Course Description
1. April 10, 15:30-16:30 Langlands’ automorphic transfer as a problem of generalising the addition operation
Langlands’ automorphic transfer from reductive groups to linear groups is equivalent to the existence of local and global Fourier transform operators induced by representations of the dual groups, of local and global functional spaces that should be stabilized by these operators and of a Poisson linear form on the global functional spaces that should be fixed by Fourier transform. Looking for a definition of non-additive Fourier transform operators leads to the crucial question of determining the Fourier transform of the operator of point-wise multiplication of functions. It has to be a generalisation of ordinary additive convolution operators.

2. April 10, 16:45-17:45 Grothendieck toposes as unifying ‘bridges’ in mathematics
The talk will explain how Grothendieck toposes can be effectively used as ‘bridges’ for transferring ideas and results across different mathematical theories. The interest of these general techniques is that they allow one to discover in classical domains new results which are established by topos-theoretic means but whose statement does not involve toposes. By way of example, these general methods will be illustrated by a topos-theoretic reinterpretation and generalisation of Stone-type dualities in topology.

3. April 11, 15:30-16:30 When do fundamental groups exist?
The talk will explain how Grothendieck toposes can be effectively used as ‘bridges’ for transferring ideas and results across different mathematical theories. The interest of these general techniques is that they allow one to discover in classical domains new results which are established by topos-theretic means but whose statement does not involve toposes. By way of example, these general methods will be illustrated by a topos-theoretic reinterpretation and generalisation of Stone-type dualities in topology.

4. April 11, 16:45-17:45 Fundamental groups and imaginary covers
This talk is based on joint work with O. Caramello. It examines the concrete construction of the new categories classified by fundamental groups as defined in the previous talk. Many classical categories – such as the categories of finite groups or finite graphs and their embeddings or their surjective homomorphisms – naturally embed into larger categories classified by fundamental groups. The new ‘imaginary’ objects which have to be added to make these categories Galoisian can be described concretely. These constructions allow one to associate new invariants – including cohomological invariants – to groups, graphs and many geometric objects.