Hybrid Bounds For Automorphic Forms On Ellipsoids Over Number Fields

We prove upper bounds for Hecke-Laplace eigenfunctions on certain Riemannian manifolds X of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are d-fold products of 2-spheres or 3-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a ('Weyl-type') saving of vol (X)(-1/6+epsilon).

Optimal control of geodesics in Riemannian manifolds

Roland Rozsnyo

The aim of this dissertation is to solve numerically the following problem, denoted by P : given a Riemannian manifold and two points a and b belonging to that manifold, find a tangent vector T at a, such that expa(T) = b, assuming that T exists. This problem is set under an optimal control formulation, which requires the definition of an objective function and a space of control, the choice of a method for the calculation of the descent direction of that function in the space of control and the use of an optimization algorithm to find its minimum, which corresponds to the solution of the original problem by construction. Several techniques are necessary to be put together, coming from the fields of geometry, numerical analysis and optimization. The first part will concern a recalling of the mathematical context in which this formulation takes place. The general principles of optimal control will also be given. In the second part, we will present an intrinsic formulation of the optimal control problem associated to P, based on Jacobi fields, which will play the role of the so called adjoint state. This derivation leads to necessary optimality conditions. We will illustrate explicitly that formulation by treating the specific case of Riemannian manifolds with constant sectional curvature. Then, we will derive the optimal control problem in coordinates, not only to check the intrinsic formulation but also to reveal how it is hidden behind the expressions in coordinates. Their use reveals some quantities whose interpretation may be given this way. Moreover, we will show that more possibilities exist to chose the cost function and the control space in coordinates. In a second step, an alternative approach will consider the Hamiltonian formulation of geodesics. This is an incursion into symplectic geometry. We will then reformulate the Riemannian optimal control problem in its Hamiltonian version. In the third part, the numerical methods used for solving P will be presented. The discretization imposes the definition of new discrete optimal control problems. The technique shows that the discrete adjoint state equation strongly depends on the numerical scheme used to solve the direct problem. We will give a collection of numerical computations in the specific case of parametric piece of surfaces, where the surface can be defined by one or several Bézier patches, each one corresponding to a chart, which is representative of a Riemannian manifold. We will compare the different numerical approaches. The last but one part will be devoted to the interesting application of wooden roof building, where the structure is made of wooden boards, with geodesic trajectories on the designed piece of surface. The Geos (Geodesic solver) software has been developed for that purpose. After having introduced some specific numerical methods used in the code, we present the Geos application interface (AI) developed as a tool for the conception of such a roof. We then show an existing wooden structure built according to that mean. Finally, we will summarize the results of our research and discuss future possible prospects.

EPFL2007

Lq,p-cohomology of Riemannian manifolds and simplicial complexes of bounded geometry

Stephen Ducret

The Lq,p-cohomology of a Riemannian manifold (M, g) is defined to be the quotient of closed Lp-forms, modulo the exact forms which are derivatives of Lq-forms, where the measure considered comes from the Riemannian structure. The Lq,p-cohomology of a simplicial complex K is defined to be the quotient of p-summable cocycles of K, modulo the coboundaries of q-summable cocycles. We introduce those two notions together with a variant for coarse cohomology on graphs, and we establish their main properties. We define the categories we work on, i.e. manifolds and simplicial complexes of bounded geometry, and we show how cohomology classes can be represented by smooth forms. The first result of the thesis is a de Rham type theorem: we prove that for an orientable, complete and (non compact) Riemannian manifold with bounded geometry (M, g) together with a triangulation K with bounded geometry, the Lq,p-cohomology of the manifold coincides with the Lq,p-cohomology of the triangulation. This is a generalization of an earlier result from Gol'dshtein, Kuz'minov and Shvedov. The second result is a quasi-isometry invariance one: we prove how this de Rham type isomorphism together with a result in coarse cohomology induces the fact that the Lq,p-cohomology of a Riemannian manifold depends only on its quasi-invariance class. This result was proved in the q = p case by Elek. We establish some consequences, such as monocity results for Lq,p-cohomology, and the quasi-isometry invariance of the existence of Sobolev inequalities.

EPFL2009

Arithmetic Bounds-Lenstra's Constant and Torsion of K-Groups

Julien Houriet

This thesis is concerned with computations of bounds for two different arithmetic invariants. In both cases it is done with the intention of proving some algebraic or arithmetic properties for number fields. The first part is devoted to computations of lower bounds for the Lenstra's constant. For a number field K the Lenstra's constant is denoted Λ(K) and defined as the length of the largest exceptional sequence in K. An exceptional sequence is a set of units in K such that for any two among them their difference is a unit as well. H.W. Lenstra showed that if Λ(K) is large enough – bigger than a constant depending on the degree and the discriminant of K – then the ring of integers of K is Euclidean with respect to the norm. Using computer software PARI/GP and some algorithms from graph theory we construct exceptional sequences in number fields having a small discriminant. These exceptional sequences yield lower bounds for Lenstra's constant which are large enough to prove the existence of 42 new Euclidean number fields of degree 8 to 12. The aim of the second part of this thesis is proving upper bounds for the torsion part of the K-groups of a number field ring of integers. A method due to C. Soulé yields bounds for the torsion of these K-groups depending on an invariant of hermitian lattices over number fields. Firstly we describe some properties of rank one hermitian lattices, especially of ideal lattices. Secondly we apply these properties to arbitrary rank hermitian lattices and this implies a significant improvement of the upper bounds for their invariants and accordingly for the torsion of K-groups. The progress mainly achieves much lower contributions of the number field attributes, particularly the degree and the absolute discriminant.

EPFL2010

Optimal control of geodesics in Riemannian manifolds

Roland Rozsnyo

EPFL2007