CIRM - Videos & books Library - Lie Theory and Generalizations

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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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Beyond endoscopy is the strategy put forward by Langlands for applying the trace formula to the general principle of functoriality. Subsequent papers by Langlands (one in collaboration with Frenkel and Ngo), together with more recent papers by Altug, have refined the strategy. They all emphasize the importance of understanding the elliptic terms on the geometric side of the trace formula. We shall discuss the general strategy, and how it pertains to these terms.[-]

Beyond endoscopy is the strategy put forward by Langlands for applying the trace formula to the general principle of functoriality. Subsequent papers by Langlands (one in collaboration with Frenkel and Ngo), together with more recent papers by Altug, have refined the strategy. They all emphasize the importance of understanding the elliptic terms on the geometric side of the trace formula. We shall discuss the general strategy, and how it ...[+]

11F66 ; 22E50 ; 22E55

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In this series of lectures, we will focus on simple Lie groups, their dense subgroups and the convolution powers of their measures. In particular, we will dicuss the following two questions.
Let G be a Lie group. Is every Borel measurable subgroup of G with maximal Hausdorff dimension equal to the group G?
Is the convolution of sufficiently many compactly supported continuous functions on G always continuously differentiable?
Even though the answer to these questions is no when G is abelian, the answer is yes when G is simple. This is a joint work with N. de Saxce. First, I will explain the history of these two questions and their interaction. Then, I will relate these questions to spectral gap properties. Finally, I will discuss these spectral gap properties.[-]

In this series of lectures, we will focus on simple Lie groups, their dense subgroups and the convolution powers of their measures. In particular, we will dicuss the following two questions.
Let G be a Lie group. Is every Borel measurable subgroup of G with maximal Hausdorff dimension equal to the group G?
Is the convolution of sufficiently many compactly supported continuous functions on G always continuously differentiable?
Even though the ...[+]

22E30 ; 28A78 ; 43A65

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In these lectures, we will report on some properties of the space of actions of a left-orderable group on the real line. We will notably describe the almost-periodic actions, the harmonic actions and their spaces.

20F60 ; 22F50 ; 37B05 ; 37E10 ; 57R30

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I will explain how certain probabilistic methods can be used to study the discrete groups of semisimple Lie groups. I will define the space of subgroups with the Chabauty topology and introduce two useful classes of random subgroups - invariant ans stationary random subgroups. In higher rank thses classes admit nice classification witch can be usesd to prove that a confined subgroup of a simple higher rank group must be a lattice.

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We introduce Margulis' and Zimmer's superrigidity. Statements give heuristics in Zimmer program, that is higher rank lattice actions on smooth manifolds. After we state the statement, we mainly focus how it interacts with group actions. Finally, we will also discuss about open questions.

22E40 ; 57M60

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Given a nilpotent Lie algebra over a characteristic zero field, one can construct a group in a universal way via the Baker-Campbell-Hausdorff formula. This integration procedure admits generalizations to dg Lie or L∞-algebras, giving in general ∞-groupoid of deformations that it encodes, as by the Lurie-Pridham correspondence, infinitesimal deformation problems are equivalent to dg Lie algebras. The recent work of Brantner-Mathew establishes a correspondence between infinitesimal deformation problems and partition Lie algebras over a positive characteristic field. In this talk, I will explain how to construct an analogue of the integration functor for certain point-set models of (spectral) partition Lie algebras, and how this integration functor can recover the associated deformation problem under some assumptions. Furthermore, I will discuss some applications of these constructions to unstable p-adic homotopy theory.[-]

Given a nilpotent Lie algebra over a characteristic zero field, one can construct a group in a universal way via the Baker-Campbell-Hausdorff formula. This integration procedure admits generalizations to dg Lie or L∞-algebras, giving in general ∞-groupoid of deformations that it encodes, as by the Lurie-Pridham correspondence, infinitesimal deformation problems are equivalent to dg Lie algebras. The recent work of Brantner-Mathew establishes a ...[+]

18M70 ; 18N40 ; 22E60 ; 55P62 ; 55U10 ; 14D15 ; 14D23

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Spherical Hecke algebra, Satake transform, and an introduction to local Langlands correspondence.
CIRM - Chaire Jean-Morlet 2016 - Aix-Marseille Université

20C08 ; 22E50 ; 11S37

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11F67 ; 11F70 ; 11F72 ; 22E55

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11F67 ; 11F70 ; 11F72 ; 22E55

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Lie Theory and Generalizations 54 résultats

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

Beyond Endoscopy and elliptic terms in the trace formula - Arthur, James Greig (Auteur de la Conférence) | CIRM H Nouveau

Dense subgroups in simple groups - Lecture 2 - Benoist, Yves (Auteur de la Conférence) | CIRM H Nouveau

Space of actions of groups on the real line - Deroin, Bertrand (Auteur de la Conférence) | CIRM H Nouveau

Random processes on symmetric spaces and discrete subgroups - Fraczyk, Mikolaj (Auteur de la Conférence) | CIRM H Nouveau

Margulis-Zimmer's super-rigidity - Lee, Homin (Auteur de la Conférence) | CIRM H Nouveau

Higher Lie theory in positive characteristic - Roca i Lucio, Victor (Auteur de la Conférence) | CIRM H Nouveau

The local Langlands correspondence: functoriality, $L$-functions, gamma functions and the epsilon factors - Prasad, Dipendra (Auteur de la Conférence) | CIRM H Nouveau

Lecture 1: Distinction and the geometric lemma - Offen, Omer (Auteur de la Conférence) | CIRM H Nouveau

Lecture 2: Distinction by a symmetric subgroup - Offen, Omer (Auteur de la Conférence) | CIRM H Nouveau