From Twistor Theory to Hyperkähler Geometry -
​​​​​​​Twistor theory is an elegant mathematical framework in which physics in our usual spacetime is reformulated using a different geometric space---namely, the space of all light rays in spacetime.  In the space of all light rays, known as the “twistor space”, the equations for light are particularly simple. This simplification motivated Roger Penrose’s hope of unifying general relativity and quantum mechanics.

Twistor theory is now sixty years old, and the most striking impacts have been mathematical. In particular, twistor ideas have deeply influenced complex and differential geometry, integrable systems, and the study of hyperkähler spaces (built on the geometry of the quaternions).  My research program centers on describing the hyperkähler geometry of certain moduli spaces from physics. This hyperkähler geometry can be more simply-described on the associated twistor space—where intricate metric and symmetry data become holomorphic objects. I’ll give a flavor of the area through a few accessible examples from twistor theory. For example, the Penrose transform turns finding spherical harmonics—solutions of Laplace’s equation on the sphere (which you’ve likely seen before as functions describing the shapes of s,p,d,f electron orbitals in a chemistry class)—into a straightforward algebra question about picking polynomials of the right degree on the twistor side.