'Positive geometries’, canonical logarithmic forms and Hodge theory

I will report on joint work with Clément Dupont, in which we define a notion of the genus of a pair (X,Y) of complex algebraic varieties, where Y is contained in X. Using this concept, we show that under some very general assumptions on X and Y, we can associate a canonical logarithmic differential form to singular chains on X whose boundary is contained in Y. This construction has many properties, including a recursive structure with respect to taking residues of forms and boundaries of chains.

This talk will first review the concept of a positive geometry in physics, before covering elements of Hodge theory and the theory of logarithmic differential forms. Then I will explain the above construction which, strangely, maps homology to differential forms. The talk will be illustrated with a large number of examples, and end with some applications in the theory of periods.