Activities This Week

Let $B/A$ be a pair of commutative rings. We propose an algebraic approach to the cotangent complex $L_{B/A}$. Using commutative semi-free DG ring resolutions of B relative to A, we construct a complex of $B$-modules $LCot_{B/A}$. This construction works more generally for a pair $B/A$ of commutative DG rings.

In the talk we will explain all these concepts. Then we will discuss the important properties of the DG $B$-module $LCot_{B/A}$. It time permits, we’ll outline some of the proofs.

It is conjectured that for a pair of rings $B/A$, our $LCot_{B/A}$ coincides with the usual cotangent complex $L_{B/A}$, which is constructed by simplicial methods. We shall also relate $LCot_{B/A}$ to modern homotopical versions of the cotangent complex.

Slides: https://sites.google.com/view/amyekut-math/home/lectures/cotangent

This is a joint work with Prof. Ilan Hirshberg and Dr. Tattwamasi Amrutam. Let $\Gamma$ be a discrete group with the property-AP. We give conditions for the inclusion of unital separable C-algebras $A \subset B$ for which every intermediate C-algebra $C$, such that $A \rtimes \Gamma \subset C \subset B \rtimes \Gamma$ is a crossed product. We shall end this talk with some natural classes of inclusion satisfying the above mentioned conditions.

Combinatorial nonpositive curvature of CAT(0) cube complexes plays a surprising role both in topological characterization of hyperbolic 3-manifolds and also in studying algebraic properties of random groups.
With Einstein, Krishna MS, Montee, and Steenbock, we introduce a new model for random quotients of free products that generalizes Gromov’s destiny model. I will discuss challenges that arise in this new setting, connections to work of Futer-Wise and Martin-Steenbock on cubulating quotients, as well as applications to residual finiteness using recent work of Einstein and Groves on relative cubulation.