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Post-edited  My favorite groups
Ghys, Etienne (Auteur de la Conférence) | CIRM (Editeur )

The world of groups is vast and meant for wandering! During this week, I will give seven short talks describing seven groups, or class of groups, that I find fascinating. These seven talks will be independent and I will have no intention of being exhaustive (this would be silly since there are uncountably many groups, even finitely generated!). In each talk, I will introduce the hero, state one or two results, and formulate one or two conjectures. The world of groups is vast and meant for wandering! During this week, I will give seven short talks describing seven groups, or class of groups, that I find fascinating. These seven talks will be independent and I will have no intention of being exhaustive (this would be silly since there are uncountably many groups, even finitely generated!). In each talk, I will introduce the hero, state one or two results, and formulate one or two c...

57S30 ; 58D05

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Post-edited  Bounds for the number of rational points on curves over global fields
Pazuki, Fabien (Auteur de la Conférence) | CIRM (Editeur )

Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the number field case and on a way to strengthen it assuming a height conjecture. During the second part we will focus on function fields of positive characteristic and describe a new result obtained in a joined work with Pacheco. Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the ...

14G05 ; 11G35

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Post-edited  On Schmidt's subspace theorem
Evertse, Jan-Hendrik (Auteur de la Conférence) | CIRM (Editeur )

Last year, I published together with Roberto Ferretti a new version of the quantitative subspace theorem, giving a better upper bound for the number of subspaces containing the solutions of the system of inequalities involved. In my lecture, I would like to discuss this improvement, and go into some aspects of its proof.

11J13 ; 11J68

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Post-edited  Formulas for the limiting distribution of traces of Frobenius
Lachaud, Gilles (Auteur de la Conférence) | CIRM (Editeur )

We discuss the distribution of the trace of a random matrix in the compact Lie group USp2g, with the normalized Haar measure. According to the generalized Sato-Tate conjecture, if A is an abelian variety of dimension g defined over the rationals, the sequence of traces of Frobenius in the successive reductions of A modulo primes appears to be equidistributed with respect to this distribution. If g = 2, we provide expressions for the characteristic function, the density, and the repartition function of this distribution in terms of higher transcendental functions, namely Legendre and Meijer functions. We discuss the distribution of the trace of a random matrix in the compact Lie group USp2g, with the normalized Haar measure. According to the generalized Sato-Tate conjecture, if A is an abelian variety of dimension g defined over the rationals, the sequence of traces of Frobenius in the successive reductions of A modulo primes appears to be equidistributed with respect to this distribution. If g = 2, we provide expressions for the cha...

11G05 ; 11G10 ; 14G10 ; 37C30

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Post-edited  Polignac numbers and the consecutive gaps between primes
Pintz, János (Auteur de la Conférence) | CIRM (Editeur )

We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, Zhang and Maynard-Tao, respectively. We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a ...

11N05 ; 11B05

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Post-edited  Which geodesic flows are left-handed?
Dehornoy, Pierre (Auteur de la Conférence) | CIRM (Editeur )

Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows are good candidates. In this conference we determine on which hyperbolic orbifolds is the geodesic flow left-handed: the answer is that yes if the surface is a sphere with three cone points, and no otherwise.
dynamical system - geodesic flow - knot - periodic orbit - global section - linking number - fibered knot
Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows ...

37C27 ; 37C15 ; 37C10 ; 57M25

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Post-edited  Extended Lagrange spaces and optimal control
Mehrmann, Volker (Auteur de la Conférence) | CIRM (Editeur )

Mathematical modeling and numerical mathematics of today is very much Lagrangian and modern automated modeling techniques lead to differential-algebraic systems. The optimal control for such systems in general cannot be obtained using the classical Euler-Lagrange approach or the maximum principle, but it is shown how this approach can be extended.
differential-algebraic equations - optimal control - Lagrangian subspace - necessary optimality conditions - Hamiltonian system - symplectic flow
Mathematical modeling and numerical mathematics of today is very much Lagrangian and modern automated modeling techniques lead to differential-algebraic systems. The optimal control for such systems in general cannot be obtained using the classical Euler-Lagrange approach or the maximum principle, but it is shown how this approach can be extended.
differential-algebraic equations - optimal control - Lagrangian subspace - necessary optimality ...

93C05 ; 93C15 ; 49K15 ; 34H05