| 1. |
- Bäcklund, Pierre, 1976-
(author)
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Studies on boundary values of eigenfunctions on spaces of constant negative curvature
- 2008
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Doctoral thesis (other academic/artistic)abstract
- This thesis consists of two papers on the spectral geometry of locally symmetric spaces of Riemannian and Lorentzian signature. Both works are concerned with the idea of relating analysis on such spaces to structures on their boundaries.The first paper is motivated by a conjecture of Patterson on the Selberg zeta function of Kleinian groups. We consider geometrically finite hyperbolic cylinders with non-compact Riemann surfaces of finite area as cross sections. For these cylinders, we present a detailed investigation of the Bunke-Olbrich extension operator under the assumption that the cross section of the cylinder has one cusp. We establish the meromorphic continuation of the extension of Eisenstein series and incomplete theta series through the limit set. Furthermore, we derive explicit formulas for the residues of the extension operator in terms of boundary values of automorphic eigenfunctions.The motivation for the second paper comes from conformal geometry in Lorentzian signature. We prove the existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.
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| 2. |
- Avelin, Helen, 1975-
(author)
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Computations of Automorphic Functions on Fuchsian Groups
- 2007
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Doctoral thesis (other academic/artistic)abstract
- This thesis consists of four papers which all deal with computations of automorphic functions on cofinite Fuchsian groups. In the first paper we develop an algorithm for numerical computation of the Eisenstein series. We focus in particular on the computation of the poles of the Eisenstein series. Using our numerical methods we study the spectrum of the Laplace-Beltrami operator as the surface is being deformed. Numerical evidence of the destruction of Γ0(5)-cusp forms is presented. In the second paper we use the algorithm described in the first paper. We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series E(z;s) on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real-valued rotation of E(z;s) as Re s=1/2, Im s → ∞ and also, on non-arithmetic groups, a complex Gaussian limit distribution for E(z;s) when Re s > 1/2 near 1/2 and Im s → ∞, at least if we allow Re s → 1/2 at some rate. Furthermore, on non-arithmetic groups and for fixed s with Re s ≥ 1/2 near 1/2, our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.In the third paper we develop algorithms for computations of Green's function and its Fourier coefficients, Fn(z;s), on Fuchsian groups with one cusp. Also an analog of a Rankin-Selberg bound for Fn(z;s) is presented.In the fourth paper we use the algorithms described in the third paper. We present some examples of numerical investigations of the value distribution of the Green's function and of its Fourier coefficients on PSL(2,Z). We also discuss the appearance of pseudo cusp forms in a numerical experiment by Hejhal.
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| 3. |
- Booker, Andrew, et al.
(author)
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Effective Computation of Maass Cusp Forms
- 2006
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In: International mathematics research notices. - 1073-7928 .- 1687-0247. ; 2006, s. 71281-
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Journal article (peer-reviewed)abstract
- We study theoretical and practical aspects of high-precision computation of Maass forms. First, we compute to over 1000 decimal places the Laplacian and Hecke eigenvalues for the first few Maass forms on PSL(2, Z)\H. Second, we give an algorithm for rigorously verifying that a proposed eigenvalue together with a proposed set of Fourier coefficients indeed correspond to a true Maass cusp form. We apply this to prove that our values for the first ten eigenvalues on PSL( 2, Z)\H are correct to at least 100 decimal places. Third, we test some algebraicity properties of the coefficients, among other things giving evidence that the Laplacian and Hecke eigenvalues of Maass forms on PSL( 2,Z)\H are transcendental.
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| 4. |
- Booker, Andrew, et al.
(author)
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Numerical computations with the trace formula and the Selberg eigenvalue conjecture
- 2007
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In: Journal für die Reine und Angewandte Mathematik. - 0075-4102 .- 1435-5345. ; 2007:607, s. 113-161
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Journal article (peer-reviewed)abstract
- We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [M. N. Huxley, Introduction to Kloostermania, in: Elementary and analytic theory of numbers, Banach Center Publ. 17, Warsaw (1985), 217–306.]. The main tool is the Selberg trace formula which, unlike previous geometric methods, allows for treatment of cases where the eigenvalue 1/4 is present. We present a few other sample applications, including the classification of even 2-dimensional Galois representations of small squarefree conductor.
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| 5. |
- Booker, Andrew R., et al.
(author)
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Twist-minimal trace formulas and the Selberg eigenvalue conjecture
- 2020
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In: Journal of the London Mathematical Society. - : Wiley. - 0024-6107 .- 1469-7750. ; 102:3, s. 1067-1134
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Journal article (peer-reviewed)abstract
- We derive a fully explicit version of the Selberg trace formula for twist‐minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2‐dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by Doud and Moore (J. Number Theory 118 (2006) 62–70) of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley (Elementary and analytic theory of numbers, Banach Center Publications 17 (PWN, Warsaw, 1985) 217–306) from 1985.
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| 6. |
- Dettmann, Carl P., et al.
(author)
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Universal Hitting Time Statistics for Integrable Flows
- 2017
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In: Journal of statistical physics. - : SPRINGER. - 0022-4715 .- 1572-9613. ; 166:3-4, s. 714-749
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Journal article (peer-reviewed)abstract
- The perceived randomness in the time evolution of "chaotic" dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for "generic" integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner's measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen.
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| 7. |
- Edwards, Samuel Charles, 1990-
(author)
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Some applications of representation theory in homogeneous dynamics and automorphic functions
- 2018
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Doctoral thesis (other academic/artistic)abstract
- This thesis consists of an introduction and five papers in the general area of dynamics and functions on homogeneous spaces. A common feature is that representation theory plays a key role in all articles.Papers I-IV are concerned with the effective equidistribution of translates of pieces of subgroup orbits in quotient spaces of semisimple Lie groups by discrete subgroups. In Paper I we focus on finite-volume quotients of SL(2,C) and study the speed of equdistribution for expanding translates orbits of horospherical subgroups. Paper II also studies the effective equidistribution of translates of horospherical orbits, though now in the setting of a quotient of a general semisimple Lie group by a lattice subgroup. Like Paper II, Paper III considers effective equidistribution in quotients of general semisimple Lie groups, but now studies translates of orbits of symmetric subgroups. In all these papers we show that the translates equidistribute with the same exponential rate as for the decay of the corresponding matrix coefficients of the translating subgroup. In Paper IV we consider the effective equidistribution of translates of pieces of horospheres in infinite-volume quotients of groups SO(n,1) by geometrically finite subgroups, and improve the dependency on the spectral gap for certain known effective equidistribution results.In Paper V we study the Fourier coefficients of Eisenstein series for generic non-cocompact cofinite Fuchsian groups. We use Zagier's renormalization of certain divergent integrals to enable use of the so-called triple product method, and then combine this with the analytic continuation of irreducible representations of SL(2,R) due to Bernstein and Reznikov.
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| 10. |
- Kleinbock, Dmitry, et al.
(author)
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A measure estimate in geometry of numbers and improvements to Dirichlet's theorem
- 2022
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In: Proceedings of the London Mathematical Society. - : John Wiley & Sons. - 0024-6115 .- 1460-244X. ; 125:4, s. 778-824
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Journal article (peer-reviewed)abstract
- Let psi$\psi$ be a continuous decreasing function defined on all large positive real numbers. We say that a real mxn$m\times n$ matrix A$A$ is psi$\psi$-Dirichlet if for every sufficiently large real number t$t$ one can find p is an element of Zm$\bm {p} \in {\mathbb {Z}}<^>m$, q is an element of Zn set minus {0}$\bm {q} \in {\mathbb {Z}}<^>n\setminus \lbrace \bm {0}\rbrace$ satisfying parallel to Aq-p parallel to mm< \psi ({t})$ and parallel to q parallel to nn<{t}$. This property was introduced by Kleinbock and Wadleigh in 2018, generalizing the property of A$A$ being Dirichlet improvable which dates back to Davenport and Schmidt (1969). In the present paper, we give sufficient conditions on psi$\psi$ to ensure that the set of psi$\psi$-Dirichlet matrices has zero or full Lebesgue measure. Our proof is dynamical and relies on the effective equidistribution and doubly mixing of certain expanding horospheres in the space of lattices. Another main ingredient of our proof is an asymptotic measure estimate for certain compact neighborhoods of the critical locus (with respect to the supremum norm) in the space of lattices. Our method also works for the analogous weighted problem where the relevant supremum norms are replaced by certain weighted quasi-norms.
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