Mathematicians studying how abstract shapes change over time face the issue that the shapes can develop “singularities” where equations break down, making it difficult to understand how the shape will continue to change.
In 1995, mathematician Tom Ilmanen posed the “multiplicity-one conjecture,” which suggested that these singularities are usually simple.
In a new development, mathematicians Richard Bamler and Bruce Kleiner have now proved that the conjecture is true.
This means that as a shape undergoes a process called “mean curvature flow,” it will usually either shrink to a point (creating a singularity that looks like the pinched-off portion of a soap bubble) or collapse to a line.
With the proof of Ilmanen’s conjecture, “we now basically have a complete understanding of the mean curvature flow of surfaces in three-dimensional spaces,” a Stanford mathematician says.
This knowledge could have significant uses in geometry and topology, and simplify some previous proofs.
