June 23, 2014 was a nice day for mathematicians Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terence Tao and Richard Taylor. They were informed that they will be receiving the inaugural Breakthrough Prizes in Mathematics, each with a cash award of USD$3,000,000.

The Breakthrough Prizes in Mathematics complement the Breakthrough Prizes in Fundamental Physics, which were inaugurated in 2012, and the Breakthrough Prizes in Life Sciences, which were inaugurated in 2013.

In future years, there will be one award in mathematics, one award in physics, and six in life sciences. Each of the eight annual awardees will receive USD$3,000,000, as in the inaugural prizes. Funding for the Breakthrough Prizes is provided jointly by Russian Internet billionaire Yuri Milner, who founded the prizes, and Facebook founder Mark Zuckerberg. The intent is to establish awards of stature comparable to or exceeding that of the Nobel prizes that are now granted in physics, chemistry, medicine, literature and peace (but not mathematics).

Here are some details of the mathematical awardees:

- Simon Donaldson, a British mathematician at Imperial College London and the Simons Center for Geometry and Physics at Stony Brook University, has published landmark results on the topology of differentiable (“smooth”) four-dimensional manifolds. He is perhaps best known for his “diagonalizability theorem,” which is that if the intersection form of a smooth, closed, simply-connected 4-manifold is positive- or negative-definite, then it is diagonalizable over the integers. Donaldson’s work has connections to string theory.
- Maxim Kontsevich, a Russian-French professor at the Institut des Hautes Etudes Scientifiques outside Paris, has already won another Breakthrough Prize, in physics. His mathematical work unifies knot theory and topological field theory with mathematical physics. For example, he introduced what is known as the moduli space of stable maps, a mathematically rigorous formulation of the Feynman integral for topological string theory. His Ph.D. work at the University of Bonn was done under the supervision of experimental mathematician Don Zagier.
- Jacob Lurie of Harvard University is the youngest recipient among the inaugural recipients at 36. He is best known for this work, beginning with his thesis in 2004, on infinity-categories and derived algebraic geometry. Derived algebraic geometry combines homotopy theory with algebraic geometry.
- Terence Tao, an Australian mathematician now at UCLA, is truly a Renaissance mathematician, with significant work in harmonic analysis, partial differential equations, combinatorics, ergodic Ramsey theory, random matrix theory and analytic number theory. Perhaps his best-known result is the Green-Tao theorem, which is that the sequence of prime numbers contains arbitrarily long arithmetic progressions, or, in other words, that given any n, there is some m and k such that the sequence (m, m+k, m+2k, …, m+k(n-1)) are all prime numbers. Subsequently Tao and Ziegler extended the result to cover polynomial progressions.
- Richard Taylor is perhaps best known as a British student of Andrew Wiles. Ater Wiles’ first but later retracted proof of Fermat’s Last Theorem, Taylor returned to Princeton to help complete the proof. In 2001, Taylor, together with Breuil, Conrad and Diamond, completed the proof of the modularity theorem, formerly known as the Taniyama-Shimura conjecture, which is that elliptic curves over the field of rational numbers are related to modular forms. Wiles, as part of the proof of Fermat’s Last Theorem, had proved the modularity theorem for the special case of semistable elliptic curves.

Some information on the Breakthough Prizes, the current winners and other details are given in a New York Times report and a Scientific American article.