{"title":"Commission Internationale de l’Enseignement Mathématique. The 2017 ICMI Awards Felix Klein and Hans Freudenthal Medals","authors":"","doi":"10.4171/lem/63-3/4-9","DOIUrl":"https://doi.org/10.4171/lem/63-3/4-9","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128098704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give elementary proof of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces $Xsubset mathbb{P}^n(mathbb{R})$. The main tool is a refinement of the simplex lemma, which essentially says that rational points on $X$ which are sufficiently close to each other must lie on a totally isotropic rational subspace of $X$.
{"title":"Rational approximation on quadrics: A simplex lemma and its consequences","authors":"D. Kleinbock, Nicolas de Saxc'e","doi":"10.4171/LEM/64-3/4-11","DOIUrl":"https://doi.org/10.4171/LEM/64-3/4-11","url":null,"abstract":"We give elementary proof of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces $Xsubset mathbb{P}^n(mathbb{R})$. The main tool is a refinement of the simplex lemma, which essentially says that rational points on $X$ which are sufficiently close to each other must lie on a totally isotropic rational subspace of $X$.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122917652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This expository paper introduces the theory of Harish-Chandra integrals, a family of special functions that express the integral of an exponential function over the adjoint orbits of a compact Lie group. Originally studied in the context of harmonic analysis on Lie algebras, Harish-Chandra integrals now have diverse applications in many areas of mathematics and physics. We review a number of these applications, present several different proofs of Harish-Chandra’s celebrated exact formula for the integrals, and give detailed derivations of the specific integral formulae for all compact classical groups. These notes are intended for mathematicians and physicists who are familiar with the basics of Lie groups and Lie algebras but who may not be specialists in representation theory or harmonic analysis.
{"title":"The Harish-Chandra integral: An introduction with examples","authors":"Colin S. McSwiggen","doi":"10.4171/lem/1017","DOIUrl":"https://doi.org/10.4171/lem/1017","url":null,"abstract":"This expository paper introduces the theory of Harish-Chandra integrals, a family of special functions that express the integral of an exponential function over the adjoint orbits of a compact Lie group. Originally studied in the context of harmonic analysis on Lie algebras, Harish-Chandra integrals now have diverse applications in many areas of mathematics and physics. We review a number of these applications, present several different proofs of Harish-Chandra’s celebrated exact formula for the integrals, and give detailed derivations of the specific integral formulae for all compact classical groups. These notes are intended for mathematicians and physicists who are familiar with the basics of Lie groups and Lie algebras but who may not be specialists in representation theory or harmonic analysis.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132541070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the Hamiltonian dynamics of a charged particle submitted to a pure magnetic field in a two-dimensional domain. We provide conditions on the magnetic field in a neighbourhood of the boundary to ensure the confinement of the particle. We also prove a formula for the scattering angle in the case of radial magnetic fields.
{"title":"Boundary effects on the magnetic Hamiltonian dynamics in two dimensions","authors":"Tho Nguyen Duc, N. Raymond, San Vũ Ngọc","doi":"10.4171/LEM/64-3/4-7","DOIUrl":"https://doi.org/10.4171/LEM/64-3/4-7","url":null,"abstract":"We study the Hamiltonian dynamics of a charged particle submitted to a pure magnetic field in a two-dimensional domain. We provide conditions on the magnetic field in a neighbourhood of the boundary to ensure the confinement of the particle. We also prove a formula for the scattering angle in the case of radial magnetic fields.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131198067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
After surveying existing proofs that every closed, orientable 3-manifold is parallelizable, we give three proofs using minimal background. In particular, our proofs use neither spin structures nor the theory of Stiefel-Whitney classes.
{"title":"Framing 3-manifolds with bare hands","authors":"R. Benedetti, P. Lisca","doi":"10.4171/LEM/64-3/4-9","DOIUrl":"https://doi.org/10.4171/LEM/64-3/4-9","url":null,"abstract":"After surveying existing proofs that every closed, orientable 3-manifold is parallelizable, we give three proofs using minimal background. In particular, our proofs use neither spin structures nor the theory of Stiefel-Whitney classes.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125602636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class, and there exist many such curves splitting the class. We show that this can be false when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class.
{"title":"Splitting Brauer classes using the universal Albanese","authors":"Wei Ho, Max Lieblich","doi":"10.4171/lem/1009","DOIUrl":"https://doi.org/10.4171/lem/1009","url":null,"abstract":"We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class, and there exist many such curves splitting the class. We show that this can be false when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129412703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is shown that under mild conditions, Benjamini-Schramm convergence of lattices in locally compact groups is equivalent to spectral convergence. Next both notions are extended to the relative case and are then expressed in terms of relative L2-theory.
{"title":"Benjamini–Schramm and spectral convergence","authors":"A. Deitmar","doi":"10.4171/LEM/64-3/4-8","DOIUrl":"https://doi.org/10.4171/LEM/64-3/4-8","url":null,"abstract":"It is shown that under mild conditions, Benjamini-Schramm convergence of lattices in locally compact groups is equivalent to spectral convergence. Next both notions are extended to the relative case and are then expressed in terms of relative L2-theory.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123981246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compute exact convergence rates in von Neumann type ergodic theorems when the acting group of measure preserving transformations is free and the means are taken over spheres or over balls defined by a word metric. Relying on the upper bounds on the spectra of Koopman operators deduced by Lubozky, Phillips, and Sarnak from Deligne's work on the Weil conjecture, we compute the exact convergence rate for the free groups (of rank $(p+1)/2$ where $pequiv 1mod 4$ is prime) of isometries of the round sphere defined by Lipschitz quaternions. We also show that any finite rank free group of automorphisms of the torus realizes the lowest possible discrepancy and prove a matching upper bound on the convergence rate.
{"title":"The exact convergence rate in the ergodic theorem of Lubotzky–Phillips–Sarnak and a universal lower bound on discrepancies","authors":"Antoine Pinochet-Lobos, C. Pittet","doi":"10.4171/lem/1003","DOIUrl":"https://doi.org/10.4171/lem/1003","url":null,"abstract":"We compute exact convergence rates in von Neumann type ergodic theorems when the acting group of measure preserving transformations is free and the means are taken over spheres or over balls defined by a word metric. Relying on the upper bounds on the spectra of Koopman operators deduced by Lubozky, Phillips, and Sarnak from Deligne's work on the Weil conjecture, we compute the exact convergence rate for the free groups (of rank $(p+1)/2$ where $pequiv 1mod 4$ is prime) of isometries of the round sphere defined by Lipschitz quaternions. We also show that any finite rank free group of automorphisms of the torus realizes the lowest possible discrepancy and prove a matching upper bound on the convergence rate.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132391552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop the theory of resolvent degree, introduced by Brauer cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilbert's 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.
{"title":"Resolvent degree, Hilbert’s 13th Problem and geometry","authors":"B. Farb, J. Wolfson","doi":"10.4171/lem/65-3/4-2","DOIUrl":"https://doi.org/10.4171/lem/65-3/4-2","url":null,"abstract":"We develop the theory of resolvent degree, introduced by Brauer cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilbert's 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128812810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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