Classifying geometric shapes defined by polynomial equations

Classification is a fundamental endeavour in mathematics: already in Antiquity, Archimedes classified regular polyhedra. I work on the classification of algebraic varieties, the abstract geometric shapes defined by polynomial equations. For example, an ordinary sphere (the surface of a football) is an algebraic variety because the spatial coordinates of each of its points satisfy a polynomial equation of degree 2 (x^2+ y^2+z^2= r^2). My research is in birational geometry, where varieties are reconstructed from building blocks with simple geometry. As varieties are defined by polynomial equations, explicit descriptions in terms of simple building blocks are of interest to engineers, physicist, and more generally to any scientist looking at phenomena modelled by equations! The range of applications of algebraic geometry both to other fields of and outside pure mathematics is very wide.

What are these simple building blocks that every geometric shape can be reconstructed from? These simple shapes are distinguished by their curvature. To fix ideas, the surface of a ball has positive curvature, that of a table has zero curvature and that of a saddle has negative curvature. I usually consider objects of higher dimensions (such as the 4-dimensional space-time in physics), and the simple shapes are called Fano varieties (positive curvature ), Calabi-Yau varieties (zero curvature ) and varieties of general type (negative curvature). Calabi-Yau varieties are especially interesting to physicists: indeed, string theory (an attempt to explain all of the particles and fundamental forces of nature in one theory) postulates that spacetime has 10 dimensions. 4 of these are the classical spacetime dimensions, while the additional 6 are unseen and form a Calabi-Yau shape.

Thanks to the work of Fields Medallist Caucher Birkar, we know that Fano shapes can be classified in each dimension. In dimension 3, the classification is complete, but in higher dimensions, it is the object of current research. By contrast, the classification of Calabi-Yau shapes is still out of reach. Yet, we can use relations between Fano and Calabi-Yau shapes to understand the geometry of Calabi-Yau shapes. This idea is at the heart of my project "Calabi-Yau Pairs and Mirror Symmetry for Fano Varieties". In return, the findings on Calabi-Yau geometry provide a new angle on the geometry of Fano shapes.