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We describe recent progress in a program to understand the classification ofthree-dimensional Fano varieties with $\mathbb{Q}$-factorial terminalsingularities using mirror symmetry. As part of this we give an improved andmore conceptual understanding of Laurent inversion, a technique that sometimesallows one to construct a Fano variety $X$ directly from a Laurent polynomial$f$ that corresponds to it under mirror symmetry.

Mirror Symmetry, Laurent Inversion and the Classification of $\mathbb{Q}$-Fano Threefolds