We summarize the definitions and the most important properties of the categories of rigid analytic motives (as defined by Ayoub) and we use them to define a new de Rham cohomology for analytic varieties in equi-characteristic p, and to study more classical cohomologies in p-adic Hodge theory. In particular, we describe the relevance (and the limitations) of a Betti-like topological realization via Berkovich spaces.
In Voevodsky’s theory of mixed motives, the Nisnevich topology plays an important role. He used the abelian category of sheaves with transfers with respect to this topology as a fundamental building block to construct his category of motives. Recently, an attempt to generalize this theory to a “non-homotopy invariant” version, called the theory of motives with modulus, was initiated by Kahn-Saito-Yamazaki. The idea is to construct the theory of motives which takes into account the behavior of motives in the neighborhood of the boundary (=modulus) of compactifications of varieties. To follow Voevodsky’s argument, we need a reasonable notion of topology which respects such boundary information. In this talk, I will introduce a topology on the category of proper modulus pairs. A proper modulus pair is just a pair of a proper variety and an effective Cartier divisor on it. The divisor can be regarded as the boundary information. This topology enables us to define the category of motives with modulus. Moreover, I will explain that this new topology is compatible with the usual Nisnevich topology, which implies the existence of an embedding of Voevodsky’s category into the category of motives with modulus. This talk is based on the joint work with Bruno Kahn, Shuji Saito and Takao Yamazaki.
Sabrina Pauli is giving a seminar talk in the geometry seminar at Duke.
Sabrina Pauli is giving a conference talk at the workshop in Real Enumerative Geometry and Beyond at Vanderbilt.
We discuss the vanishing of the negative S1-stable and P1-stable motivic homotopy groups over a base scheme S, and the Morel’s connectivity theorem. We discuss the ordinal Morel’s proof and the generations. We compare the formulas for motivically fibrant replacements in S1-stable and P1-stable cases that implies the vanishing, and touch the techniques of unramified sheaves and of transfers used in many significant researches to control the interaction of the Nisnevich and A1-localisations.
In this talk I will discuss joint work with Niny Arcilla Maya, Morgan Opie, Kirsten Wickelgren, and Inna Zakharevich in which we define a compactly supported motivic Euler characteristic and show it agrees with previous definitions as well as defines a homomorphism from K_0(Var/k) to the Grothendieck-Witt group of k in characteristic 0. In the first half of the seminar I will talk about the A^1 Euler characteristic and results of Levine-Raskit and Bachmann-Wickelgren on how to compute it. In the second half of the seminar I will talk about our computation using the Hochschild Homology of a smooth scheme, the benefits of defining the Euler characteristic in this way, and if time allows an example of how one might try to go about proving the main theorem in a different way.
Paul Arne Østvær is giving a seminar talk at the University of Zurich with the title "Inverting spherical Bott elements".
In his unpublished "Notes on framed correspondences" (2001–2003), V. Voevodsky invented a category of framed correspondences Fr∗(k), framed presheaves and framed sheaves. The major goal was to find a new approach to motivic stable homotopy theory, which would be more friendly for computational purposes. This programme was realised by G. Garkusha jointly with the speaker. I plan to give a survey of major achievements.
Sabrina Pauli is giving a seminar talk at the University of Southern California about Lines on Quintic Threefolds.
Sabrina Pauli is giving a seminar talk at the University of South Carolina about Lines on Quintic Threefolds.
Paul Arne Østvær is giving a seminar talk at Radboud University Nijmegen with the title "Introduction to homotopy groups of motivic spheres".
The doctoral candidate will defend his dissertation: Calculations of Motivic Invariants.
M.Sc. Jonas Irgens Kylling will give a trial lecture on the following topic: "The Grothendieck–Lefschetz trace formula".
Sabrina Pauli is giving a conference talk at the conference First Joint Meeting Brazil-France in Mathematics at IMPA in Rio de Janeiro.
In joint work with Elden Elmanto and Paul Arne Oestvaer we extend etale descent results of Thomason, Levine and Elmanto-Levine-Spitzweck-Oestvaer. Specifically, over quite general base schemes, we construct self-maps of motivic Moore spectra whose telescopes satisfy etale hyperdescent. We also show that etale localization is smashing in our context, and consequently recover all the aforementioned etale descent results. In this talk I will give an overview of the proof of these results: I will explain our methods for constructing the self-maps, our use of the six functors formalism to reduce to the case of fields, and our use of the slice spectral sequence to reduce to Levine's etale descent theorem.