Recently, some of the most profound advances in algebra and geometry have been inspired by ideas from physics. Maxim Kontsevich has led the way in a number of these developments.
Beginning with Heisenberg’s introduction of quantum mechanics, the mathematical process of quantization – that is, of passing from classical to quantum mechanics – has been a central theme. One version, known as deformation quantization, has for its natural setting classical spaces known as Poisson manifolds. Their exact quantization had been carried out in special cases, but in general this proved to be a formidable problem. It was resolved brilliantly by Kontsevich, who used ideas from quantum field theory.