Terry Tao has recently made substantial progress on the 3x+1 conjecture. Our own Professor Lagarias is the international go-to expert on this conjecture. Check out this Quanta magazine piece about this progress, which includes Jeff’s comments, here.
On the subject of Tao, another Quanta article beautifully conveys the excitement of mathematical discoverywhile telling the story of three particle physicists who serendipitously discovered some “new basic mathematics” while calculating neutrino oscillations, and the collaboration with Terry Tao that ensued. Basically, the result says that the components of an orthonormal eigenbasis for a real symmetric (or Hermitian) matrix can be described by a formula involving only the eigenvalues of the matrix and certain submatrices. See Tao’s blog for a concise description of the precise statement. Their preprint is worth a look too: after a readable introduction, which includes examples that even a Math 217 student could understand, it provides three different proofs. [Spoiler alert: as it turns out, the physicists had actually re-discovered a 1934 result Karl Löwner (Karl Löwner. Über monotone Matrixfunktionen. Math. Z., 38(1):177–216, 1934), a good reminder of the value of being able to read mathematics in German.]