John von Neumann’s 84-year-old problem solved by graduate students

A famous math problem that has puzzled mathematicians for decades has met its solution by Cornell Univ. researchers. Graduate students Yash Lodha and Justin Moore, the University’s professor of mathematics, detailed a geometric solution for the von Neumann-Day problem, first introduced by John von Neumann back in 1929.


Lodha presented his solution at the London Mathematical Society’s Group Theory symposium and recently submitted the work to a journal. The inspiration for Lodha’s work originated in the early 20th century, when mathematicians first proved that a ball that exists in three-dimensional space can be chopped into a finite number of pieces and can be reassembled, like a jigsaw puzzle, into two balls, each the size of the original ball. It was eventually known as the Banach-Tarski paradox. von Neumann attributed the algebraic properties of the symmetries in the problemto their inherent natures towards the spheres. He isolated this property and mathematicians soon described it as a “non-amenability” property.


Advances in mathematics are very rare and most of the time they’re incrementally build off of previous work. In order complete this work, among his most valuable insights was one first described by the late Bill Thurston, another one of Cornell’s professors from the past. The discourse involved a way of expressing the group in a different light, as a “continued fractions model.”


Lodha’s work also stems from the work of Nicolas Monod, who constructed a geometrically oriented counterexample to the von Neumann-Day problem. Lodha and Moore’s contribution was to isolate a finitely presented subgroup. Further work from the group could make the solution to the von Neumann-Day problem even stronger: by isolating stronger finite conditions for proving that the group has a finite number of rules.


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