Events for December 8, 2025

In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation.

The Hodge Conjecture asserts that for particularly nice types of spaces called projective manifolds (equivalently, what algebraic geometers call smooth projective algebraic varieties), the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles.

Despite the topic looking far away from actuarial science, there may be some applications, in particular, through discrete Hodge theory on graphs and simplicial complexes, which turns the geometric ideas above into linear-algebraic tools you can compute with.