### Bipartite matching, invariance, and regularity of optimal transport

Probability Seminar

11th February 2022, 3:30 pm – 4:30 pm

Fry Building, 2.04 (also on zoom)

The bipartite matching problem is one of the classical random optimization problems. The macroscopic behaviour is well understood since the work of Ajtai, Komlos, Tusnady, Talagrand and others in the 80s and early 90s. A few years ago, Caracciolo et al. proposed a new ansatz, based on a linearisation of the Monge-Ampère equation to the Poisson equation, to get refined estimates for this problem.

I will show how one can use their ansatz to prove that in dimension two certain thermodynamic limits of the optimal bipartite matchings do not exist. A key tool is a harmonic approximation result for optimal couplings between arbitrary measures.

This is based on joint work with Michael Goldman, Francesco Mattesini, and Felix Otto.

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