Why is the Higgs boson so light?
In a triumph of theoretical and experimental physics, the Higgs boson, first predicted to exist in 1964, was discovered at the Large Hadron Collider in July 2012. On the one hand, its mass of 125 GeV was comfortably within the range predicted by properties and masses of other Standard Model particles. On the other hand, it was 14-16 orders of magnitude too light, which is about as bad as estimating the mass of an 18-wheeler truck by the mass of the moon! The problem is that the Higgs boson is a scalar particle, whose mass is unstable to quantum corrections. The heaviest particles which could possibly exist, whether or not we've discovered them, will tend to drag the mass of the Higgs upwards toward them. Our ignorance about the highest-energy processes in fundamental physics, which take place at the Grand Unified scales of 10^16 GeV or the Planck scale of 10^18 GeV, prevent us from ruling out heavy particles at these scales. So the "natural" value for the Higgs boson mass should be 10^16-10^18 GeV, and the fact that its actual mass is so much lighter is a sign of a fine-tuning, comparable to finding two people of such precisely equal weights to balance a see-saw to within the radius of a proton. Either you have to be extremely lucky, or there's some deeper reason at work.
One elegant proposal for a deeper reason is supersymmetry, where the would-be quantum corrections to the Higgs mass are precisely canceled by other particles which contribute an equal and opposite correction. These new particles are called superpartners, and are a main target for both the LHC and dark matter experiments. But interestingly, the precise value of the Higgs mass (and the fact that we haven't discovered these superpartners yet) means that the cancellation can't be perfect: supersymmetry must be spontaneously broken in our universe.
My research focuses on supersymmetry from both the bottom up and the top down. From the bottom up, I'm interested in finding explicit theoretical realizations of the spectra of supersymmetric particles which are consistent with the measured Higgs mass and other nice properties of supersymmetry, like gauge coupling unification. With colleagues at MIT, I've worked on a model called auxiliary gauge mediation, where superpartners acquire mass as a secondary effect from interactions of new gauge groups disguised as global symmetries of the Standard Model. From the top down, spontaneously-broken supersymmetry implies that there must be a massless fermion, the goldstino, which gets absorbed by the gravitino (superpartner of the graviton) when supersymmetry is combined with general relativity. Assuming nothing else about the spectrum or properties of superpartners, Jesse Thaler, Dan Roberts, and I tried to determine if there were unambiguous signatures of supersymmetry during inflation. The jury is still out on this question, but along the way we constructed a new "constrained superfield" describing inflation which has found use in other, more formal aspects of supergravity and inflation.