Cubical structures appear naturally in several situations, sometimes clandestinely. In particular, we will consider such structures related to Feynman categories and discuss their origins. In this context, they appear for instance in coproduct structures, in a W-construction and relative W-constructions given via push-forwards, which give natural cubical complexes that are homotopy equivalent to moduli spaces.

A complex polynomial in n variables satisfies the (Hurwitz) half-plane property if its value is nonzero when its inputs all have positive real part. This classical definition is the start of an interesting story that factors through the closely related theory of stable polynomials, due to Borcea and Brändén [Duke J. Math 2008] and leads to the notion of Lorentzian polynomials, recently introduced by Brändén and Huh [Annals of Math 2020]. Lorentzian polynomials also have an elementary definition, though subtle properties and close links to (statistical) negative dependence, matroid theory, and discrete convexity.

In this two-part Basic Notions seminar, Graham plans to spend a few minutes saying why he thinks this is interesting. Then together we will watch a recording of an introductory lecture that June Huh gave at the IAS in 2019. We will break in the middle, since the lecture is 90 minutes long. We will resume in the second week and conclude with some informal discussion.