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The authors will classify Rohlin flows on von Neumann algebras up to strong cocycle conjugacy. This result provides alternative approaches to some preceding results such as Kawahigashi's classification of flows on the injective type II_1 factor, the classification of injective type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III_0 factors. Several concrete examples are also studied.

Von Neumann algebras. --- Conjugacy classes. --- Group theory.

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Reversibility is a thread woven through many branches of mathematics. It arises in dynamics, in systems that admit a time-reversal symmetry, and in group theory where the reversible group elements are those that are conjugate to their inverses. However, the lack of a lingua franca for discussing reversibility means that researchers who encounter the concept may be unaware of related work in other fields. This text is the first to make reversibility the focus of attention. The authors fix standard notation and terminology, establish the basic common principles, and illustrate the impact of reversibility in such diverse areas as group theory, differential and analytic geometry, number theory, complex analysis and approximation theory. As well as showing connections between different fields, the authors' viewpoint reveals many open questions, making this book ideal for graduate students and researchers. The exposition is accessible to readers at the advanced undergraduate level and above.

Conjugacy classes. --- Group theory. --- Automorphisms. --- Dynamics. --- Reverse mathematics.

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Group theory --- Group schemes (Mathematics) --- Linear algebraic groups. --- Conjugacy classes. --- Group schemes (Mathematics). --- Groupes algébriques linéaires

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Topological groups. Lie groups --- Finite groups --- Lie groups --- Groupes finis --- Groupes de Lie --- Conjugacy classes --- Classes of conjugate elements --- Group theory --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Finite --- Modules (Algebra) --- Groepentheorie --- Algèbres de Lie. --- Groupes (Théorie des). --- Lie (Algebra's van).

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Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods. By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.

Mellin transform. --- Convolutions (Mathematics) --- Sequences (Mathematics) --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Convolution transforms --- Transformations, Convolution --- Distribution (Probability theory) --- Functions --- Integrals --- Transformations (Mathematics) --- Transform, Mellin --- Integral transforms --- ArtinГchreier reduced polynomial. --- Emanuel Kowalski. --- EulerАoincar formula. --- Frobenius conjugacy class. --- Frobenius conjugacy. --- Frobenius tori. --- GoursatЋolchinВibet theorem. --- Kloosterman sheaf. --- Laurent polynomial. --- Legendre. --- Pierre Deligne. --- Ron Evans. --- Tannakian category. --- Tannakian groups. --- Zeeev Rudnick. --- algebro-geometric. --- autodual objects. --- autoduality. --- characteristic two. --- connectedness. --- dimensional objects. --- duality. --- equidistribution. --- exponential sums. --- fiber functor. --- finite field Mellin transform. --- finite field. --- finite fields. --- geometrical irreducibility. --- group scheme. --- hypergeometric sheaf. --- interger monic polynomials. --- isogenies. --- lie-irreducibility. --- lisse. --- middle convolution. --- middle extension sheaf. --- monic polynomial. --- monodromy groups. --- noetherian connected scheme. --- nonsplit form. --- nontrivial additive character. --- number theory. --- odd characteristic. --- odd prime. --- orthogonal case. --- perverse sheaves. --- polynomials. --- pure weight. --- semisimple object. --- semisimple. --- sheaves. --- signs. --- split form. --- supermorse. --- theorem. --- theorems.