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Dia de Geometria

Dia de Geometria / Geometry Day -- June 19, 2018

Schedule (see below for the abstracts)


10:00-11:00 Fátima Silva Leite (Univ. Coimbra, Portugal), What is geometric control theory?

11:00-12:00 Emmanuel Trelat (Univ. Sorbonne, France), Spectral analysis of sub-Riemannian Laplacians and Weyl measure.


14:00-15:00 V. Jurdjevic (Univ. Toronto, Canada), Jacobi's geodesic problem and integrable Hamiltonian systems on Lie algebras.

15:00-16:00 Irina Markina (Univ. Bergen, Norway), Sub-Riemannian structures on spheres S3 and S7.




Fátima Silva Leite (Univ. Coimbra, Portugal)


Title: What is geometric control theory?


Abstract: In this lecture we present some of the basic concepts of mathematical control theory, namely controllability and optimal control, and give special emphasis to geometric control where tools from differential geometry play an important role. A control system is a family of dynamical systems parameterized by the controls, and so it can be seen as a family of vector fields. The most basic theoretical tool of the geometric viewpoint is the Lie bracket.
Controllability is related to the ability to reach a state from any other state in the system, using the available controls. Optimal control deals with the possibility to do it in the best possible way. These concepts will be illustrated with simple examples.




Emmanuel Trelat (Univ. Sorbonne, France)


Title: Spectral analysis of sub-Riemannian Laplacians and Weyl measure.


Abstract: In a series of works on sub-Riemannian geometry with Yves Colin de Verdière and Luc Hillairet, we study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration occurs on the shief generated by Lie brackets of length r-1, where r is the degree of nonholonomy. In the singular case, like Martinet or Grushin, the situation is more involved but we obtain small-time asymptotic expansions of the heat kernel and the Weyl law in some cases.




V. Jurdjevic (Univ. Toronto, Canada)


Title: Jacobi's geodesic problem and integrable Hamiltonian systems on Lie algebras.


Abstract: Jacobi's Geodesic problem of 1835 asked for a curve x(t) on the ellipsoid En={x\in Rn+1:\sum_{i=0}^n \frac{1}{a_i}x_i^2}=1 that connects two given points of En whose length L=\int_0^1\sqrt{\sum_{i=0}^n (\frac{dx_i}{dt})^2} dt is minimal.
Jacobi found a system of coordinates, known ever since as the elliptic coordinates, in terms of which the associated Hamilton-Jacobi partial differential equation is separable. This discovery led to a class of integrable systems by the method of separation of variables -- known as the inverse method of Jacobi. (Landau, Lifschitz, Mechanics).

To explain our interest in this result we will go to the papers of J. Moser who in 1975 wrote the Hamiltonian equations for Jacobi's problem in the coordinates (x,p) of the cotangent bundle of the ambient space Rn+1 as follows:
\frac{dx}{dt}=p,\frac{dp}{dt}=-\frac{(p,A^{-1}p)}{2||A^{-1}x||^{2}}A^{-1}x. (1)
Remarkably, functions Fk=pk2+\sum_{j=0,j\neq k}^{n}\frac{(x_{j}p_{k}-x_{k}p_{j})^2}{(a _{k}-a _{j})},k=0,...,n, A=diag(a_0,\dots,a_n) are constant along the solutions of (1). Even more remarkably, these functions are in involution relative to the Poisson bracket in the ambient space, hence Jacobi's problem is Liouville integrable.

This lecture is principally motivated by Arnold's query: Are there any hidden symmetries that account for the integrability of Jacobi's problem? This question seems particularly pertinent, since a generic ellipsoid does not seem to have any obvious symmetries, and integrability is invariably connected with symmetries.

We will tackle this question in a somewhat indirect manner: we will first show that an elliptic geodesic problem on the unit sphere, that is the metric of the form \sqrt{\sum_{i=1}^n a_i\dot x_i^2} is equivalent to Jacobi's geodesic problem on the ellipsoid. Then we will note that the cotangent bundle of the sphere appears as a coadjoint orbit on the space of symmetric matrices. Finally we will show that the integrals of motion for the elliptic geodesic problem on the sphere correspond to the spectral invariants of an isospectral Hamiltonian on a Lie group G restricted to a particular coadjoint orbit (in our case the cotangent bundle of the sphere), which in turn yields an easy correspondence with the integrals of motion for Jacobi's problem on the ellipsoid.

Along the way we will show that our quest for the answer to Arnold's query sheds new light on the famous discoveries of S.M. Manakov, A.T. Fomenko, A.S. Mischenko, V.V. Trofimov and O. Bogoyavlensky on the integrability of n- dimensional mechanical tops.



Irina Markina (Univ. Bergen, Norway)


Title: Sub-Riemannian structures on spheres S3 and S7.


Abstract: In the talk we describe various sub-Riemannian structures on 3- and 7-dimensional spheres. The sub-Riemannian structures on S3, related to the right action of Lie group over itself, the one inherited from the natural complex structure of the open unit ball in C2 and the geometry that appears when considering it as a principal bundle via the Hopf map coincide. The sub-Riemannian structures on S7 with distributions of rank 4, can be obtained, for instance, by the quaternion Hopf map, as a span of Clifford vector fields or as the natural quaternion structure of the open unit ball in H2. The structures are different and we will discuss the bracket generating properties of the obtained distributions.



Place:   Sala 5.5
Start Date:   2018-06-19
Start Time:   10:00
Research Groups: -Geometry
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© 2012 Centre for Mathematics, University of Coimbra, funded by

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