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Lecture Video

SES # Date Title Video
1 April 10, 2017
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, 2017
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-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.
3 April 11, 2017
When do fundamental groups exist ?
The talk will present an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts : this unifies and generalises Grothendieck's theory of ‘Galoisian categories’ and Fraïssé's construction in model theory. This theory allows one to construct fundamental groups in many classical contexts such as finite groups, finite graphs, motives and many more. We will in particular present an approach based on it for investigating the independence from l of l-adic cohomology.
4 April 11, 2017
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.