Hot Topics: Thin Groups and Super-strong Approximation

Will focus on recent developments concerning quantitative aspects of 'thin groups'. These are discrete subgroups of semisimple Lie groups which are both (i.e. Zariski dense) and (i.e. of infinite co-volume). This dual nature leads to intricate questions. Many new ideas and techniques, arising in particular from arithmetic combinatorics, have been involved in the study of such groups, leading for instance to far-reaching generalizations of the strong approximation theorem in which congruence quotients are shown to exhibit a spectral gap. A variety of experts from group theory, number theory, ergodic theory and harmonic analysis will present the accomplishments to date to a broad audience and discuss directions for further study.

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