Boundedness and Moduli problems in birational geometry

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Algebraic geometry is a sophisticated area of mathematics dating back to the mid 19th-century, that links algebra and geometry with many parts of mathematics and theoretical physics. The basic objects, called algebraic varieties, are the common zero sets of polynomial functions, which are higher dimensional analogues to the ellipses and hyperbolas of antiquity. The subject has key applications in very many branches of modern mathematics, science and technology. One of the main goals in algebraic geometry is to classify algebraic varieties. These can often be decomposed into simpler shapes that act as fundamental building blocks in the classification. But how many different shapes appear in each class of building blocks? Calabi-Yau varieties, characterised as flat from the point of view of Ricci curvature, are one of 3 types of building blocks of algebraic varieties. Calabi-Yau 3-folds and 4-folds have formed the focus of interest of string theorists over recent decades. A better understanding of the geometry and the classification of Calabi-Yau varieties would advance string theory in fundamental ways, and would provide many new examples and models to study. Since they are building blocks for constructions in geometry and theoretical physics, understanding how many Calabi-Yau varieties there are is a question of fundamental importance. The problem is to know whether the shapes of Calabi-Yau varieties come in just finitely many families in any fixed dimension - a property that goes under the name of boundedness. This very difficult question remains wide open. While this problem has long been considered to be out of reach, recent developments make powerful techniques available to investigate new aspects of it. The aim of this research project is to show that there is essentially a finite number of families of Calabi-Yau varieties with some extra piece of structure -- an elliptic fibration -- in any dimension.