Seminar in Mathematical Modeling and Analysis - Ben Scott

The seminars in Mathematical Modeling and Analysis are aimed at researchers, employees, and students.

This week's seminar is given by Ben Scott, University of Chicago.

Title: Regularity of the tropical Monge-Ampère equation on the boundary of the 3-simplex

Abstract: In a recent paper, Jakob Hultgren, Mattias Jonsson, Enrica Mazzon, and Nick McCleerey presented existence and uniqueness results for a Monge-Ampère equation adapted to integral tropical manifolds. Specifically, they define and solve such an equation on the boundary of an n-simplex, doing so by employing ideas from the theory of optimal transport. Motivated by the SYZ conjecture of mirror symmetry, they also indicate the potential to further use optimal transport in investigating the behavior of the solution near the singular set. In this talk I will detail the later work done by Jonsson, McCleerey, Neil Patram, and myself which answers this question in dimension 2. Our method draws on both optimal transport regularity theory for the Monge-Ampère equation as well as a family of models for special Kähler metrics due to Martin Callies and Andriy Haydys. After describing how a solution to the Monge-Ampère equation enables the construction of necessary geometric data, I will show how the aforementioned models reduce to a single case, thus giving an exact description of the solution’s singular behavior.

Contact Antti Perälä to receive the Zoom link.