The structure and growth of Hochschild (co)homology

This project will combine methods from commutative algebra, representation theory and rational homotopy theory to improve our understanding of Hochschild homology and cohomology, especially the open problem of determining their growth.

At the project's core is the deep interplay between Hochschild cohomology and the cotangent complex, a bridge that will be exploited in both directions.

I will use techniques pioneered in his solution of Vasconcelos' conjecture, which were further developed in my work with Iyengar to drastically improve our knowledge on the cotangent complex.

Concretely, the first objective is to show that non-complete intersection rings exhibit exponential growth in their Hochschild homology; through the theory of free loop spaces this will be applied to Vigu-Poirrier's conjecture on rationally hyperbolic spaces, and to Gromov's closed geodesic problem.

Second, the same novel methods will also be used to shed light on the long out of reach Second Conjecture of Quillen on the cotangent complex.

Third, I will develop the theory of natural operations on Hochschild cohomology, filling a gap in the state-of-the-art and adding a tool to be applied in the first two objectives.

Each of these problems directly impacts our understanding of the homological behaviour of complete intersection rings, and will indirectly be used to develop and unify the theory of ""non-commutative complete intersection rings"" which mirror their behaviour.

The proposed project will be hosted a world focal point for homotopical methods in algebra, and supervised by two leading experts in algebra and topology; it will raise my research profile to the top level, establishing my position as a leading figure at the intersection of commutative algebra, non-commutative algebra, and topology.