Grothendieck群与加法不变量的分类

分圆迹映射$K \rightarrow TC$与Grothendieck迹公式在非交换几何上的类比

Analogy between the cyclotomic trace map $K \rightarrow TC$ and the Grothendieck trace formula via noncommutative geometry

在这篇文章中，我们提出了一个分类程序，以捕捉之间的类比结晶Grothendieck-Lefschetz迹公式和分圆迹映射$K\rightarrow TC$从代数$K$ -理论的拓扑循环同调$TC$。首先，我们把这类方案归为非交换方案的$(2, \infty)$ -范畴。这给出了一个方案的有理点集的分类。然后，我们对Crystalline Grothendieck-Lefschetz迹公式进行了分类，并在$\mathbf{F}_{p}$上的非对易格式下找到了Crystalline上同调的一个类似形式.我们的类比表明，在非交换设置$l\neq p$的$l$ -进上同调迹公式的分类存在。最后，我们写下相应的字典。

Heisenberg分类与Hilbert格式

Heisenberg categorification and Hilbert schemes

给定SL（2，C）的一个有限子群G，我们定义一个加法2-范畴H^G，它的Grothendieck群同构于Heisenberg代数的一个积分形式。我们构造了H^G在C^2/G的最小分解上的点的Hilbert方案上的相干层的导出范畴上的作用。

等变凝聚层、Soergel双模与仿射Hecke代数的范畴化

Equivariant Coherent Sheaves, Soergel Bimodules, and Categorification of Affine Hecke Algebras

本文利用Soergel的范畴仿射Hecke代数给出了Grothendieck分解下的某些等变相干层的描述。作为一个应用，我们推导出这些相干层范畴的范畴的反常层的Bezrukavnikov-云的工作中考虑的关系，推广Arkhipov Bezrukavnikov的结果。此外，我们还证明了Riche层和Bezrukavnikov-Riche层上的弱辫群作用可以提升为严格辫群作用.

Universal additive Chern classes and a GRR-type theorem

We construct a functor, from the category of schemes to the category of graded rings, that is an initial object for having a theory of Chern classes with an additive first Chern class. For any scheme $X$, the graded ring that our functor associates to $X$ is related to the associated graded ring of the $γ$-filtration on the Grothendieck ring of finite rank locally free sheaves on $X$ via a Grothendieck-Riemann-Roch type theorem.

稳定双有理不变量复形

Complexes of stable birational invariants

本文引入了一个新的稳定双有理不变量，以链复形同伦类型的形式分类化动机体积，并由此构造了回缩有理性的动机障碍及一类普遍的不变量。

Additive invariants of orbifolds

In this article, using the recent theory of noncommutative motives, we compute the additive invariants of orbifolds (equipped with a sheaf of Azumaya algebras) using solely "fixed-point data". As a consequence, we recover, in a unified and conceptual way, the original results of Vistoli concerning algebraic K-theory, of Baranovsky concerning cyclic homology, of the second author with Polishchuk concerning Hochschild homology, and of Baranovsky-Petrov and Caldararu-Arinkin (unpublished) concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topologic…