Symmetry of optimizers by examples

The course is intended especially for Master’s students, PhD students, and postdoctoral researchers, but everyone interested is warmly welcome to attend.

If a variational problem has a symmetry, it is natural to ask whether an optimizer has the same symmetry. Symmetries offer clear advantages for computing optimizers and determining the optimal value of a variational problem.

In general, however, optimizers do not necessarily reflect the underlying symmetry: symmetry may be broken. The mathematical analysis must therefore focus on relevant, non-trivial examples and the techniques used to study them.

Key examples will include the sharp Sobolev inequality and its generalization, the Caffarelli–Kohn–Nirenberg inequalities, as well as sharp Hardy–Littlewood–Sobolev inequalities and Beckner’s inequality.

Particular emphasis will be placed on techniques such as the use of the isoperimetric inequality and rearrangements, flows, and Obata-type identities, as well as the analysis of the second variation operator for proving symmetry breaking.

If time permits, sharp spinor inequalities will also be discussed.

Please feel free to forward this announcement to anyone who might be interested.