Seminar on Combinatorics, Lie Theory, and Topology
# Independence, negative dependence and Lorentzian polynomials

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Europe/Rome

Aula Magna (Department of Mathematics)
### Aula Magna

#### Department of Mathematics

Description

Matroid theory is a discrete axiomatization of independence in algebra and graph theory, while negative dependence models repelling particles in statistical physics, and repelling random variables in probability theory. Many problems on independence display strong negative dependence properties. Several conjectures on negative dependence were formulated in the 1960's and 1970's, but negative dependence resisted all attempts of building a working theory around it for a long time. However in recent years two different successful approaches to prove negative dependence inequalities have been developed. One using Hodge theory and the other using the geometry of zeros of multivariate polynomials. The theory of Lorentzian polynomials merges these two approaches. In this talk we will introduce Lorentzian polynomials and give applications to matroid theory and the random cluster model.

The talk is based on joint work with June Huh and Jonathan Leake.