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.
Grothendieck proved that the small etale site is invariant under universal homeomorphism of schemes and calls this the "remarkable equivalence." The statement is false for Nisnevich/etale sheaves on big sites. However, after the inverting the residual characteristics, it turns out that the stable motivic homotopy category is. We will try to give a complete proof of this theorem, state some applications and future directions. This is joint work with A. A. Khan.
Since Suslin and Voevodsky's introduction of finite correspondences, several alternate correspondence categories have been constructed in order to provide different linear approximations to the motivic stable homotopy category. In joint work with Andrei Druzhinin, we provide an axiomatic approach to a class of correspondence categories that are defined by an underlying cohomology theory. For such cohomological correspondence categories, one can prove strict homotopy invariance and cancellation properties, resulting in a well behaved associated derived category of motives.
In this talk, I will discuss how moduli spaces of Morse flow trees in Legendrian contact homology (LCH) can be oriented in a coherent and computable manner, obtaining a Morse-theoretic way to compute LCH with integer coefficients. This is built on the machinery of capping disks, and I will briefly explain how different systems of capping disks affect the orientations. This, in turn, uses the fact that an exact Lagrangian cobordism with cylindrical Legendrian ends induces a morphism between the LCH-complexes of the ends, which can be proven to hold also with integer coefficients.
This is the first in a series of four talks which aims at an introduction to the theory of motives for rigid-analytic varieties as developed by Ayoub. In the first talk, I will mostly discuss the motivations for defining and studying rigid-analytic varieties and formulate some results (by Ayoub and Vezzani) that can be proved for the categories of motives of rigid-analytic varieties. In particular, I will formulate the recent rigidity theorem for rigid-analytic motives, proved by Bambozzi and Vezzani. While the first talk should mainly convey ideas and motivation, the remaining three talks will give more details to understand the proof of the rigidity theorem.
The classical Cayley-Dickson construction produces a sequence of algebras, including the quaternion and octonion algebras, from which we get H-space structures on the three- and seven-spheres by taking unit spheres, and hence we get the quaternionic and octonionic Hopf fibrations. I will describe a version of the Cayley-Dickson construction that works directly with the unit spheres, using homotopy type theory. Homotopy type theory can (conjecturally) be seen as an internal language to reason about higher toposes, giving rise to a kind of synthetic homotopy theory. Indeed, this version of the Cayley-Dickson construction works in any higher topos.
This talk discusses a few properties of cones with respect to a single endomorphism of the unit in the motivic stable homotopy category.
I will discuss the "isotropic motivic category". This "local" version of Voevodsky motivic category (with finite coefficients), obtained from the "global" one by, roughly speaking, annihilating the motives of anisotropic varieties, has many remarkable properties. Considering such "local" versions for all finitely generated extensions of a ground field, permits to read global information in a rather simple form. For appropriate (so-called, "flexible") fields, "isotropic motives" are more reminiscent of their topological counterparts. In particular, "isotropic Chow groups" hypothetically coincide with Chow groups modulo numerical equivalence (with finite coefficients) and so should be finite-dimensional (checked in various cases). On the other hand, the "isotropic motivic cohomology" ring of a point doesn't depend on a field and encodes Milnor's operations.
In preparation for the MHE seminar "log motives over a field", we give an introduction to ongoing work on motives for log schemes over fields. This is joint with Doosung Park and Paul Arne Østvær.
There are several cohomology theories over a field like Hodge cohomology theory that are not A1-invariant but still having other fundamental properties like the Projective bundle formula. These are not representable in DM. I will explain how to extend DM to include them using log geometry and cube-invariance. Some fundamental properties like Gysin triangles and blow-up triangles will be also discussed. This is joint with Federico Binda and Paul Arne Østvær.
I will talk about how to prove an arithmetic refinement of the Yau-Zaslow formula by replacing the classical Euler characteristic in Beauville's argument by a variant of Levine's motivic Euler characteristic. We derive several similar formulas for other related invariants, including Saito's determinant of cohomology, and a generalisation of a formula of Kharlamov and Rasdeaconu on counting real rational curves on real K3 surfaces. Joint work with Frank Neumann.
Unlike most cohomology theories in algebraic geometry, algebraic K-theory does not satisfy descent with respect to arbitrary blow-up squares. We explain why the only obstruction is the failure of the Mayer-Vietoris property for unions of closed subschemes. Since this obstruction vanishes after forcing A^1-homotopy invariance, this gives a direct new proof of Cisinski's theorem that homotopy K-theory does satisfy cdh descent.
Subtle Stiefel-Whitney classes have been introduced by Smirnov and Vishik as a tool for classifying quadratic forms. Following this path, in this talk, I will introduce subtle characteristic classes for Hermitian forms, coming from the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split Hermitian form of a quadratic extension. Moreover, I will discuss the connection between these new classes and the subtle Stiefel-Whitney ones, deduce information on the kernel invariant for quadratic forms divisible by a 1-fold Pfister form, show that these classes see the triviality of Hermitian forms and express the motive of the torsor associated to a Hermitian form in terms of its subtle characteristic classes.
This talk is about an extension of classical GW-theory to the world of stable infinity categories. I will discuss the relations of the new GW-theory to K-theory and L-theory via the 'fundamental cofibre sequence' and its relations to previous definitions of GW. A key feature of the new theory is that we do not need to invert 2 and it can even be applied to ring spectra, not just discrete rings. This is joint work with Calmès, Dotto, Harpaz, Hebestreit, Land, Nardin, Nikolaus and Steimle.