Random Geometry and Yang-Mills Gauge Theory

In 2000, the Clay Institute offered $1 million for a mathematical construction of 4D Yang–Mills gauge theory. That problem remains unsolved, but there has been spectacular progress in recent years on many related 2D and 4D problems.

In this Presidential Lecture, Scott Sheffield will survey these developments. It all begins with 1+1=2, which implies that two non-parallel lines in the plane (co-dimension 1) meet at a point (co-dimension 2). Less trivially, any two paths through a square (one top to bottom, one left to right) intersect somewhere.

Similarly, 2+2=4 implies that two fully non-parallel 2D planes in 4D meet at a point (interpret one dimension as time and imagine moving lines in 3D colliding like light sabers) and that knotted loops in 3D cannot be disentangled without tearing the rope.

Further implications include the self-duality of 1-forms (in 2D) and 2-forms (in 4D), the conformal invariance of certain Gaussian fields, and the self-duality of cellular spanning trees, along with many deep and exciting results about random curves and surfaces, spin systems and connections. But how will this help with our remaining open problems?