a geometric action on a $\operatorname{CAT}(0)$ cube complex that has an equivariant “fibering” over the Davis building of $G$ . This characterization will be used in forthcoming work of the first author to give a commensurability classification of the groups quasi-isometric to certain RAAGs.

  2. Carathéodory’s metrics on Teichmüller spaces and $L$ -shaped pillowcases

    Markovic, Vladimir
    One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each $\mathcal{T}_{g}$ , the Teichmüller space of a closed surface of genus $g\ge2$ . The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential...

  3. Canonical growth conditions associated to ample line bundles

    Witt Nyström, David
    We propose a new construction which associates to any ample (or big) line bundle $L$ on a projective manifold $X$ a canonical growth condition (i.e., a choice of a plurisubharmonic (psh) function well defined up to a bounded term) on the tangent space $T_{p}X$ of any given point $p$ . We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows one to recover all the infinitesimal Okounkov bodies of $L$ at $p$ . The construction is inspired by toric geometry and the theory of Okounkov bodies; in the toric case, the...

  4. The colored HOMFLYPT function is $q$ -holonomic

    Garoufalidis, Stavros; Lauda, Aaron D.; Lê, Thang T. Q.
    We prove that the HOMFLYPT polynomial of a link colored by partitions with a fixed number of rows is a $q$ -holonomic function. By specializing to the case of knots colored by a partition with a single row, it proves the existence of an $(a,q)$ superpolynomial of knots in $3$ -space, as was conjectured by string theorists. Our proof uses skew-Howe duality that reduces the evaluation of web diagrams and their ladders to a Poincaré–Birkhoff–Witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram. The result is a concrete and algorithmic web evaluation...

  5. Totaro’s question on zero-cycles on torsors

    Gordon-Sarney, R.; Suresh, V.
    Let $G$ be a smooth connected linear algebraic group, and let $X$ be a $G$ -torsor. Totaro asked: If $X$ admits a zero-cycle of degree $d\geq1$ , then does $X$ have a closed étale point of degree dividing $d$ ? While the literature contains affirmative answers in some special cases, we give examples to show that the answer is negative in general.

  6. Current fluctuations of the stationary ASEP and six-vertex model

    Aggarwal, Amol
    Our results in this article are twofold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time $T$ , and show that they are of order $T^{1/3}$ along a characteristic line. Upon scaling by $T^{1/3}$ , we establish that these fluctuations converge to the long-time height fluctuations of the stationary Kardar–Parisi–Zhang (KPZ) equation, that is, to the Baik–Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and Spohn from 2005...

  7. The Abelianization of the real Cremona group

    Zimmermann, Susanna
    We present the Abelianization of the group of birational transformations of $\mathbb{P}^{2}_{\mathbb{R}}$ .

  8. On the marked length spectrum of generic strictly convex billiard tables

    Huang, Guan; Kaloshin, Vadim; Sorrentino, Alfonso
    In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard map from the (maximal) marked length spectrum of the domain.

  9. Spectral asymptotics for sub-Riemannian Laplacians, I: Quantum ergodicity and quantum limits in the 3-dimensional contact case

    Colin de Verdière, Yves; Hillairet, Luc; Trélat, Emmanuel
    This is the first paper of a series in which we plan to study spectral asymptotics for sub-Riemannian (sR) Laplacians and to extend results that are classical in the Riemannian case concerning Weyl measures, quantum limits, quantum ergodicity, quasimodes, and trace formulae. Even if hypoelliptic operators have been well studied from the point of view of PDEs, global geometrical and dynamical aspects have not been the subject of much attention. As we will see, already in the simplest case, the statements of the results in the sR setting are quite different from those in the Riemannian one. ¶ Let us...

  10. Quantitative nonorientability of embedded cycles

    Young, Robert
    We introduce an invariant linked to some foundational questions in geometric measure theory and provide bounds on this invariant by decomposing an arbitrary cycle into uniformly rectifiable pieces. Our invariant measures the difficulty of cutting a nonorientable closed manifold or mod- $2$ cycle in $\mathbb{R}^{n}$ into orientable pieces, and we use it to answer some simple but long-open questions on filling volumes and mod- $\nu$ currents.

  11. Full-rank affine invariant submanifolds

    Mirzakhani, Maryam; Wright, Alex
    We show that every $\operatorname{GL}(2,R)$ orbit closure of translation surfaces is a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. We use this result to give applications related to polygonal billiards. For example, we exhibit infinitely many rational triangles whose unfoldings have dense $\operatorname{GL}(2,R)$ orbit.

  12. The C∗-algebra of a minimal homeomorphism of zero mean dimension

    Elliott, George A.; Niu, Zhuang
    Let $X$ be an infinite metrizable compact space, and let $\sigma:X\toX$ be a minimal homeomorphism. Suppose that $(X,\sigma)$ has zero mean topological dimension. The associated C∗-algebra $A=\mathrm{C}(X)\rtimes_{\sigma}\mathbb{Z}$ is shown to absorb the Jiang–Su algebra $\mathcal{Z}$ tensorially; that is, $A\cong A\otimes\mathcal{Z}$ . This implies that $A$ is classifiable when $(X,\sigma)$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang–Su algebra.

  13. The C∗-algebra of a minimal homeomorphism of zero mean dimension

    Elliott, George A.; Niu, Zhuang
    Let $X$ be an infinite metrizable compact space, and let $\sigma:X\toX$ be a minimal homeomorphism. Suppose that $(X,\sigma)$ has zero mean topological dimension. The associated C∗-algebra $A=\mathrm{C}(X)\rtimes_{\sigma}\mathbb{Z}$ is shown to absorb the Jiang–Su algebra $\mathcal{Z}$ tensorially; that is, $A\cong A\otimes\mathcal{Z}$ . This implies that $A$ is classifiable when $(X,\sigma)$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang–Su algebra.

  14. On the Lagrangian structure of transport equations: The Vlasov–Poisson system

    Ambrosio, Luigi; Colombo, Maria; Figalli, Alessio
    The Vlasov–Poisson system is an important nonlinear transport equation, used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of classical solutions is limited to dimensions $d\leq3$ under strong assumptions on the initial data, whereas weak solutions are known to exist under milder conditions. However, in the setting of weak solutions it is unclear whether the Eulerian description provided by the equation physically corresponds to a Lagrangian evolution of the particles. In this article we develop several general tools concerning the Lagrangian structure of transport equations with nonsmooth vector fields, and we apply these...

  15. On the Lagrangian structure of transport equations: The Vlasov–Poisson system

    Ambrosio, Luigi; Colombo, Maria; Figalli, Alessio
    The Vlasov–Poisson system is an important nonlinear transport equation, used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of classical solutions is limited to dimensions $d\leq3$ under strong assumptions on the initial data, whereas weak solutions are known to exist under milder conditions. However, in the setting of weak solutions it is unclear whether the Eulerian description provided by the equation physically corresponds to a Lagrangian evolution of the particles. In this article we develop several general tools concerning the Lagrangian structure of transport equations with nonsmooth vector fields, and we apply these...

  16. Equidistribution in $\operatorname{Bun}_{2}(\mathbb{P}^{1})$

    Shende, Vivek; Tsimerman, Jacob
    Fix a finite field. The set of $\operatorname{PGL}_{2}$ bundles over $\mathbb{P}^{1}$ is in bijection with the natural numbers, and carries a natural measure assigning to each bundle the inverse of the number of automorphisms. A branched double cover $\pi:C\to\mathbb{P}^{1}$ determines another measure, given by counting the number of line bundles over $C$ whose image on $\mathbb{P}^{1}$ has a given sheaf of endomorphisms. We show the measures induced by a sequence of such hyperelliptic curves tends to the canonical measure on the space of $\operatorname{PGL}_{2}$ bundles. ¶ This is a function field analogue of Duke’s theorem on the equidistribution of Heegner...

  17. Equidistribution in $\operatorname{Bun}_{2}(\mathbb{P}^{1})$

    Shende, Vivek; Tsimerman, Jacob
    Fix a finite field. The set of $\operatorname{PGL}_{2}$ bundles over $\mathbb{P}^{1}$ is in bijection with the natural numbers, and carries a natural measure assigning to each bundle the inverse of the number of automorphisms. A branched double cover $\pi:C\to\mathbb{P}^{1}$ determines another measure, given by counting the number of line bundles over $C$ whose image on $\mathbb{P}^{1}$ has a given sheaf of endomorphisms. We show the measures induced by a sequence of such hyperelliptic curves tends to the canonical measure on the space of $\operatorname{PGL}_{2}$ bundles. ¶ This is a function field analogue of Duke’s theorem on the equidistribution of Heegner...

  18. Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

    Le Masson, Etienne; Sahlsten, Tuomas
    We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and Le Masson. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Le Masson, and Lindenstrauss on discrete graphs. It...

  19. Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

    Le Masson, Etienne; Sahlsten, Tuomas
    We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and Le Masson. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Le Masson, and Lindenstrauss on discrete graphs. It...

  20. Hasse principle for three classes of varieties over global function fields

    Tian, Zhiyu
    We give a geometric proof that the Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces, smooth cubic hypersurfaces of dimension at least $4$ in characteristic at least $7$ , and smooth complete intersections of two quadrics, which are of dimension at least $3$ , in odd characteristics.

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