Groups of order p^3 are mixed Tate

A natural place to study the Chow ring of the classifying space BG, for G a linear algebraic group, is Voevodsky's triangulated category of motives, inside which Morel and Voevodsky, and Totaro have defined motives M(BG) and M^c(BG), respectively. We show that, for any group G of order p^3 over a field of characteristic not p which contains a primitive p^2-th root of unity, the motive M(BG) is a mixed Tate motive. We also show that, for a finite group G over a field of characteristic zero, M(BG) is a mixed Tate motive if and only M^c(BG) is a mixed Tate motive.Comment: 17 page

The categorical DT/PT correspondence and quasi-BPS categories for local surfaces

We construct semiorthogonal decompositions of Donaldson-Thomas (DT) categories for reduced curve classes on local surfaces into products of quasi-BPS categories and Pandharipande-Thomas (PT) categories, giving a categorical analogue of the numerical DT/PT correspondence for Calabi-Yau 3-folds. The main ingredient is a categorical wall-crossing formula for DT/PT quivers (which appear as Ext-quivers in the DT/PT wall-crossing) proved in our previous paper. We also study quasi-BPS categories of points on local surfaces and propose conjectural computations of their K-theory analogous to formulas already known for the three dimensional affine space.Comment: 42 page

Categorical and K-theoretic Donaldson-Thomas theory of C3\mathbb{C}^3 (part II)

Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three dimensional affine space and in the categorical Hall algebra of the two dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory. We first prove a categorical analogue of Davison's support lemma, namely that complexes in the quasi-BPS categories for coprime length and weight are supported over the small diagonal in the symmetric product of the three dimensional affine space. The categorical support lemma is used to determine the torsion-free generator of the torus equivariant K-theory of the quasi-BPS category of coprime length and weight. We next construct a bialgebra structure on the torsion free equivariant K-theory of quasi-BPS categories for a fixed ratio of length and weight. We define the K-theoretic BPS space as the space of primitive elements with respect to the coproduct. We show that all localized equivariant K-theoretic BPS spaces are one dimensional, which is a K-theoretic analogue of the computation of (numerical) BPS invariants of the three dimensional affine space.Comment: 45 pages, to appear in Forum Math. Sigm

The local categorical DT/PT correspondence

In this paper, we prove the categorical wall-crossing formula for certain quivers containing the three loop quiver, which we call DT/PT quivers. These quivers appear as Ext-quivers for the wall-crossing of DT/PT moduli spaces on Calabi-Yau 3-folds. The resulting formula is a semiorthogonal decomposition which involves quasi-BPS categories studied in our previous papers, and we regard it as a categorical analogue of the numerical DT/PT correspondence. As an application, we prove a categorical DT/PT correspondence for sheaves supported on reduced plane curves in the affine three dimensional space.Comment: 35 page

Deformed dimensional reduction

Since its first use by Behrend, Bryan, and Szendr\H{o}i in the computation of motivic Donaldson-Thomas (DT) invariants of $\mathbb{A}_{\mathbb{C}}^3$, dimensional reduction has proved to be an important tool in motivic and cohomological DT theory. Inspired by a conjecture of Cazzaniga, Morrison, Pym, and Szendr\H{o}i on motivic DT invariants, work of Dobrovolska, Ginzburg, and Travkin on exponential sums, and work of Orlov and Hirano on equivalences of categories of singularities, we generalize the dimensional reduction theorem in motivic and cohomological DT theory and use it to prove versions of the Cazzaniga-Morrison-Pym-Szendr\H{o}i conjecture in these settings.Comment: 40 page

On Rationally Ergodic and Rationally Weakly Mixing Rank-One Transformations

We study the notions of weak rational ergodicity and rational weak mixing as defined by Jon Aaronson. We prove that various families of infinite measure-preserving rank-one transformations possess (or do not posses) these properties, and consider their relation to other notions of mixing in infinite measure.Comment: Some typos corrected, a reference added, arguments expande

Ergodicity and Conservativity of products of infinite transformations and their inverses

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation $T$ in the class, the cartesian product $T\times T$ of the transformation with itself is ergodic, but the product $T\times T^{-1}$ of the transformation with its inverse is not ergodic. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.Comment: Added references and revised some arguments; removed old section 6; main results unchange