between mathematics and physics is the reformulation of the geometric Langlands program in terms of quantum field theory. A very recent example involves the computation of scattering amplitudes in gauge theories, motivated by the practical problem of computing backgrounds for the Large Hadron Collider. These computations use the tools of algebraic geometry and some methods from geometric number theory. Ideas of topology are very important in many areas of physics. Most notably, topological quantum field theories of the Chern-Simons form are crucial to understanding some phases of condensed matter systems. These are being actively explored because they offer a promising avenue for constructing quantum computers.
Example 3: Kinetic Theory
Kinetic theory is a good example of the interaction between areas of the mathematical sciences that have traditionally been seen as core and those that had been seen as applied. The theory was proposed by Maxwell and Boltzmann to describe the evolution of rarefied gases (not dense enough to be considered a “flow,” not dispersed enough to be just a system of particles, a dynamical system). Mathematically, the Boltzmann equation involves the spatial interaction (collisions) of probability densities of particles travelling at different velocities. The analytical properties of solutions—their existence, regularity, and stability, and the phenomenon of shock formation—were little understood until approximately 30 years ago. Hilbert and Carleman worked on these problems for many years with little success, and attempts to understand the analytical aspects of the equation—existence, regularity, stability of solutions, as well as possible shock formation—had not advanced very far. In the 1980s, the equation arose as part of the need for the modeling of the reentry dynamics of space flight through the upper atmosphere and it was taken up again by the mathematical community, particularly in France. That gave rise to 20 years of remarkable development, from the celebrated work of Di Perna-Lions (1988) showing the existence of solutions, to the recent contributions of Villani and his collaborators. In the meantime, the underlying idea of the modeling of particles interacting at a rarefied scale appeared in many other fields in a more complex way: sticky particles, intelligent particles, and so on, in the modeling of semiconductors, traffic flow, flocking, and social behavior, particularly in phenomena involving decision making.
The sorts of connections exemplified here are powerful, because they establish alternative modes by which mathematical concepts may be explored. They often inspire further work because surprising connections hint at deeper relationships. It is clear that the mathematical sciences have benefited in recent years from valuable, and perhaps surprising, connections within the discipline itself. For example, the Langlands program in