Mostrando recursos 1 - 20 de 54

  1. Monodromy dependence and connection formulae for isomonodromic tau functions

    Its, A. R.; Lisovyy, O.; Prokhorov, A.
    We discuss an extension of the Jimbo–Miwa–Ueno differential $1$ -form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola, generalizing a previous construction by B. Malgrange. We show how this $1$ -form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection...

  2. The critical height is a moduli height

    Ingram, Patrick
    Silverman defined the critical height of a rational function $f(z)$ of degree $d\geq2$ in terms of the asymptotic rate of growth of the Weil height along the critical orbits of $f$ . He also conjectured that this quantity was commensurate to an ample Weil height on the moduli space of rational functions degree $d$ . We prove that conjecture.

  3. Integration of oscillatory and subanalytic functions

    Cluckers, Raf; Comte, Georges; Miller, Daniel J.; Rolin, Jean-Philippe; Servi, Tamara
    We prove the stability under integration and under Fourier transform of a concrete class of functions containing all globally subanalytic functions and their complex exponentials. This article extends the investigation started by Lion and Rolin and Cluckers and Miller to an enriched framework including oscillatory functions. It provides a new example of fruitful interaction between analysis and singularity theory.

  4. Almost sure multifractal spectrum of Schramm–Loewner evolution

    Gwynne, Ewain; Miller, Jason; Sun, Xin
    Suppose that $\eta$ is a Schramm–Loewner evolution ( $\operatorname{SLE}_{\kappa}$ ) in a smoothly bounded simply connected domain $D\subset{\mathbf{C}}$ and that $\phi$ is a conformal map from $\mathbf{D}$ to a connected component of $D\setminus\eta([0,t])$ for some $t\gt 0$ . The multifractal spectrum of $\eta$ is the function $(-1,1)\to[0,\infty)$ which, for each $s\in(-1,1)$ , gives the Hausdorff dimension of the set of points $x\in\partial\mathbf{D}$ such that $|\phi'((1-\epsilon)x)|=\epsilon^{-s+o(1)}$ as $\epsilon\to0$ . We rigorously compute the almost sure multifractal spectrum of $\operatorname{SLE}$ , confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the almost sure...

  5. Galois and Cartan cohomology of real groups

    Adams, Jeffrey; Taïbi, Olivier
    Suppose that $G$ is a complex, reductive algebraic group. A real form of $G$ is an antiholomorphic involutive automorphism $\sigma$ , so $G(\mathbb{R})=G(\mathbb{C})^{\sigma}$ is a real Lie group. Write $H^{1}(\sigma,G)$ for the Galois cohomology (pointed) set $H^{1}(\operatorname{Gal}(\mathbb{C}/\mathbb{R}),G)$ . A Cartan involution for $\sigma$ is an involutive holomorphic automorphism $\theta$ of $G$ , commuting with $\sigma$ , so that $\theta\sigma$ is a compact real form of $G$ . Let $H^{1}(\theta,G)$ be the set $H^{1}(\mathbb{Z}_{2},G)$ , where the action of the nontrivial element of $\mathbb{Z}_{2}$ is by $\theta$ . By analogy with the Galois group, we refer to $H^{1}(\theta,G)$ as the Cartan...

  6. Group cubization

    Osajda, Damian
    We present a procedure of group cubization: it results in a group whose features resemble some of those of a given group and which acts without fixed points on a $\operatorname{CAT}(0)$ cubical complex. As a main application, we establish the lack of Kazhdan’s property (T)for Burnside groups.

  7. Odd degree number fields with odd class number

    Ho, Wei; Shankar, Arul; Varma, Ila
    For every odd integer $n\geq3$ , we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_{n}$ whose class number is odd. To do so, we study the class groups of families of number fields of degree $n$ whose rings of integers arise as the coordinate rings of the subschemes of ${\mathbb{P}}^{1}$ cut out by integral binary $n$ -ic forms. By obtaining upper bounds on the mean number of $2$ -torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to $1$ as $n$ tends to...

  8. Regularization under diffusion and anticoncentration of the information content

    Eldan, Ronen; Lee, James R.
    Under the Ornstein–Uhlenbeck semigroup $\{U_{t}\}$ , any nonnegative measurable $f:\mathbb{R}^{n}\to\mathbb{R}_{+}$ exhibits a uniform tail bound better than that implied by Markov’s inequality and conservation of mass. For every $\alpha\geq e^{3}$ , and $t\gt 0$ , ¶ \[\gamma_{n}(\{x\in\mathbb{R}^{n}:U_{t}f(x)\gt \alpha\int f\,d\gamma_{n}\})\leq C(t)\frac{1}{\alpha}\sqrt{\frac{\log\log\alpha}{\log\alpha}},\] where $\gamma_{n}$ is the $n$ -dimensional Gaussian measure and $C(t)$ is a constant depending only on $t$ . This confirms positively the Gaussian limiting case of Talagrand’s convolution conjecture (1989). This is shown to follow from a more general phenomenon. Suppose that $f:\mathbb{R}^{n}\to\mathbb{R}_{+}$ is semi-log-convex in the sense that for some $\beta\gt 0$ , for all $x\in\mathbb{R}^{n}$ , the eigenvalues of...

  9. A geometric characterization of toric varieties

    Brown, Morgan V.; McKernan, James; Svaldi, Roberto; Zong, Hong R.
    We prove a conjecture of Shokurov which characterizes toric varieties using log pairs.

  10. Picard–Lefschetz oscillators for the Drinfeld–Lafforgue–Vinberg degeneration for $\operatorname{SL}_{2}$

    Schieder, Simon
    Let $G$ be a reductive group, and let $\operatorname{Bun}_{G}$ denote the moduli stack of $G$ -bundles on a smooth projective curve. We begin the study of the singularities of a canonical compactification of $\operatorname{Bun}_{G}$ due to Drinfeld (unpublished), which we refer to as the Drinfeld–Lafforgue–Vinberg compactification $\overline{\operatorname{Bun}}_{G}$ . For $G=\operatorname{GL}_{2}$ and $G=\operatorname{GL}_{n}$ , certain smooth open substacks of this compactification have already appeared in the work of Drinfeld and Lafforgue on the Langlands correspondence for function fields. The stack $\overline{\operatorname{Bun}}_{G}$ is, however, already singular for $G=\operatorname{SL}_{2}$ ; questions about its singularities arise naturally in the geometric Langlands program, and form...

  11. A $p$ -adic Waldspurger formula

    Liu, Yifeng; Zhang, Shouwu; Zhang, Wei
    In this article, we study $p$ -adic torus periods for certain $p$ -adic-valued functions on Shimura curves of classical origin. We prove a $p$ -adic Waldspurger formula for these periods as a generalization of recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic $p$ -adic $L$ -function of Rankin–Selberg type. At a character of positive weight, the $p$ -adic $L$ -function interpolates the central critical value of the complex Rankin–Selberg $L$ -function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding $p$ -adic torus...

  12. $\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of strata

    Apisa, Paul
    The object of this article is to study $\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of strata of Abelian differentials. The main result is that all higher-rank affine invariant submanifolds in hyperelliptic components are branched covering constructions; that is, every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of Abelian differentials. This result implies the finiteness of algebraically primitive Teichmüller curves in all hyperelliptic components for genus greater than two. A classification of all $\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of strata (up to computing connected components and up...

  13. The Breuil–Mézard conjecture when $l\neq p$

    Shotton, Jack
    Let $l$ and $p$ be primes, let $F/\mathbb{Q}_{p}$ be a finite extension with absolute Galois group $G_{F}$ , let $\mathbb{F}$ be a finite field of characteristic $l$ , and let ¶ \[\overline{\rho}:G_{F}\rightarrow \operatorname{GL}_{n}(\mathbb{F})\] be a continuous representation. Let $R^{\square}(\overline{\rho})$ be the universal framed deformation ring for $\overline{\rho}$ . If $l=p$ , then the Breuil–Mézard conjecture (as recently formulated by Emerton and Gee) relates the mod $l$ reduction of certain cycles in $R^{\square}(\overline{\rho})$ to the mod $l$ reduction of certain representations of $\operatorname{GL}_{n}(\mathcal{O}_{F})$ . We state an analogue of the Breuil–Mézard conjecture when $l\neq p$ , and we prove it whenever $l\gt...

  14. 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.

  15. 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...

  16. 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...

  17. 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...

  18. 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.

  19. 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...

  20. The Abelianization of the real Cremona group

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

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