Tag - Motives

Thomas Haines: Pavings of convolution fibres and applications

1st March, 2024

A convolution morphism is the geometric analogue of the convolution of functions in a Hecke algebra. The properties of fibres of convolution morphisms are used in a variety of ways in the geometric Langlands programme and in the study of Schubert varieties. I will explain a very general result about cellular pavings of fibres of convolution morphisms in the setting of partial affine flag varieties, as well as applications related to the very purity and parity vanishing of cohomology of Schubert varieties over finite fields, structure constants for parahoric Hecke algebras, and the (motivic) geometric Satake equivalence.

Tess Bouis: Motivic Cohomology of Mixed Characteristic Schemes

14th February, 2024

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-A1-invariant in general, but it recovers classical motivic cohomology on smooth schemes over a Dedekind domain after A1-localisation. The construction relies on the syntomic cohomology of Bhatt-Morrow-Scholze and the cdh-local motivic cohomology of Bachmann-Elmanto-Morrow, and generalises the construction of Elmanto-Morrow in the case of schemes over a field.

Federico Binda: Motivic monodromy and p-adic cohomologies 

16th May, 2023

In this talk, I will discuss some recent advances in the theory of motives in the context of rigid analytic geometry. Building on work of Ayoub, Bondarko, we provide an equivalence between the category of “unipotent” rigid analytic motives over a non-archimedean field and the category of “monodromy maps” M → M (−1) of algebraic motives over the residue field. This allows us to build a unified framework for the study of monodromy operators and weight filtrations of cohomology theories for varieties over a local field. As an application, we give a streamlined definition of Hyodo–Kato cohomology without recourse to log-geometry, as predicted by Fontaine, and we produce an induced Clemens–Schmid chain complex.

Bhargav Bhatt: p-adic motives II

7th April, 2023

In the 1960s, Grothendieck dreamt that algebraic varieties can be linearized in a universal way, leading to his philosophy of motives. Subsequent ideas of many mathematicians (especially Beilinson and Deligne) led to a beautiful conjectural framework surrounding the notion of a motive. In the last decade, thanks to the discovery of perfectoid geometry and subsequent developments, some aspects of this framework have also been realized unconditionally in the context of p-adic motives on p-adic varieties. In these lectures, I will survey some of this landscape, with an emphasis on the concrete applications that have guided the theoretical developments.

Bhargav Bhatt: p-adic motives I

7th April, 2023

In the 1960s, Grothendieck dreamt that algebraic varieties can be linearized in a universal way, leading to his philosophy of motives. Subsequent ideas of many mathematicians (especially Beilinson and Deligne) led to a beautiful conjectural framework surrounding the notion of a motive. In the last decade, thanks to the discovery of perfectoid geometry and subsequent developments, some aspects of this framework have also been realized unconditionally in the context of p-adic motives on p-adic varieties. In these lectures, I will survey some of this landscape, with an emphasis on the concrete applications that have guided the theoretical developments.

Toni Mikael Annala: Stable Homotopy without Homotopy

30th January, 2023

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 deeper aspects of such theories, such as cohomology operations that act on the cohomology groups. In this talk, I will discuss ongoing efforts, joint with Ryomei Iwasa and Marc Hoyois, to set up a workable theory of non-homotopy invariant stable motivic homotopy theory, with the goal of providing effective tools of studying cohomology theories in algebraic geometry by geometric means.

Grigory Garkusha: Homological Algebra for Enriched Grothendieck Categories

8th November, 2022

Enriched Grothendieck categories naturally occur in algebraic geometry, where associated abelian categories rarely have projectives but have plenty of information encoded by enriched category theory. In this talk general properties of derived categories for Grothendieck categories of enriched functors and various recollements of such categories will be presented. Applications are given for Voevodsky's triangulated categories of motives.

Aleksander Horawa: Motivic action on coherent cohomology of Hilbert modular varieties

3rd February, 2022

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.