AGNT Ical Atom

The Kazhdan–Lusztig isomorphism, relating the affine Hecke algebra of a p-adic group to the equivariant K-theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne–Langlands conjectures concerning the classification of tamely ramified irreducible representations. In this talk, I will recall the statement of the Kazhdan–Lusztig isomorphism. I will also introduce the relative Langlands duality and propose a conjectural relative version of the Kazhdan–Lusztig isomorphism. I will focus on specific examples which we can prove.

Chromatic homotopy theory aims to study cohomology theories through a hierarchy of simpler layers, organized by a notion called height. In this talk I will introduce the basic ideas behind this viewpoint and explain two approaches to analyzing these monochromatic layers: the classical K(n)-local category, which is closely related to one-dimensional formal group laws, and the T(n)-local or telescopic category, which is more directly tied to periodic phenomena in the stable homotopy groups of spheres. I will then describe a framework for understanding periodicity inside the chromatic layers, and explain how this allows one to lift Picard elements from the K(n)-local setting to the telescopic setting. Finally, I will present an application to chromatic Galois theory, leading to the construction of a first example of a non-abelian Galois extension in the T(n)-local world.

Locally analytic representations of p-adic Lie groups with Q_p coefficients are powerful tools in p-adic Hodge theory and the p-adic Langlands program. This perspective reveals important differential structures, such as the Sen and Casimir operators. A few years ago, Rodrigues Jacinto and Rodriguez Camargo developed a “solid” version of this theory using the language of condensed mathematics, which provides more robust homological tools (comparison theorems, spectral sequences…) for studying these representations. This talk will present work that extends this solid theory to a much broader class of mixed characteristic coefficients, such as F_p((X)) or Z_p[[X]]<p/x>, as well as semilinear representations. I will conclude by exploring how these ideas relate to mixed characteristic phenomena in p-adic Hodge theory and the Langlands program.

We will describe a general strategy for proving the algebraicity of the Hodge Weil classes on abelian varieties of Weil type. The latter are even dimensional abelian varieties admitting a suitable embedding of a CM number field in their rational endomorphism ring. We will describe the implementation of the strategy for abelian varieties of dimension 4 and 6, and why it implies the Hodge conjecture for abelian varieties of dimension at most 5.