Seminars Sorted by Series

In this talk, I will specify a Turing machine T and prove the following about it. 1. On input C/K a smooth projective hyperbolic curve over a number field, if T halts, then its output is C(K). 2. The Hodge, Tate, and Fontaine-Mazur conjectures imply...

Let X/k be a smooth scheme over a perfect field k of characteristic p > 0 and let W be the Witt ring of k. Inspired by Drinfeld’s “stacky” approach to prismatic cohomology, Bhatt, Lurie, and others have shown that a lifting of X/k to W induces a...

Borel proved that any holomorphic map from an affine complex algebraic curve to a Shimura variety (with sufficient level structure) must be algebraic. We will discuss a p-adic analogue of this theorem. This is joint work with Abhishek Oswal and...

I will explain how ideas from the (geometric) Langlands program help solve the following purely algebraic problem: describe the ring of conjugation-invariant functions on the scheme of commuting pairs in a complex reductive group. The answer was...

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...

In a recent article, Vladimir Drinfeld proposed (using prismatic F-gauges) new "Shimurian" analogs of the stack of n-truncated Barsotti-Tate groups. I will give an overview of Drinfeld’s work and the relation to the prismatic Dieudonné theory of...

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.

Recent interactions between condensed mathematics and K-theory have led us to revisit the topic of (nonconnective) algebraic K-theory of topological algebras. In this talk, among recent developments, I will focus on the ring of continuous functions...

I will discuss duality theorems for p-adic proétale cohomology of rigid analytic Stein spaces, in both arithmetic and geometric cases. This is based on a joint work with Pierre Colmez and Wieslawa Niziol.

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...

In topology, the choice of a spin structure on an oriented surface gives a quadratic refinement of the intersection form on mod 2 homology. There is a similar story about 3-manifolds. I will report on joint work (in progress) with Artane Siad where...

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...

Davenport-Heilbronn and Bhargava counted cubic, quartic, and quintic fields by first counting a larger set of orbits in some prehomogenous vector space and then sieving out the orbits corresponding to fields. Several mathematicians have studied...

We report on proofs of some recent conjectures of Drinfeld on the algebraicity of formal stacks associated with minuscule cocharacters of smooth group schemes over Z_p. These stacks are constructed using the theory of syntomification as developed by...

We describe a new theory of sheaves of rings of differential operators on the Witt-vectors on a smooth scheme characteristic p. Roughly speaking, these sheaves are to the de Rham-Witt complex as the usual sheaf of differential operators is to the de...

For p a prime number, Bhatt–Lurie and Drinfeld defined the prismatization of a p-complete commutative ring R. I will overview a different perspective on prismatization, using homotopy theory and in particular topological Hochschild homology. This...

In this talk I want to explain some surprising features of the pro-etale cohomology of rigid-analytic varieties, and how they can be explained by a six functor formalism with values in solid quasi-coherent sheaves on the Fargues-Fontaine curve. 

This...

p-adic heights have been a rich source of explicit functions vanishing on rational points on a curve. In this talk, we will outline a new construction of canonical p-adic heights on abelian varieties from p-adic adelic metrics, using p-adic Arakelov...

We will discuss de Rham and Hyodo-Kato flip-flopping for dual towers of rigid analytic spaces. The main tools are comparison theorems expressing the two cohomologies as pro-étale cohomologies of corresponding relative period sheaves that, by...

Given a quasi projective family S of complex algebraic varieties, its Hodge locus is the locus of points of S where the corresponding variety admits exceptional Hodge classes (conjecturally: exceptional algebraic cycles). In this talk, I will survey...

Let S be a connected smooth rigid analytic variety over a p-adic field K and let T be a p-adic local system over S. A celebrated theorem of Liu and Zhu says that if V is de Rham at one classical point, then V is globally de Rham. When S has good...

The resolutions of Slodowy slices $\tilde{S}_{e}$ are symplectic varieties that contain the Springer fiber $(G/B)_{e}$ as a Lagrangian subvariety. In joint work with R. Bezrukavnikov, M. McBreen and Z. Yun, we construct analogues of these spaces for...

I will describe the main ideas that go into the proof of the (unramified, global) geometric Langlands conjecture. All of this work is joint with Gaitsgory and some parts are joint with Arinkin, Beraldo, Chen, Faergeman, Lin, and Rozenblyum. I will...