Symmetries are ubiquitous throughout mathematics and scientific disciplines ranging from physics, to chemistry, to biology. Every mathematical structure comes equipped with a corresponding notion of automorphisms or symmetries, traditionally described by the theory of groups and their representations.

Symmetry often provides powerful constraints that make it possible to solve problems that would otherwise be intractable. A key example is the theory of integrable systems, namely systems of nonlinear differential equations that admit infinitely many symmetries, allowing for explicit analytic solutions. These structures frequently persist in the corresponding quantum theories, leading to the notion of quantum integrability. The study of integrable models has revealed a rich landscape of mathematical structures associated with symmetry, including quantum groups and Hopf algebras.

More recently, developments in theoretical physics have shown that the traditional framework of symmetries must be even further extended to capture phenomena appearing in many modern examples, such as supersymmetry and categorical symmetries, which require new mathematical structures involving tools from category theory, super geometry, and topology. In the context of quantum phases of matter, such generalized notions of symmetries have allowed researchers to characterize exotic new topological phases of matter and their phase transitions in terms of patterns of spontaneous symmetry breaking.

This summer school will provide an extensive introduction to symmetries in the contexts described above, introduce the mathematical structures underlying them, and discuss applications both within mathematics and in theoretical physics.

The school will be arranged in the format of a flipped classroom, meaning that all the lectures will be given by the students to each other.

We are grateful to Prof. Carlo Beenakker (Leiden University) with his help in the school organization.

The school will consist of several stages:

Assignment of topics. Several months before the beginning of the school, the participants will be provided with the detailed plan of the school: what is the tentative content of each lecture of each course, with the abundant amount of references provided. Within one week they will have to select their top-5 favourite lectures. Based on this rank, every participant will get one lecture to prepare.

Preparatory work. There will be 1 month to start learning the material and prepare the first draft of the plan of the lecture, which should be an up-to-2-page long document containing main definitions, statements and formulas from the lecture. Submitting this document by the deadline is an essential requirement for participation in the school. Although participants are encouraged to prepare together in the groups assigned to one topic, everyone has to write a plan on their own. Then everyone will get written feedback by the mentors on their plans, and will have to improve it. This process will go until the convergence or start of the school.

School. Every course will consist of 5 lectures and 5 tutorials. Every day there will be 2 lectures in the mornings, taught by participants, 60 to 90 minutes long, with 30 minutes long breaks in between. After lunch there will be tutorials, based on the materials of the lectures, led by the lecturers of the day and mentors.