When mathematicians fold a space along its symmetries, they often create a sharp crease—a singularity—where the geometry breaks down, like the centre of a black hole in physics. This project builds new tools to answer a deceptively simple question: how much does the original, crumpled space still reveal after you smooth out those creases? The problem matters because singularities are unavoidable in many areas of mathematics and theoretical physics. To study them, researchers replace each singularity with a patch of smooth geometry—a “resolution”—but they want to add as little extra structure as possible. Two major conjectures, the Generalized McKay Correspondence and the Crepant Resolution Conjecture, have guided this work for decades, but they remain unproven in full generality. This is fundamental, curiosity-driven research with no immediate practical application. The project’s innovation is to bring techniques from symplectic topology—a field usually concerned with the geometry of motion and energy—into a problem traditionally tackled by algebraic geometry. If successful, the work could reshape how mathematicians understand the deep links between different kinds of geometry. Past fundamental work on singularities and symmetries has, for example, underpinned developments in string theory and the mathematics of quantum field theory; this project may open similar unexpected pathways.
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A large portion of Mathematics is concerned with the study of symmetries. One often wishes to consider two objects as being the same if they are related by a symmetry. When we start identifying points of a space that are related by symmetries, we usually end up with a space that has badly behaved regions called singularities where the space looks like it was folded in on itself. For example in physics the centre of a black hole is a singularity in space. In order to be able to study the space, Mathematicians often need to create a nicer space first, by replacing the singularities with smooth regions. This procedure is called a "resolution of singularities". This usually creates new topology: for example a singular point may get replaced by several new spheres. The goal is to create as little new topology as possible, because one wants the new smooth space to be as similar as possible to the original singular space. Such resolutions are called "crepant". Mathematicians and Physicists have been fascinated for decades by this passage from a singular space to a crepant resolution. It led to two major conjectures called the "Generalized McKay Correspondence" and the "Crepant Resolution Conjecture". Both conjectures try to answer the fundamental question "How much does the resolution of singularities remember about the properties of the original singular space?". This EPSRC project builds new tools to advance our understanding of this fundamental question, and it aims to prove versions of those conjectures. Prof. Ritter (University of Oxford) will lead the project as the PI with the help of a PDRA, and Prof. McLean (The State University of New York at Stony Brook) will assist in the project as a collaborator. The proposal builds upon the existing collaboration by Ritter and McLean on the McKay Correspondence, as well as recent work by McLean on birational Calabi-Yau manifolds. The study of resolutions of singularities has excited much research in the area of Algebraic Geometry. The innovative ingredient of this proposal is to bring into play insights from another research field, Symplectic Topology. The output of the proposal will open up new research directions that are expected to have a profound influence on Geometry. The impact of this research is likely to be far-reaching due to its fundamental and interdisciplinary nature.
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