A Breakthrough in Understanding Geometric Shapes

In mathematics, an n-dimensional manifold is a concept that extends the idea of curves (1-dimensional) and surfaces (2-dimensional) to higher dimensions. Formally, manifolds are topological spaces where every point has a neighborhood that resembles an open subset of n-dimensional Euclidean space. Manifolds play a central role in various areas of mathematics, including analysis, optimization, geometry, and topology. They are also essential in the formulation of many physical theories; a clear example is the study of spacetime in general relativity, which is modeled by a four-dimensional manifold.

In 1968, mathematician John Milnor (Fields Medal 1962, Wolf Prize 1989, Abel Prize 2011) made a famous conjecture in Riemannian Geometry. He proposed that if a manifold is not negatively curved, in an averaged sense (nonnegative Ricci curvature), then it could not have infinitely many holes (the fundamental group is finitely generated). This conjecture has influenced many mathematical studies for decades. Recently, Elia Bruè of Bocconi's Department of Decision Sciences, Aaron Naber (Northwestern University), and Daniele Semola (Swiss Federal Polytechnic, Zurich) found an example that disproves Milnor's conjecture. They created a seven-dimensional manifold with nonnegative Ricci curvature where the fundamental group is not finitely generated. This is surprising because it shows that such shapes can be much more complex than previously thought. This result raises new questions, such as whether similar phenomena can happen in lower dimensions or with different geometric constraints. These questions could lead to new insights and discoveries in geometry.

Key Ideas in Simple Terms:

1. Ricci Curvature:
   • Think of Ricci curvature as a measure of how a shape bends. It plays a central role in the Einstein equations of general relativity.
2. Fundamental Group:
   • An algebraic object that encodes the hole structure of a shape. If it is finitely generated, the hole structure of the shape has finite complexity, meaning there are only finitely many holes.
3. The Counterexample:
   • The researchers constructed a seven-dimensional shape with infinitely many holes. This shape still has nonnegative Ricci curvature, contradicting Milnor's conjecture.