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In 1900, David Hilbert asked whether each locally euclidean topological group admits a Lie group structure. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the PeterWeyl Theorem, the Pontryaginvan Kampen Duality Theorem for locally compact abelian groups, and finally the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book “Hilbert’s Fifth Problem and Related Topics” by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao.It is not possible to describe briefly the richness of the topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 “The Structure of Compact Groups” by Karl H. Hofmann and Sidney A. Morris, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and Pavel Zalesskii (2012). The 2007 book “The Lie Theory of Connected ProLie Groups” by Karl Hofmann and Sidney A. Morris, demonstrates how powerful Lie Theory is in exposing the structure of infinitedimensional Lie groups.The study of free topological groups initiated by A.A. Markov, M.I. Graev and S. Kakutani, has resulted in a wealth of interesting results, in particular those of A.V. Arkhangelʹskiĭ and many of his former students who developed this topic and its relations with topology. The book “Topological Groups and Related Structures” by Alexander Arkhangelʹskii and Mikhail Tkachenko has a diverse content including much material on free topological groups. Compactness conditions in topological groups, especially pseudocompactness as exemplified in the many papers of W.W. Comfort, has been another direction which has proved very fruitful to the present day.
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For the 250th birthday of Joseph Fourier, born in 1768 in Auxerre, France, this MDPI Special Issue will explore modern topics related to Fourier Analysis and Heat Equation. Modern developments of Fourier analysis during the 20th century have explored generalizations of Fourier and Fourier–Plancherel formula for noncommutative harmonic analysis, applied to locallycompact, nonAbelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups. One should add the developments, over the last 30 years, of the applications of harmonic analysis to the description of the fascinating world of aperiodic structures in condensed matter physics. The notions of model sets, introduced by Y. Meyer, and of almost periodic functions, have revealed themselves to be extremely fruitful in this domain of natural sciences. The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, subRiemannian manifolds, and Lie groups. In parallel, in geometric mechanics, JeanMarie Souriau interpreted the temperature vector of Planck as a spacetime vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties. One last comment concerns the fundamental contributions of Fourier analysis to quantum physics: Quantum mechanics and quantum field theory. The content of this Special Issue will highlight papers exploring noncommutative Fourier harmonic analysis, spectral properties of aperiodic order, the hypoelliptic heat equation, and the relativistic heat equation in the context of Information Theory and Geometric Science of Information.
WeylHeisenberg group  affine group  Weyl quantization  Wigner function  covariant integral quantization  Fourier analysis  special functions  rigged Hilbert spaces  quantum mechanics  signal processing  nonFourier heat conduction  thermal expansion  heat pulse experiments  pseudotemperature  GuyerKrumhansl equation  higher order thermodynamics  Lie groups thermodynamics  homogeneous manifold  polysymplectic manifold  dynamical systems  nonequivariant cohomology  Lie group machine learning  SouriauFisher metric  Born–Jordan quantization  shorttime propagators  timeslicing  Van Vleck determinant  thermodynamics  symplectization  metrics  nonequilibrium processes  interconnection  discrete multivariate sine transforms  orthogonal polynomials  cubature formulas  nonequilibrium thermodynamics  variational formulation  nonholonomic constraints  irreversible processes  discrete thermodynamic systems  continuum thermodynamic systems  fourier transform  rigid body motions  partial differential equations  Lévy processes  Lie Groups  homogeneous spaces  stochastic differential equations  harmonic analysis on abstract space  heat equation on manifolds and Lie Groups
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