D. B. A. Epstein, V. Markovic, The Logarithmic Spiral: A Counterexample to the K = 2 Conjecture, Annals of Mathematics, Second Series, Vol. 161, No. 2 (Mar., 2005 ...

We construct a counterexample to the "hot spots" conjecture; there exists a bounded connected planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with ...

Arithmetic Geometry and Manin's Conjecture Publication Trend The graph below shows the total number of publications each year in Arithmetic Geometry and Manin's Conjecture.

It would take another year and a half for them to assemble a true counterexample to the periodic tiling conjecture.

But they’re related. Greenfeld and Tao planned to come up with a discrete counterexample to the conjecture that they could then modify to work in the continuous case as well.