Dolbeault cohomology is a fundamental cohomological invariant for complex manifolds. This analytic invariant is connected to de Rham cohomology by means of a spectral sequence, called the Frölicher spectral sequence. In this talk, I will explore this connection from a multiplicative viewpoint: using homotopy-theoretical methods, I will describe how products (and higher products) behave in the Frölicher spectral sequence. I will also review an extension of the theory to the case of almost complex manifolds and talk about some open problems in complex geometry that may be addressed using homotopy theory.
I believe I can keep the whole talk accessible to junior mathematicians as the homotopy techniques are quite basic. It would not harm to know a bit about A-infinity structures. Maybe I can recommend to read Section 1 of https://arxiv.org/pdf/1202.3245.pdf.