Harald Skarke (Vienna): Abundant sets of internal spaces for string theory

String model building usually relies on compactification on a Calabi‐Yau (CY) threefold or involves a construction known as F‐theory which combines the data of the internal space and of some background fields into those of a CY fourfold.
The most fertile construction method for CY manifolds comes from a branch of algebraic geometry known as toric geometry, where families of CY n‐folds are associated to (n+1)‐ dimensional polytopes that have a certain property called reflexivity.
I will explain the concepts introduced above. Then I will outline how we managed to classify all 476,800,776 reflexive 4‐polytopes almost 20 years ago, thereby creating what is still the world's largest list of CY threefolds.
Finally I will report on recent work on the classification of a particular class of reflexive 5‐polytopes (there are 322,383,760,930), which resulted in the largest existing database for CY fourfolds.