Dr Alexander Kasprzyk - Maximally mutable Laurent polynomials

Recent progress has been made classifying Fano manifolds via mirror symmetry: to each Fano manifold X we associate a Laurent polynomial mirror f, such that certain key invariants of X and f agree. If this method is to be applied systematically in high dimensions, a fundamental question which needs answering is "Which Laurent polynomials are mirrors to Fano manifolds?" Several successful ansatzs have been proposed over the past decade, giving a partial answer to this question. The most recent of these -- the Minkowski ansatz -- successfully recovered the three-dimensional classification of Mori--Mukai. But each ansatz has limitations. For example, the Minkowski ansatz is strictly three-dimensional. Here I will describe a new family of Laurent polynomials, called rigid maximally mutable, that conjecturally give a complete answer to this question.