Leo Herr

My work is in algebraic geometry. Algebraic geometry studies solutions to polynomial equations, like y = x² or the circle x² + y² = 1. These are called varieties. Famous problems like Fermat’s last theorem are all about points on varieties.

I study varieties using extra combinatorial data called logarithmic structures which enrich and compactify ordinary varieties as a middleman between schemes and tropical geometry. These tropical spaces look like beehives and tell us about the logarithms of the original variety. They describe degenerations, compactifications, and boundaries of varieties.

I use log geometry to count the number of curves on the variety meeting certain conditions. These “Gromov-Witten invariants” are used in physics because curves in the variety can be the paths of particles hurtling through spacetime. To find the Gromov-Witten invariants of a variety, one can gradually break it up into smaller pieces and compute them for each piece. Then log geometry helps you glue them back into the invariants of the original variety. These numbers can also be computed using K theory, for which my coauthors and I invented “log quantum K theory.”

My other work is in number theory (the Hasse problem) and deformations of algebras after L. Illusie.