This talk aims to summarize a project I was involved in during the past 5 years, with the hope of explaining our most complete understanding so far, as well as challenges and open problems. The main messages of this project are summarized below; I plan to describe, through examples, many of the concepts they refer to, and the evolution of ideas leading to them. No special background is assumed.

The dichotomy between structure and randomness plays an important role in areas such as combinatorics and number theory.

There is growing interest in looking at operations on quantum cohomology that take into account symmetries in the holomorphic spheres (such as the quantum Steenrod powers, using a Z/p-symmetry). In order to prove relations between them, one needs to generalise this to include equivariant operations with more marked points, varying domains and different symmetry groups. We will look at the general method of construction of these operations, as well as two distinct examples of relations between them.

While the trend in machine learning has tended towards more complex hypothesis spaces, it is not clear that this extra complexity is always necessary or helpful for many domains. In particular, models and their predictions are often made easier to understand by adding interpretability constraints. These constraints shrink the hypothesis space; that is, they make the model simpler. Statistical learning theory suggests that generalization may be improved as a result as well. However, adding extra constraints can make optimization (exponentially) harder.