Eisenstein proved, in 1852, that if a function f(z) is
algebraic, then its Taylor expansion at a point has coefficients
lying in some finitely-generated Z-algebra. I will explain ongoing
joint work with Josh Lam which studies the extent to which
the...
By Deligne's Hodge theory, the integral cohomology groups
H^n(X^h, Z) of the C-analytification of a separated scheme X of
finite type over C are provided with a mixed Hodge structure,
functorial in X. Given a non-Archimedean field K isomorphic
to...
Because of the existence of approximate p-power roots, a
perfectoid algebra over Q_p admits no continuous derivations, and
thus the natural Kahler tangent space of a perfectoid space over
Q_p is identically zero. However, it turns out that many...
The Breuil-Mezard Conjecture predicts the existence of
hypothetical "Breuil-Mezard cycles" that should govern congruences
between mod p automorphic forms on a reductive group G. Most of the
progress thus far has been concentrated on the case G = GL...
I will explain what the question means and how to make it
precise. Then I will give a conjectural answer. This is based on
joint work with Peter Scholze.
Let K be a finite extension of Qp. The Emerton-Gee stack for GL2
is a stack of etale (phi, Gamma)-modules of rank two. Its reduced
part, X, is an algebraic stack of finite type over a finite field,
and can be viewed as a moduli stack of two...
P-adic non abelian Hodge theory, also known as the p-adic
Simpson correspondence, aims at describing p-adic local systems on
a smooth rigid analytic variety in terms of Higgs bundles. I will
explain in this talk why the « Hodge-Tate stacks »...
Let X be a smooth projective variety over the field of complex
numbers. The classical Riemann-Hilbert correspondence supplies a
fully faithful embedding from the category of perverse sheaves on X
to the category of algebraic D_X-modules. In this...
Let X be a smooth projective variety over the complex numbers.
Let M be the moduli space of irreducible representations of the
topological fundamental group of X of a fixed rank r. Then M is a
finite type scheme over the spectrum of the integers Z...
Many cohomology theories in algebraic geometry, such as
crystalline and syntomic cohomology, are not homotopy invariant.
This is a shame, because it means that the stable motivic homotopy
theory of Morel--Voevodsky cannot be employed in studying
the...
