Olivier Martin works with geometric surfaces called algebraic varieties, which are sets of points where certain polynomials cancel out. A simple example is the unit circle, which lies in a plane and is described by the equation x^2 + y^2 – 1 = 0. This makes it an algebraic curve. By exploring polynomials of higher degree and with more variables, we can generate a wide range of surfaces found in fields as diverse as mathematics, cryptography, computer vision, and theoretical physics.

The concept of birational equivalence plays a crucial role in understanding these surfaces. Two algebraic varieties are said to be birational if they are nearly identical. Imagine the shadow cast by a charging cable; this shadow, while a projection, retains the essential properties of the shape of the cable and is thus birational to it. Olivier Martin aims to identify examples of surfaces that cannot be simplified in this way, specifically those that are not birational to surfaces with isolated singularities in three-dimensional space. This research has the potential to provide valuable insights into the geometric properties of algebraic surfaces in three-dimensional space.