Representation Degrees, Character Values and Applications

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In recent years there has been a growing use of Representation Theory in Asymptotic Group Theory and related topics.On the one hand, there is the rather new subject of representation growth, somewhat analogous to the classical study ofsubgroup growth. On the other hand, the solutions of various seemingly unrelated problems - such as random walks on groups,or random generation of simple groups, or word maps on groups - require new results in representation theory, involving character degreesand values. In this proposal we plan to study several quantitative aspects of representation theory, and to applythem in a wide variety of such contexts.