The Bieri--Neumann--Strebel--Renz invariants $\Sigma^q(X)$ of a connected, finite-type CW-complex $X$ are the vanishing loci for the Novikov--Sikorav homology of $X$ in degrees up to $q$. These invariants live in the unit sphere inside $H^1(X,\mathbb{R})$; this sphere can be thought of as parametrizing all free abelian covers of $X$, while the $\Sigma$-invariants keep track of the geometric finiteness properties of those covers. On the other hand, the characteristic varieties $V^q(X) \subset H^1(X,\mathbb{C}^{*})$ are the non-vanishing loci in degree $q$ for homology with coefficients in rank $1$ local systems. After explaining these notions and providing motivation, I will describe a rather surprising connection between these objects, to wit: each BNSR invariant $\Sigma^q(X)$ is contained in the complement of the tropicalization of $V^{\le q}(X)$. I will conclude with some examples and applications pertaining to complex geometry, group theory, and low-dimensional topology.

The Dundas-Goodwillie-McCarthy (DGM) theorem asserts that the difference between the K-theory of a ring and its thickening is the same as the difference in topological cyclic homology (TC). This has had spectacular applications in computations for K-theory as well as inspiring recent developments in p-adic Hodge theory. According to Bondal and van den Bergh, the category of perfect complexes on a qcqs scheme is equivalent to the category of perfect complexes on a $A_{\infty}$-ring. Therefore, the DGM theorem is applicable in this geometric context.

However, the category of perfect complexes on algebraic stacks do not enjoy this "monogenic generation" property. In joint work with Vova Sosnilo. we proved a version of the DGM theorem where this is applicable in many cases. I will explain how the proof works, whose new input comes from Bondarko's theory of weights (aka co-t-structures), and also some applications.

Joint work with Ajneet Dhillon. See arXiv:2007.14327 [math.AG]. The classical Skolem--Noether Theorem by Giraud shows us (1) how we can assign to an Azumaya algebra A on a scheme X a cohomological Brauer class in H^2(X,G_m) and (2) how Azumaya algebras correspond to twisted vector bundles. The Derived Skolem--Noether Theorem by Lieblich generalizes this result to weak algebras in the derived 1-category locally quasi-isomorphic to derived endomorphism algebras of perfect complexes. We show that in general for a co-family of presentable monoidal quasi-categories with descent over a quasi-category with a Grothendieck topology, there is a fibre sequence giving in particular the above correspondences. For a totally supported perfect complex E over a quasi-compact and quasi-separated scheme X, the long exact sequence on homotopy group sheaves splits giving equalities pi_i(Aut Perf E,id_E)=pi_i(Aut_Alg Perf REnd E, id_REnd E) for i >= 1. Further applications include complexes in Derived Algebraic Geometry, module spectra in Spectral Algebraic Geometry and ind-coherent sheaves and crystals in Derived Algebraic Geometry in characteristic 0.