Speaker: Dennis Gaitsgory (Harvard)
Title: An introduction to the quantum Langlands program
Abstract: The mini-course will be devoted to the study of local and global quantum geometric Langands correspondence.
The starting point is the fundamental local equivalence, which is an equivalence between the Kazhdan-Lusztig category for a given reductive group G and the Whittaker category for its Langlands dual.We will show how this local equivalence, combined with local-to-global methods determines the unramified categorical quantum Langlands equivalence for a global curve. We will then outline the ideas of the local quantum geometric Langlands
(those are 2-categorical in nature).

Speaker: Yakov Kremnitzer (Oxford)
Title: Global analytic geometry and the field with one element
Abstract: This mini course will describe an approach to analytic geometry over Banach rings and over the "field with one element". It will cover the basics of relative algebraic geometry, Banach and bornological rings and modules, bornological Tannakian duality, analytic geometry and the field with one element.

Speaker: Bertrand Toen (Toulouse)
Title: Matrix factorizations, trace formula and conductor formula
Abstract: In this series of lecture we propose an approach to the so-called Bloch's conductor formula by means of non-commutative and derived methods. We will start by some reminders on non-commutative spaces and their cohomology and eventually will explain how these can be constructed by methods from the stable homotopy theory of schemes à la Voevodsky-Morel. We then focus on the statement and the proof of the trace formula for non-commutative spaces. In the last lecture we apply this trace formula for non-commutative spaces coming from matrix factorizations and apply this to Bloch's conductor conjecture.