DOMOSCHOOL 2022 Edition – Gauge Theory and the Langlands Program – Mellerio Rosmini College of Domodossola, July 18/22

The Scientific board and the Organizing Commiteee Welcome all to the third edition of the DomoSchool, the International Alpine School in Mathematics and Physics at Collegio Rosmini in Domodossola. Domoschool 2022 Edition will consists of THREE courses, one of 14 hours and two of 3 hours each for a total of 20 hours.

The Langlands program is originally born as a relation between infinite dimensional representations of the general linear group and the allowed fields of numbers with degree n.  It is certainly an extremely rich program of geometrization involving mainly number theory and algebraic aspects of representation theory and algebraic geometry. Nevertheless, after a simple observation by Sir Michael Atiyah, who noticed a relation between Langlands duality among Lie groups and the Montonen-Olive duality in non abelian Yang-Mills theory, Kapustin and Witten, in 2006, proposed a new viewpoint on Langlands program, using methods of quantum field theory.

Their construction gave a geometric analysis perspective, based on the analysis of a system of non linear partial differential equations defining the minimization of a functional that can be understood as a suitable complexification of the Yang-Mills functional. This set of differential equations, known as Kapustin-Witten equations, constraint the linear connections on the adjoint bundle over a four dimensional manifold, and can be understood as a sort of real version of the Hitchin equations over a Riemann surface. If one consider a four manifold with a three dimensional boundary, the solution of the differential system respecting suitable singular conditions on the boundary (the Nahm poles) are conjecturally related to knot invariants on the boundary three manifold. Indeed, they can also be related to the Chern-Simons fluxes, where the complexified Chern-Simons functional characterizes the boundary configurations.

The aim of the school is to introduce PhD students, postdocs and young researchers, both in mathematics and in physics, to this topic.

Courses will be conducted at PhD level by world leading experts.

Moreover, students will have the opportunity to give 20 minutes talks presenting their research on themes related to the main topics of the courses.

Informal discussions will be strongly encouraged, and specific time slots (short talks) will be reserved for this important part of the school.

The school is mainly dedicated to PhD students, postdocs and young researchers, but interaction with and among professors will be also encouraged. 

Due to Covid-19 restriction the capacity of the classroom has been reduced, attendance will be limited to a maximum of 35 participants. Please note that it is mandatory for participants to be able to show a valid document certifying vaccination or recovery from Covid-19 (in UE super green pass every day of lessons)