| Summary: | This review celebrates the generous gift by Ronald and Maxine Linde for the remodeling of the Caltech mathematics department and the author is very grateful to the editors of this volume for the invitation to contribute. We attempt to survey recent results and open problems connected to Lieb–Thirring inequalities. In view of several excellent existing reviews [132, 17, 113, 91, 108] as well as highly recommended textbooks [134, 136], we sometimes put our focus on developments during the past decade. © 2020 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes. Partial support through US National Science Foundation grant DMS-1363432 is acknowledged. The author would like to thank all his collaborators on the topic of Lieb–Thirring inequalities and, in particular, A. Laptev, S. Larson, M. Lewin, E. H. Lieb, P. T. Nam and T. Weidl for helpful remarks on a preliminary version of this review. Submitted - 2007.09326.pdf
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