scholarly journals Geodesic planes in geometrically finite acylindrical -manifolds

2021 ◽  
pp. 1-40
Author(s):  
YVES BENOIST ◽  
HEE OH
Keyword(s):  
Closed Geodesic ◽  
Duke Math ◽  
Geometrically Finite ◽  
Convex Cocompact ◽  
Invent Math ◽  
Convex Core

Abstract Let M be a geometrically finite acylindrical hyperbolic $3$ -manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$ . These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math.209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J., to appear, Preprint, 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $3$ -manifold $M_0$ , the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $M_0$ . We construct a counterexample of this phenomenon when $M_0$ is non-arithmetic.

scholarly journals On the minimal distance between elliptic fixed points for geometrically finite Fuchsian groups

2016 ◽  
Vol 12 (06) ◽  
pp. 1663-1668
Author(s):  
Joshua S. Friedman
Keyword(s):  
Lower Bound ◽  
Fixed Points ◽  
Fuchsian Group ◽  
Closed Geodesic ◽  
Universal Constant ◽  
Fuchsian Groups ◽  
Hyperbolic Distance ◽  
Minimal Distance ◽  
Geometrically Finite ◽  
Upper Half Plane

Let [Formula: see text] be a geometrically finite Fuchsian group acting on the upper half-plane [Formula: see text]. Let [Formula: see text] denote the set of elliptic fixed points of [Formula: see text] in [Formula: see text]. We give a lower bound on the minimal hyperbolic distance between points in [Formula: see text]. Our bound depends on a universal constant and the length of the smallest closed geodesic on [Formula: see text].

scholarly journals Fractional parts of Dedekind sums

2016 ◽  
Vol 12 (05) ◽  
pp. 1137-1147
Author(s):  
William D. Banks ◽  
Igor E. Shparlinski
Keyword(s):  
Number Theory ◽  
Uniform Distribution ◽  
Exponential Sums ◽  
Kloosterman Sums ◽  
Duke Math ◽  
Dedekind Sums ◽  
Bilinear Forms ◽  
Recent Improvement ◽  
Double Exponential ◽  
Invent Math

Using a recent improvement by Bettin and Chandee to a bound of Duke, Friedlander and Iwaniec [Bilinear forms with Kloosterman fractions, Invent. Math. 128 (1997) 23–43] on double exponential sums with Kloosterman fractions, we establish a uniformity of distribution result for the fractional parts of Dedekind sums [Formula: see text] with [Formula: see text] and [Formula: see text] running over rather general sets. Our result extends earlier work of Myerson [Dedekind sums and uniform distribution, J. Number Theory 28 (1988) 233–239] and Vardi [A relation between Dedekind sums and Kloosterman sums, Duke Math. J. 55 (1987) 189–197]. Using different techniques, we also study the least denominator of the collection of Dedekind sums [Formula: see text] on average for [Formula: see text].

scholarly journals SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE

2020 ◽  
Vol 8 ◽  
Author(s):  
DANIEL LE ◽  
BAO V. LE HUNG ◽  
BRANDON LEVIN ◽  
STEFANO MORRA
Keyword(s):  
Representation Theory ◽  
Galois Representations ◽  
Galois Representation ◽  
Modular Representation ◽  
Duke Math ◽  
Modular Representation Theory ◽  
Deformation Rings ◽  
Invent Math

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $p$ . This is a generalization to $\text{GL}_{3}$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $n$ -dimensional Galois representations’, Duke Math. J. 149(1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math. 212(1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $\text{GL}_{3}(\mathbb{F}_{q})$ .

scholarly journals ON THE EXCEPTIONAL SPECIALIZATIONS OF BIG HEEGNER POINTS

2016 ◽  
Vol 17 (1) ◽  
pp. 207-240 ◽  
Author(s):  
Francesc Castella
Keyword(s):  
Duke Math ◽  
Heegner Points ◽  
Hida Family ◽  
Derivative Formula ◽  
Heegner Cycles ◽  
Hida Families ◽  
Invent Math

We extend the $p$-adic Gross–Zagier formula of Bertolini et al. [Generalized Heegner cycles and $p$-adic Rankin $L$-series, Duke Math. J.162(6) (2013), 1033–1148] to the semistable non-crystalline setting, and combine it with our previous work [Castella, On the $p$-adic variation of Heegner points, Preprint, 2014, arXiv:1410.6591] to obtain a derivative formula for the specializations of Howard’s big Heegner points [Howard, Variation of Heegner points in Hida families, Invent. Math.167(1) (2007), 91–128] at exceptional primes in the Hida family.

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

scholarly journals EXTREME VALUES OF GEODESIC PERIODS ON ARITHMETIC HYPERBOLIC SURFACES

2020 ◽  
pp. 1-36
Author(s):  
Bart Michels
Keyword(s):  
Lower Bound ◽  
Closed Geodesic ◽  
Extreme Values ◽  
Theta Series ◽  
Hyperbolic Surface ◽  
Quadratic Fields ◽  
Maass Forms ◽  
Real Quadratic Fields ◽  
Central Values ◽  
Real Quadratic

Abstract Given a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

scholarly journals An Algebraic Brascamp–Lieb Inequality

2021 ◽  
Author(s):  
Jennifer Duncan
Keyword(s):  
General Class ◽  
Recent Progress ◽  
Algebraic Groups ◽  
Hölder’S Inequality ◽  
Duke Math ◽  
Rational Maps ◽  
Affine Invariant ◽  
Linear Maps ◽  
Nonlinear Maps ◽  
Funct Anal

AbstractThe Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions $$f_j\in L^{q_j}(\mathbb {R}^{n_j})$$ f j ∈ L q j ( R n j ) , for $$j=1,\ldots ,m$$ j = 1 , … , m , under some corresponding linear maps $$B_j$$ B j . This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where $$B_j$$ B j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

scholarly journals The local-to-global property for Morse quasi-geodesics

2021 ◽  
Author(s):  
Jacob Russell ◽  
Davide Spriano ◽  
Hung Cong Tran
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Type Theorem ◽  
Hyperbolic Groups ◽  
Global Property ◽  
Mapping Class ◽  
Relatively Hyperbolic Groups ◽  
Convex Cocompact ◽  
Relatively Hyperbolic ◽  
Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

scholarly journals Laurent phenomenon and simple modules of quiver Hecke algebras

2019 ◽  
Vol 155 (12) ◽  
pp. 2263-2295 ◽  
Author(s):  
Masaki Kashiwara ◽  
Myungho Kim
Keyword(s):  
Simple Module ◽  
Hecke Algebras ◽  
Duke Math ◽  
Cluster Algebras ◽  
Coordinate Ring ◽  
Simple Modules ◽  
Laurent Phenomenon ◽  
Cluster Variable ◽  
Quantum Cluster

In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$, then $[S]$ is a cluster monomial in $[\mathscr{C}]$. If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$, then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$.

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