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

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  • 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)
    • Nikita Karpenko
      Univeristy of Alberta, Canada
    Chow ring of generic flag varieties
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    Let $G$ 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, Germany
    Motivic Virtual Fundamental Classes
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    Let $B$ 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, USA
    Vanishing in motivic stable stems
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    Recent 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.
    • Jose Pablo Pelaez Menaldo
      UNAM, Mexico City, Mexico
    A triangulated approach to the the Bloch-Beilinson filtration
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    We will present an approach to the Bloch-Beilinson filtration in the context of Voevodsky's triangulated category of motives.
    • Kirsten Wickelgren
      Georgia Tech, USA
    Motivic Euler numbers and an arithmetic count of the lines on a cubic surface
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    A 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.
    • Daniel Juan Pineda
      UNAM, Morelia, Mexico
    On NIl groups of the quaternion group
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    We will describe the Nil groups of the ring $\mathbb{Z}Q_8$, the integral group ring of the quaternion group, we will give applications for the calculation of $K$ theory groups of some infinite groups.
    • Ben Antieau
      University of Illinois at Chicago, USA
    Negative and homotopy $K$-theory of ring spectra and extensions of the theorem of the heart
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    Barwick 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, USA
    A derived zeta-function
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    Motivic 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.