Mostrando recursos 1 - 20 de 5.569

  1. Higher $K$ -theory via universal invariants

    Tabuada, Gonçalo
    Using the formalism of Grothendieck's derivators, we construct the universal localizing invariant of differential graded (dg) categories. By this we mean a morphism $\mathcal{U}_l$ from the pointed derivator $\mathsf{HO}(\mathsf{dgcat})$ associated with the Morita homotopy theory of dg categories to a triangulated strong derivator $\mathcal{M}_{\rm dg}^{\rm loc}$ such that $\mathcal{U}_l$ commutes with filtered homotopy colimits, preserves the point, sends each exact sequence of dg categories to a triangle, and is universal for these properties. ¶ Similarly, we construct the universal additive invariant of dg categories, that is, the universal morphism of derivators $\mathcal{U}_a$ from $\mathsf{HO}(\mathsf{dgcat})$ to a strong triangulated derivator $\mathcal{M}_{\rm dg}^{\rm...

  2. Tug-of-war with noise: A game-theoretic view of the $p$ -Laplacian

    Peres, Yuval; Sheffield, Scott
    Fix a bounded domain $\Omega \subset {\mathbb R}^d$ , a continuous function $F:\partial \Omega \rightarrow {\mathbb R}$ , and constants $\epsilon >0$ and $1 \lt p,q \lt \infty$ with $p^{-1} + q^{-1} = 1$ . For each $x \in \Omega$ , let $u^\epsilon(x)$ be the value for player I of the following two-player, zero-sum game. The initial game position is $x$ . At each stage, a fair coin is tossed, and the player who wins the toss chooses a vector $v \in \overline{B}(0,\epsilon)$ to add to the game position, after which a random noise vector with mean zero and variance...

  3. Nonsmooth classical points on eigenvarieties

    Bellaïche, Joël
    We construct nonsmooth points on unitary eigenvarieties. More precisely, we construct points so that the local ring of the irreducible components of the eigenvariety through this point is nonsmooth and not even a unique factorization domain (UFD). We use those points to construct geometrically several independent extensions in the relevant Selmer group

  4. The characteristic polynomial of a random unitary matrix: A probabilistic approach

    Bourgade, P.; Hughes, C. P.; Nikeghbali, A.; Yor, M.
    In this article, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin-Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [8] using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such...

  5. Geodesics and commensurability classes of arithmetic hyperbolic $3$ -manifolds

    Chinburg, T.; Hamilton, E.; Long, D. D.; Reid, A. W.
    We show that if $M$ is an arithmetic hyperbolic $3$ -manifold, the set $\mathbb{Q}L(M)$ of all rational multiples of lengths of closed geodesics of $M$ both determines and is determined by the commensurability class of $M$ . This implies that the spectrum of the Laplace operator of $M$ determines the commensurability class of $M$ . We also show that the zeta function of a number field with exactly one complex place determines the isomorphism class of the number field

  6. Boundary properties of Green functions in the plane

    Baranov, Anton; Hedenmalm, Håkan
    We study the boundary properties of the Green function of bounded simply connected domains in the plane. Essentially, this amounts to studying the conformal mapping taking the unit disk onto the domain in question. Our technique is inspired by a 1995 article of Jones and Makarov [11]. The main tools are an integral identity as well as a uniform Sobolev embedding theorem. The latter is in a sense dual to the exponential integrability of Marcinkiewicz-Zygmund integrals. We also develop a Grunsky identity, which contains the information of the classical Grunsky inequality. This Grunsky identity is the case where $p=2$ of...

  7. Microlocalization of rational Cherednik algebras

    Kashiwara, Masaki; Rouquier, Raphaël
    We construct a microlocalization of the rational Cherednik algebras $H$ of type $S_n$ . This is achieved by a quantization of the Hilbert scheme $\mathrm{Hilb}^n{\mathbb C}^2$ of $n$ points in ${\mathbb C}^2$ . We then prove the equivalence of the category of $H$ -modules and that of modules over its microlocalization under certain conditions on the parameter

  8. Algebraic cycles and completions of equivariant $K$ -theory

    Edidin, Dan; Graham, William
    Let $G$ be a complex, linear algebraic group acting on an algebraic space $X$ . The purpose of this article is to prove a Riemann-Roch theorem (Theorem 6.5) that gives a description of the completion of the equivariant Grothendieck group $G_0(G,X) \otimes {\mathbb C}$ at any maximal ideal of the representation ring $R(G) \otimes {\mathbb C}$ in terms of equivariant cycles. The main new technique for proving this theorem is our nonabelian completion theorem (Theorem 5.3) for equivariant $K$ -theory. Theorem 5.3 generalizes the classical localization theorems for diagonalizable group actions to arbitrary groups

  9. Linear manifolds in the moduli space of one-forms

    Möller, Martin
    We study closures of ${\rm GL}^+_2(\mathbb{R})$ -orbits in the total space $\Omega M_g$ of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that in the generic stratum, such manifolds are the whole stratum, the hyperelliptic locus, or parameterize curves whose Jacobian has additional endomorphisms. This follows from a cohomological description of the tangent bundle to $\Omega M_g$ . For nongeneric strata, similar results can be shown by a case-by-case inspection. We also propose to study a notion of linear manifold that comprises Teichmüller curves, Hilbert modular surfaces, and the...

  10. Convex foliated projective structures and the Hitchin component for ${\rm PSL}_4(\mathbf{R})$

    Guichard, Olivier; Wienhard, Anna
    In this article, we give a geometric interpretation of the Hitchin component $\mathcal{T}^4(\Sigma) \subset {\rm Rep}(\pi_1(\Sigma), {\rm PSL}_4(\mathbf{R}))$ of a closed oriented surface of genus $g\geq 2$ . We show that representations in $\mathcal{T}^4(\Sigma)$ are precisely the holonomy representations of properly convex foliated projective structures on the unit tangent bundle of $\Sigma$ . From this, we also deduce a geometric description of the Hitchin component $\mathcal{T}(\Sigma, {\rm Sp}_4(\mathbf{R}))$ of representations into the symplectic group

  11. A twisted invariant Paley-Wiener theorem for real reductive groups

    Delorme, Patrick; Mezo, Paul
    Let $G^+$ be the group of real points of a possibly disconnected linear reductive algebraic group defined over $\mathbb{R}$ which is generated by the real points of a connected component $G^\prime$ . Let $K$ be a maximal compact subgroup of the group of real points of the identity component of this algebraic group. We characterize the space of maps $\pi\mapsto \mathrm{tr}(\pi(f))$ , where $\pi$ is an irreducible tempered representation of $G^+$ and $f$ varies over the space of smooth, compactly supported functions on $G^\prime$ which are left and right $K$ -finite. This work is motivated by applications to the twisted...

  12. The spectral decomposition of shifted convolution sums

    Blomer, Valentin; Harcos, Gergely
    Let $\pi_1$ , $\pi_2$ be cuspidal automorphic representations of ${\rm PGL}_2(\mathbb{R})$ of conductor $1$ and Hecke eigenvalues $\lambda_{\pi_{1, 2}}(n)$ , and let $h>0$ be an integer. For any smooth compactly supported weight functions $W_{1, 2}:\mathbb{R}^\times\to\mathbb{C}$ and any $Y>0$ , a spectral decomposition of the shifted convolution sum \[ \sum_{m\pm n=h}\frac{\lambda_{\pi_1}(|m|)\lambda_{\pi_2}(|n|)}{\sqrt{|mn|}} W_1\Big(\frac{m}{Y}\Big)W_2\Big(\frac{n}{Y}\Big) \] is obtained. As an application, a spectral decomposition of the Dirichlet series \[ \sum_{\substack{m,n\geq 1 m-n=h}} \frac{\lambda_{\pi_1}(m)\lambda_{\pi_2}(n)}{(m+n)^{s}} \Big(\frac{\sqrt{mn}}{m+n}\Big)^{100} \] is proved for $\mathfrak{R}s > 1/2$ with polynomial growth on vertical lines in the $s$ -aspect and uniformity in the $h$ -aspect

  13. Deviation of ergodic averages for rational polygonal billiards

    Athreya, J. S.; Forni, G.
    We prove a polynomial upper bound on the deviation of ergodic averages for almost all directional flows on every translation surface, in particular, for the generic directional flow of billiards in any Euclidean polygon with rational angles

  14. Commuting Hamiltonians and Hamilton-Jacobi multi-time equations

    Cardin, Franco; Viterbo, Claude
    The aim of this article is twofold. First, we show that the $C^0$ -limit of a pair of commuting Hamiltonians commutes. This means, on the one hand, that if the limit of the Hamiltonians is smooth, the Poisson bracket of their limit still vanishes and, on the other hand, that we may define “commutation” for $C^0$ -functions. The second part of this article deals with solving multi-time Hamilton-Jacobi equations using variational solutions. This extends the work of Barles and Tourin [BT] in the viscosity case to include the case of $C^0$ -Hamiltonians, and it removes their convexity assumption, provided that...

  15. On the explicit construction of higher deformations of partition statistics

    Bringmann, Kathrin
    The modularity of the partition-generating function has many important consequences: for example, asymptotics and congruences for $p(n)$ . In a pair of articles, Bringmann and Ono [11], [12] connected the rank, a partition statistic introduced by Dyson [18], to weak Maass forms, a new class of functions that are related to modular forms and that were first considered in [14]. Here, we take a further step toward understanding how weak Maass forms arise from interesting partition statistics by placing certain $2$ -marked Durfee symbols introduced by Andrews [1] into the framework of weak Maass forms. To do this, we construct...

  16. Propagation of singularities for the wave equation on edge manifolds

    Melrose, Richard; Vasy, András; Wunsch, Jared
    We investigate the geometric propagation and diffraction of singularities of solutions to the wave equation on manifolds with edge singularities. This class of manifolds includes, and is modeled on, the product of a smooth manifold and a cone over a compact fiber. Our main results are a general diffractive theorem showing that the spreading of singularities at the edge only occurs along the fibers and a more refined geometric theorem showing that for appropriately regular (nonfocusing) solutions, the main singularities can only propagate along geometrically determined rays. Thus, for the fundamental solution with initial pole sufficiently close to the edge,...

  17. Surfaces with boundary: Their uniformizations, determinants of Laplacians, and isospectrality

    Kim, Young-Heon
    Let $\Sigma$ be a compact surface of type $(g, n)$ , $n > 0$ , obtained by removing $n$ disjoint disks from a closed surface of genus $g$ . Assuming that $\chi(\Sigma) \lt 0$ , we show that on $\Sigma$ , the set of flat metrics that have the same Laplacian spectrum of the Dirichlet boundary condition is compact in the $C^\infty$ -topology. This isospectral compactness extends the result of Osgood, Phillips, and Sarnak [OPS3, Theorem 2] for surfaces of type $(0,n)$ whose examples include bounded plane domains. ¶ Our main ingredients are as follows. We first show that the determinant...

  18. Cohomological goodness and the profinite completion of Bianchi groups

    Grunewald, F.; Jaikin-Zapirain, A.; Zalesskii, P. A.
    The concept of cohomological goodness was introduced by J.-P. Serre in his book on Galois cohomology [31]. This property relates the cohomology groups of a group to those of its profinite completion. We develop properties of goodness and establish goodness for certain important groups. We prove, for example, that the Bianchi groups (i.e., the groups ${\rm PSL}(2,{\cal O})$ , where ${\cal O}$ is the ring of integers in an imaginary quadratic number field) are good. As an application of our improved understanding of goodness, we are able to show that certain natural central extensions of Fuchsian groups are residually finite....

  19. Riemann-Hilbert problem associated to Frobenius manifold structures on Hurwitz spaces: Irregular singularity

    Shramchenko, Vasilisa
    Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius manifold structures on Hurwitz spaces (real doubles, deformations) is described at the level of the corresponding Riemann-Hilbert problems

  20. Knot homology via derived categories of coherent sheaves, I: The $\mathfrak{sl}(2)$ -case

    Cautis, Sabin; Kamnitzer, Joel
    Using derived categories of equivariant coherent sheaves, we construct a categorification of the tangle calculus associated to $\mathfrak{sl}(2)$ and its standard representation. Our construction is related to that of Seidel and Smith [SS] by homological mirror symmetry. We show that the resulting doubly graded knot homology agrees with Khovanov homology (see [Kh1])

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