Mathematicians Make a Breakthrough on Gauss’s Riddle, Unsolved for 200 Years

A recent article in Scientific American spotlights a major mathematical breakthrough that “helps resolve a long-standing mystery” originating from ideas posed by Carl Friedrich Gauss in 1801. Ishan Levy, Visiting Professor in the School of Mathematics, has developed, alongside Harvard mathematician Aaron Landesman, a new framework that makes significant progress on the Cohen-Lenstra conjecture, which “claimed to be able to determine the average—not the exact—length of a cycle [of quadratic forms] before it reset.”

“That conjecture stood, generally accepted but unproven, for more than 40 years,” before Levy and Landesman’s work “found a new framework that goes a long way toward proving it.”

Their success builds on earlier work co-authored by Robert and Luisa Fernholz Professor Akshay Venkatesh and Craig Westerland, Member (2004–05) in the School of Mathematics, alongside the University of Wisconsin–Madison’s Jordan Ellenberg. The article describes Venkatesh and his collaborators’ foundational effort as a “tour de force” that combined aspects of number theory, algebraic geometry, topology, a combinatorics section, homological algebra, and probability.

Though those attempts hit technical roadblocks, Levy and Landesman successfully “patched” the proof using techniques from homotopy theory. Describing the remarkable scope of Levy and Landesman’s new solution, Scientific American notes that “the proof flows among different branches of math, drawing insights from across the mathematical kingdom.”