Seminars Sorted by Series

Let SO(3,R) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure \mu. Confirming a conjecture by Breuillard and Green, we show that if A is an open subset of SO(3,R) with sufficiently small measure, then...

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This is the organizational meeting for a learning seminar during the fall term on topics related to non-abelian p-adic Hodge theory. 

With every bounded prism Bhatt and Scholze associated a cohomology theory of formal p-adic schemes. The prismatic cohomology comes equipped with the Nygaard filtration and the Frobenius endomorphism. The Bhatt-Scholze construction has been advanced...

Multiplier ideals in characteristic zero and test ideals in positive characteristic are fundamental objects in the study of commutative algebra and birational geometry in equal characteristic.  We introduced a mixed characteristic version of the...

This talk will introduce background, achievements and challenges in the quest to find non-Archimedean versions of the celebrated Corlette-Simpson correspondence, which on Kähler manifolds relates representations of the fundamental group to certain...

I will discuss the following conjecture: an irreducible $\bar{Q}$$_{\ell}$-local system L on a smooth complex algebraic variety S arises in cohomology of a family of varieties over S if and only if L can be extended to an etale local system over...

In this talk, we introduce the analytic de Rham stack for rigid varieties over $Q_p$ (and more general analytic stacks). This object is an analytic incarnation of the (algebraic) de Rham stack of Simpson, and encodes a theory of analytic D-modules...

Cohomology of classifying space/stack of a group G is the home which resides all characteristic classes of G-bundles/torsors. In this talk, we will try to explain some results on Hodge/de Rham cohomology of BG where G is a $p$-power order...

Etale cohomology of $F_p$-local systems does not behave nicely on general smooth p-adic rigid-analytic spaces; e.g., the $F_p$-cohomology of the 1-dimensional closed unit ball is infinite. 

However, it turns out that the situation is much better if...

The notion of $p$-adic shtukas are introduced by Scholze in his Berkeley lectures on $p$-adic geometry. They are closely related to $p$-divisible groups when their ``legs" are bounded by some minuscule cocharacter. But compared to $p$-divisible...

In this talk, I will first describe how classical Dieudonne module of finite flat group schemes and $p$-divisible groups can be recovered from crystalline cohomology of classifying stacks. Then, I will explain how in mixed characteristics, using...

I will present a new theory of motivic cohomology for general (qcqs) schemes. It is related to non-connective algebraic K-theory via an Atiyah-Hirzebruch spectral sequence. In particular, it is non-$A^1$-invariant in general, but it recovers...

I will present two settings where $q$-De Rham and prismatic vector bundles can be described in terms of modules over an appropriate ring of $q$-twisted differential operators and also the relation with former results. 

This is a joint work with...

Motivated by the desire to express in terms of de Rham data the pro-étale cohomology with non-trivial $\mathbb{Q}_p$-coefficients of rigid spaces $X$, defined over $\mathbb{Q}_p$ or $\mathbb{C}_p$, I will explain how to define D-modules on the...

Topological Hochschild homology is an important invariant, closely related to algebraic K-theory, and can be seen as a noncommutative analogue of de Rham chains.

In this talk, I will describe various computations of the ring/monoid of endomorphisms...

In this talk, I will present a computation of the image of the Hodge-Tate logarithm map (defined by Heuer) in the case of smooth Stein varieties. When the variety is the affine space, Heuer has proved that this image is equal to the group of closed...

A few years ago, Bhatt-Morrow-Scholze introduced an invariant of $p$-adic formal schemes called syntomic cohomology, which has a close relationship to (étale-localized) algebraic $K$-theory. In a recent paper, Antieau-Mathew-Morrow-Nikolaus showed...

Abstract: I will discuss work in progress developing the theory of log prismatic cohomology and related theories using the Cartier-Witt stack.   

Given a family of motives, the de Rham realization (a certain vector bundle with integrable "Gauss-Manin" connection) can be compared to the crystalline realizations for various primes p, but the resulting Frobenius structures cannot be directly...

Let X be a proper, smooth rigid space and G a commutative rigid group. We study the relationship between G-representations of the fundamental group of X and G-Higgs bundles on X. This is joint work with Ben Heuer and Mingjia Zhang.

We develop on a new strategy based on point-set topology, which allows us to produce a purely p-adic statement for the crystallinity properties of rigid flat connections. 

Joint with Michael Groechenig.

Lusztig's theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. In this talk, we will describe a construction which extends this "depth zero" picture to...

Let $E$ be a finite degree extension of $Qp$. Given a mod p representation of the absolute Galois group of E we construct a sheaf on a punctured absolute Banach-Colmez space that should give the first step in the construction of the mod p local...

A theorem of Borel says that any holomorphic map from a complex algebraic variety to a smooth arithmetic variety is automatically an algebraic map. The key ingredient is to show that any holomorphic map from the (poly) punctured disc to the Baily...

Let $f:Y \rightarrow X$ be a finite covering map of complex algebraic varieties. The essential dimension of f is the smallest integer e such that, birationally, f arises as the pullback of a covering $Y^{'} \rightarrow X^{'}$ of dimension e, via a...

The minimal model program for 3-folds has been developed only in characteristics $p \geq 5$. A key difficulty at small primes is that the singularities occurring in the minimal model program need not be Cohen-Macaulay, as they are in characteristic...

I will talk about (very much in progress) joint work with Mark Kisin on a Hodge—Newton style inequality for the mod p Breuil—Kisin modules arising from crystalline Galois representations.

With every bounded prism Bhatt and Scholze associated a cohomology theory of formal p-adic schemes. The prismatic cohomology comes equipped with the Nygaard filtration and the Frobenius endomorphism. The Bhatt-Scholze construction has been advanced...

The recent work of Drinfeld and Bhatt-Lurie led to a new geometric approach to p-adic cohomology theories, analogously to what was done earlier in Hodge theory by Simpson. This stacky perspective gives in particular a new approach to p-adic non...

In joint work in progress with Anschütz and Le Bras we aim to construct a 6-functor formalism for quasicoherent sheaves on the relative Fargues-Fontaine curve over rigid-analytic varieties (and even general v-stacks), providing new insights into the...

In chromatic homotopy theory, an object like the sphere spectrum $S^0$ is studied by means of its "localizations", much as an abelian group can be localized at each prime p.  Remarkably, the "primes" $K(n)$ in the homotopy setting correspond to...

A smooth, oriented n-manifold is called a homotopy sphere if it is homeomorphic, but not necessarily diffeomorphic, to the standard n-sphere. In dimensions $n>4$, one often studies the group Θn of homotopy spheres up to orientation-preserving...

Abstract: Since the original conjectures of Beilinson and Lichtenbaum in the 80s, several versions of motivic cohomology have been introduced and developed, notably by Voevodsky. Most classically, Bloch's higher Chow groups provide the accepted...

Abstract: Multiplier ideals and test ideals are ways to measure singularities in characteristic zero and p > 0 respectively.  In characteristic zero, multiplier ideals are computed by a sufficiently large blowup by comparing the canonical module of...

Abstract: The p-adic Simpson correspondence aims to give a non-abelian generalisation of the Hodge-Tate decomposition. Following an idea of Faltings, it should relate pro-étale vector bundles on smooth rigid spaces over Cp to Higgs bundles. In this...

Abstract: Recently the work of Fargues--Scholze provides a geometrization of the local Langlands conjecture. It is natural to ask if in this context any form of local-global compatibility can be stated/verified. We discuss some expectations and...

Abstract: In recent work with Antieau and Nikolaus we use prismatic cohomology to compute algebraic K-theory of Z/pn and similar rings. Our approach is based on a new description of absolute prismatic cohomology, which can be made completely...

Abstract: Given an étale Zp-local system of rank n on an algebraic variety X, continuous cohomology classes of the group GLn(Zp) give rise to classes in (absolute) étale cohomology of the variety with coefficients in Qp. These characteristic classes...

Abstract: We study prismatic crystals and their cohomology by using q-Higgs modules (= a q-analogue of p-connections). When the base is lying over the q-crystalline prism, they are locally described in terms of q-Higgs modules and the associated...

Abstract: Chromatic homotopy theory relates certain important questions in homotopy theory to the theory of formal groups. 

Recent advancements in p-adic geometry can be thus used to study questions in homotopy theory. I will discuss how this...

Abstract: We will explain how to construct a Kirillov model for Emerton's completed cohomology of the tower of modular curves. The trickiest part is to prove injectivity of this model.  This is joint work with Shanwen Wang.

Abstract: I will discuss the formulation and sketch the proofs of duality theorems for the geometric and arithmetic p-adic pro-étale cohomology of Stein spaces. This is based on a joint work with Pierre Colmez and Sally Gilles.

Abstract: Let p be a prime number. Emerton introduced the p-adically completed cohomology, which admits a representation of some p-adic group and can be thought of as some spaces of p-adic automorphic forms. In this talk, I want to explain that for...

Abstract: For a reductive group $G$, its $B_{d}R^{+}$-affine Grassmannian is defined as the étale (equivalently, v-) sheafification of the presheaf quotient $LG/L^{+}G$ of the $B_{d}R$-loop group $LG$ by the $B_{d}R^{+}$-loop subgroup $L^{+}G$. We...