Mostrando recursos 1 - 20 de 66

  1. The $p$ -curvature conjecture and monodromy around simple closed loops

    Shankar, Ananth N.
    The Grothendieck–Katz $p$ -curvature conjecture is an analogue of the Hasse principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$ -curvature vanishes modulo $p$ , for almost all primes $p$ . We prove that if the variety is a generic curve, then every simple closed loop on the curve has finite monodromy.

  2. Analytic torsion and R-torsion of Witt representations on manifolds with cusps

    Albin, Pierre; Rochon, Frédéric; Sher, David
    We establish a Cheeger–Müller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all noncompact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion, and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a...

  3. A minimization problem with free boundary related to a cooperative system

    Caffarelli, Luis A.; Shahgholian, Henrik; Yeressian, Karen
    We study the minimum problem for the functional \begin{equation*}\int_{\Omega}(\vert\nabla\mathbf{u}\vert^{2}+Q^{2}\chi_{\{\vert\mathbf{u}\vert\gt 0\}})\,dx\end{equation*} with the constraint $u_{i}\geq0$ for $i=1,\ldots,m$ , where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain and $\mathbf{u}=(u_{1},\ldots,u_{m})\in H^{1}(\Omega;\mathbb{R}^{m})$ . First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set $\{\vert u\vert\gt 0\}$ is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary $\Omega\cap\partial\{\vert u\vert\gt 0\}$ .

  4. Independence of $\ell$ for the supports in the decomposition theorem

    Sun, Shenghao
    In this article, we prove a result on the independence of $\ell$ for the supports of irreducible perverse sheaves occurring in the decomposition theorem, as well as for the family of local systems on each support. It generalizes Gabber’s result on the independence of $\ell$ of intersection cohomology to the relative case.

  5. Universal dynamics for the defocusing logarithmic Schrödinger equation

    Carles, Rémi; Gallagher, Isabelle
    We consider the Schrödinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time, and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data and remain when an extra power-like nonlinearity is present in the equation. One...

  6. On finiteness properties of the Johnson filtrations

    Ershov, Mikhail; He, Sue
    Let $\Gamma$ be either the automorphism group of the free group of rank $n\geq4$ or the mapping class group of an orientable surface of genus $n\geq12$ with at most $1$ boundary component, and let $G$ be either the subgroup of $\mathrm{IA}$ -automorphisms or the Torelli subgroup of $\Gamma$ . For $N\in\mathbb{N}$ denote by $\gamma_{N}G$ the $N$ th term of the lower central series of $G$ . We prove that ¶ (i) any subgroup of $G$ containing $\gamma_{2}G=[G,G]$ (in particular, the Johnson kernel in the mapping class group case) is finitely generated; ¶ (ii) if $N=2$ or $n\geq8N-4$ and $K$ is any subgroup...

  7. Integer homology $3$ -spheres admit irreducible representations in $\operatorname{SL}(2,\mathbb{C})$

    Zentner, Raphael
    We prove that the fundamental group of any integer homology $3$ -sphere different from the $3$ -sphere admits irreducible representations of its fundamental group in $\operatorname{SL}(2,\mathbb{C})$ . For hyperbolic integer homology spheres, this comes with the definition; for Seifert-fibered integer homology spheres, this is well known. We prove that the splicing of any two nontrivial knots in $S^{3}$ admits an irreducible $\operatorname{SU}(2)$ -representation. Using a result of Kuperberg, we get the corollary that the problem of $3$ -sphere recognition is in the complexity class $\mathsf{coNP}$ , provided the generalized Riemann hypothesis holds. To prove our result, we establish a topological...

  8. Bohr sets and multiplicative Diophantine approximation

    Chow, Sam
    In two dimensions, Gallagher’s theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fiber version of Gallagher’s theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes, and Velani. The idea is to find large generalized arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin–Schaeffer theorem for the problem at hand, via the geometry of numbers.

  9. Hodge theory of classifying stacks

    Totaro, Burt
    We compute the Hodge and the de Rham cohomology of the classifying space $\mathit{BG}$ (defined as étale cohomology on the algebraic stack $\mathit{BG}$ ) for reductive groups $G$ over many fields, including fields of small characteristic. These calculations have a direct relation with representation theory, yielding new results there. The calculations are closely analogous to, but not always the same as, the cohomology of classifying spaces in topology.

  10. On the conservativity of the functor assigning to a motivic spectrum its motive

    Bachmann, Tom
    Given a $0$ -connective motivic spectrum $E\in\mathbf{SH}(k)$ over a perfect field $k$ , we determine $\underline{h}_{0}$ of the associated motive $ME\in\mathbf{DM}(k)$ in terms of $\underline{\pi}_{0}(E)$ . Using this, we show that if $k$ has finite $2$ -étale cohomological dimension, then the functor $M:\mathbf{SH}(k)\to\mathbf{DM}(k)$ is conservative when restricted to the subcategory of compact spectra and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual $2$ -étale cohomological dimension by considering what we call real motives.

  11. Uniform rectifiability from Carleson measure estimates and $\mathbf{\varepsilon}$ -approximability of bounded harmonic functions

    Garnett, John; Mourgoglou, Mihalis; Tolsa, Xavier
    Let $\Omega\subset{\mathbb{R}}^{n+1}$ , $n\geq1$ , be a corkscrew domain with Ahlfors–David regular boundary. In this article we prove that $\partial\Omega$ is uniformly $n$ -rectifiable if every bounded harmonic function on $\Omega$ is $\varepsilon$ -approximable or if every bounded harmonic function on $\Omega$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $\Omega={\mathbb{R}}^{n+1}\setminusE$ and $E$ is Ahlfors–David regular. Our results establish a conjecture posed by Hofmann, Martell, and Mayboroda, in which they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability, one in terms of the so-called $S\lt N$ estimates...

  12. The Sard conjecture on Martinet surfaces

    Belotto da Silva, André; Rifford, Ludovic
    Given a totally nonholonomic distribution of rank $2$ on a $3$ -dimensional manifold, we investigate the size of the set of points that can be reached by singular horizontal paths starting from the same point. In this setting, by the Sard conjecture, that set should be a subset of the so-called Martinet surface of $2$ -dimensional Hausdorff measure zero. We prove that the conjecture holds in the case where the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces, and we show that the result holds true under an assumption of nontransversality of the distribution...

  13. 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...

  14. 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.

  15. 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.

  16. 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...

  17. 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...

  18. 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.

  19. 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...

  20. 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...

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