Detecting blow-ups of Fano varieties via their Laurent polynomial mirrors

Mirror symmetry gives a correspondence between certain Fano varieties and Laurent polynomials, translating the classification of Fano varieties up to deformation into a combinatorial problem. I will present a set of combinatorial conditions Phi on pairs of Laurent polynomials (f,g) which imply the existence of mirror Fano varieties X_f and X_g related by a blow-up map X_g \to X_f. These criteria generalise the relationship between fans of toric varieties related by toric blowup; I will explain how in some key examples. Time permitting, I will discuss a new approach to constructing mirrors to Laurent polynomials, which is the main idea in the proof that Laurent polynomials in two variables satisfying the conditions Phi have mirrors related by blowing up in one point. This is based on upcoming joint work with Mark Gross.