# Recent Trends in Algebraic Cycles, Algebraic K-Theory and Motives

Organizers:

- Guillermo Cortiñas (Universidad de Buenos Aires)
- E. Javier Elizondo (Universidad Nacional Autónoma de México)
- James Lewis (University of Alberta at Edmonton)
- Paulo Lima-Filho (Texas A&M)
- Chuck Weibel (Rutgers University at New Brunswick)

Schedule:

- Tuesday, Jul 25 [McGill U., McConnell Engineering Building, Room 304]
- 11:45 Nikita Karpenko (Univeristy of Alberta, Canada), Chow ring of generic flag varieties
- 14:15 Marc Levine (University of Essen, Germany), Motivic Virtual Fundamental Classes
- 15:45 Kyle Ormsby (Reed College, USA), Vanishing in motivic stable stems
- 17:00 Jose Pablo Pelaez Menaldo (UNAM, Mexico City, Mexico), A triangulated approach to the the Bloch-Beilinson filtration
- Wednesday, Jul 26 [McGill U., McConnell Engineering Building, Room 304]
- 11:15 Kirsten Wickelgren (Georgia Tech, USA), Motivic Euler numbers and an arithmetic count of the lines on a cubic surface
- 13:45 Daniel Juan Pineda (UNAM, Morelia, Mexico), On NIl groups of the quaternion group
- 14:45 Ben Antieau (University of Illinois at Chicago, USA), Negative and homotopy $K$-theory of ring spectra and extensions of the theorem of the heart
- 16:15 Inna Zakharevich (Cornell University, USA), A derived zeta-function

- Nikita Karpenko

Univeristy of Alberta, CanadaChow ring of generic flag varietiesLet $G$ be a split semisimple algebraic group over a field $k$ and let $X$ be the flag variety (i.e., the variety of Borel subgroups) of $G$ twisted by a generic $G$-torsor. We study the conjecture that the canonical epimorphism of the Chow ring of $X$ onto the associated graded ring of the topological filtration on the Grothendieck ring of $X$ is an isomorphism. Since the topological filtration in this case is known to coincide with the computable gamma filtration, this conjecture indicates a way to compute the Chow ring. - Marc Levine

University of Essen, GermanyMotivic Virtual Fundamental ClassesLet $B$ be a reasonable base-scheme and $Z$ a quasi-projective $B$-scheme. Relying on the Grothendieck 6-functor formalism for the motivic stable homotopy category, we define an object $C^{st}_{Z/B}$ in the motivic stable homotopy category $\text{SH}(B)$, which we call the*intrinsic stable normal cone*of $Z$ over $B$. For a motivic ring spectrum $\mathcal{E}$, we construct a fundamental class $[C^{st}_{Z/B}]_\mathcal{E}$ in $\mathcal{E}^{0,0}(C^{st}_{Z/B})$ and use this to construct for each perfect obstruction theory $\phi:E\to L_{Z/B}$ a virtual fundamental class $[Z,\phi]^{vir}_\mathcal{E}\in \mathcal{E}^{0,0}(\pi_{Z!}\Sigma^{E^\vee}1_Z)$. Here $\pi_Z:Z\to B$ is the structure morphism and we assume that $B$ is affine. There are also $G$-equivariant versions of these constructions for $G$ a ``tame'' algebraic group over $B$. Taking $B=\text{Spec} k$ and $\mathcal{E}=H\mathbb{Z}$, the spectrum representing motivic cohomology, we recover the definition of the fundamental class $[C_{Z/B}]\in \text{CH}_0(C_{Z/B})$ of the intrinsic normal cone $C_{Z/B}$ of $Z$ and the virtual fundamental class $[Z,\phi]^{vir}\in \text{CH}_r(Z)$, $r=\text{rank}E$, as defined by Behrend-Fantechi. Taking $\mathcal{E}=EM(K^{MW}_*)$, we get a virtual fundamental class $[Z,\phi]^{vir}_{K^{MW}_*}\in \tilde{\text{CH}}_r(Z,\text{det}^{-1}E)$, with $\tilde{\text{CH}}$ the Chow-Witt theory of Barge-Morel and Fasel. In case $r=0$, $\text{det}E=\mathcal{O}_Z$, and $Z$ projective over $k$, we can push this class forward to get a Grothendieck-Witt degree $\tilde{deg}[Z,\phi]^{vir}_{K^{MW}_*}\in \text{GW}(k)$. - Kyle Ormsby

Reed College, USAVanishing in motivic stable stemsRecent work of Röndigs-Spitzweck-Østvær sharpens the connection between the slice and Novikov spectral sequences. Using classical vanishing lines for the $E_2$-page of the Adams-Novikov spectral sequence and the work of Andrews-Miller on the $\alpha_1$-periodic ANSS, I will deduce some new vanishing theorems in the bigraded homotopy groups of the $\eta$-complete motivic sphere spectrum. In particular, I will show that the $m$-th $\eta$-complete Milnor-Witt stem is bounded above (by an explicit piecewise linear function) when $m \equiv 1$ or $2 \pmod{4}$, and then lift this result to integral Milnor-Witt stems (where an additional constraint on $m$ appears). This is joint work with Oliver Röndigs and Paul Arne Østvær. - Kirsten Wickelgren

Georgia Tech, USAMotivic Euler numbers and an arithmetic count of the lines on a cubic surfaceA celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, the number of lines depends on the surface, but work of Finashin-Kharlamov, Okonek-Teleman, and Segre shows that a certain signed count is always 3. We extend this count to an arbitrary field using A1-homotopy theory: we define an Euler number in the Grothendieck-Witt group for a relatively oriented algebraic vector bundle as a sum of local degrees, and then generalize the count of lines to a cubic surface over an arbitrary field. This is joint work with Jesse Leo Kass. - Ben Antieau

University of Illinois at Chicago, USANegative and homotopy $K$-theory of ring spectra and extensions of the theorem of the heartBarwick proved that the $K$-theory of a stable infinity-category with a bounded $t$-structure agrees with the $K$-theory of its heart in non-negative degrees. Joint work with David Gepner and Jeremiah Heller extends this to an equivalence of nonconnective K-theory spectra when the heart satisfies certain finiteness conditions such as noetherianity. Applications to negative $K$-theory and homotopy $K$-theory of ring spectra are provided, which were the original motivation for our work. - Inna Zakharevich

Cornell University, USAA derived zeta-functionMotivic measures can be thought of as homomorphisms out of the Grothendieck ring of varieties. Two well-known such measures are the Larsen--Lunts measure (over $\mathbf{C}$) and the Hasse--Weil zeta function (over a finite field). In this talk we will show how to lift the Hasse--Weil zeta function to a map of $K$-theory spectra which restricts to the usual zeta function on $K_0$. As an application we will show that the Grothendieck spectrum contains nontrivial elements in the higher homotopy groups.