Inequalities play a fundamental role in Functional Analysis and it is widely recognized that finding them, especially sharp estimates, is an art. E. H. Lieb has discovered a host of inequalities that are enormously useful in mathematics as well as in physics. His results are collected in this book which should become a standard source for further research. Together with the mathematical proofs the author also presents numerous applications to the calculus of variations and to many problems of quantum physics, in particular to atomic physics.

This volume contains the proceedings of the QMATH13: Mathematical Results in Quantum Physics conference, held from October 8-11, 2016, at the Georgia Institute of Technology, Atlanta, Georgia. In recent years, a number of new frontiers have opened in mathematical physics, such as many-body localization and Schrodinger operators on graphs. There has been progress in developing mathematical techniques as well, notably in renormalization group methods and the use of Lieb-Robinson bounds in various quantum models. The aim of this volume is to provide an overview of some of these developments. Topics include random Schrodinger operators, many-body fermionic systems, atomic systems, effective equations, and applications to quantum field theory. A number of articles are devoted to the very active area of Schrodinger operators on graphs and general spectral theory of Schrodinger operators. Some of the articles are expository and can be read by an advanced graduate student.