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A projective variety is called two regular if it is defined by quadrics and all matrices in the minimal free resolutions of its homogeneous coordinate ring have linear entries. In an objective sense these are "the simplest" projective varieties and perhaps for this very reason they are ubiquitous in algebraic geometry. In this talk I will explain several novel contexts of interest for the SIAGA community where these varieties play a prominent role. In the process we will describe other properties which characterize two-regular varieties highlighting the fruitful interplay between classical and convex algebraic geometry.
