Related papers: Geometrization of the local Langlands corresponden…

We prove that the (twisted orbifold) motives of the moduli spaces of $\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles of coprime rank and degree on a smooth projective curve over an algebraically closed field in which the rank is…

Vincent Lafforgue has recently made a spectacular breakthrough in the setting of the global Langlands correspondence for global fields of positive characteristic, by constructing the `automorphic--to--Galois' direction of the correspondence…

We define an $\infty$-category of rational motives for inverse limits of algebraic stacks, so-called pro-algebraic stacks. We show that it admits a $6$-functor formalism for certain classes of morphisms. On pro-schemes, we show that this…

Let $F$ be a non-archimedean local field of residual characteristic $p$, $\ell\neq p$ be a prime number, and $\mathrm{W}_F$ the Weil group of $F$. We classify the indecomposable $\mathrm{W}_F$-semisimple Deligne…

In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of…

Given a locally free coherent sheaf on a smooth algebraic surface, we consider the Quot-scheme parametrizing zero-dimensional quotients of the sheaf and find the corresponding motivic class in the Grothendieck ring of algebraic varieties.

An introduction is given to the logic of sheaves of structures and to set theoretic forcing constructions based on this logic. Using these tools, it is presented an alternative proof of the independence of the Continuum Hypothesis; which…

We construct smooth presentations of algebraic stacks that are local epimorphisms in the Morel-Voevodsky $\mathbb{A}^1$-homotopy category. As a consequence we show that the motive of a smooth stack (in Voevodsky's triangulated category of…

A brief survey is given of the classical Langlands correspondence between n-dimensional representations of Galois groups of local and global fields of dimension 1 and irreducible representations of the groups GL(n). A generalization of the…

We study the determinant of certain etale sheaves constructed via middle convolution in order to realize special linear groups regularly as Galois groups over the rationals.

We show in many cases the existence of adjoints to extension of scalars on categories of motivic nature, in the framework of field extensions. This is to be contrasted with the more classical situation where one deals with a finite type…

Exactly solvable mirror pairs of Calabi-Yau threefolds of hypersurface type exist in the class of Gepner models that include nondiagonal affine invariants. Motivated by the string modular interpretation established previously for models in…

We give explicit formulas for the local multiplicative convolution functors which express the local monodromies of the convolution of two $\ell$-adic sheaves on the torus ${\mathbb G}_m$ over the algebraic closure of a finite field in terms…

Let $F$ be a non-archimedean locally compact field. We study a class of Langlands-Shahidi pairs $({\bf H},{\bf L})$, consisting of a quasi-split connected reductive group $\bf H$ over $F$ and a Levi subgroup $\bf L$ which is closely related…

We show that representations of convolution algebras such as Lustzig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in (affine) type A can be realised in terms of certain equivariant motivic sheaves…

We prove the equivalence between the categories of motives of rigid analytic varieties over a perfectoid field $K$ of mixed characteristic and over the associated (tilted) perfectoid field $K^{\flat}$ of equal characteristic. This can be…

Let $X$ be a smooth connected projective algebraic curve over an algebraically closed field, and let $S$ be a finite nonempty closed subset in $X$. We study deformations of $\overline{\mathbb F}_\ell$-sheaves. The universal deformation…

In this expository paper we provide a geometric proof of the local Langlands Correspondence for the groups $\operatorname{GL}_{1}$ defined over $p$-adic fields $K$. We do this by redeveloping the theory of proalgebraic groups and use this…

We prove the version of Joyce-Song formula for the Behrend function identities in the motivic setting. The main method we use is the proof of Kontsevich-Soibelman conjecture about the motivic Milnor fibers by Q. T. Le, who uses the method…

In this paper we investigate the structure of algebraic cobordism of Levine-Morel as a module over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it. We show that the associated graded groups of algebraic…