I wanted to start a thread on brilliant mathematical problems that are being solved. Since the millenium problems are the most famous, I titled the thread that way. However, that doesn't stop us from posting other incredibly important theories that are being proved: #1 P versus NP # 2 The Hodge conjecture # 3 The Poincare conjecture (proven) # 4 The Riemann hypothesis # 5 Yang-Mills existence and mass gap # 6 Navier-Stokes existence and smoothness # 7 The Birch and Swinnerton-Dyer conjecture http://www.esi2.us.es/~mbilbao/claymath.htm
An important and beautiful result has just been proven that is not a millenium problem. It has major implications for mathematics as well as philosophy. I strongly recommend the whole article for philosophers. Mathematicians Bridge Finite-Infinite Divide A surprising new proof is helping to connect the mathematics of infinity to the physical world. With a surprising new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary. The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: “finitistic” ones, which can be proved without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption — not evident in nature — that infinite objects exist. Mapping and understanding this division is “at the heart of mathematical logic,” said Theodore Slaman, a professor of mathematics at the University of California, Berkeley. This endeavor leads directly to questions of mathematical objectivity, the meaning of infinity and the relationship between mathematics and physical reality... https://www.quantamagazine.org/20160524-mathematicians-bridge-finite-infinite-divide/