In 2000, the Clay Institute offered one million dollars 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.

It all starts with 1+1=2. The fact that 1+1=2 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 rope.

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