We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in ℙ4.

I briefly give an overview on the key ML frameworks involved in this analysis (neural networks, auto-differentiation). This talk is mainly based on 2012.04656.

An $\mathbb R$-tree is a metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of orders underlying $\mathbb R$-trees be characterised by the fact that every branch is order-isomorphic to a real interval? In the first part of the talk, I answer this question in the negative: there is a branchwise-real tree order which is not continuously gradable. In the second part, I show that a branchwise-real tree order is continuously gradable if and only if every embedded well-stratified (i.e. set-theoretic) tree is $\mathbb R$-gradable. This tighter link with set theory is put to work in the third part answering a number of refinements of the main question, yielding several independence results.