We define an analogue of the Casimir element for a graded affine Hecke algebra $ \mathbb{H} $, and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology H^D (X) of an $ \mathbb{H} $-module X, and show that H^D (X) carries a representation of a canonical double cover of the Weyl group $ \widetilde{W} $. Our main result shows that the $ \widetilde{W} $-structure on the Dirac cohomology of an irreducible $ \mathbb{H} $-module X determines the central character of X in a precise way. This can be interpreted as p-adic analogue of a conjecture of Vogan for Harish-Chandra modules. We also apply our results to the study of unitary representations of $ \mathbb{H} $.

We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov–Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop–Gromov volume comparison theorem and Perelman’s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these.

Moreover, we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the Green function that we prove. This is parallel to the fact that Perelman’s monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li–Yau.

In [CM4] one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross-sections of Einstein manifolds.

Finally, there are obvious parallelisms between our monotonicity and the positive mass theorem of Schoen–Yau and Witten.

The tip multifractal spectrum of a 2-dimensional curve is one way to describe the behavior of the uniformizing conformal map of the complement near the tip. We give the tip multifractal spectrum for a Schramm–Loewner evolution (SLE) curve, we prove that the spectrum is valid with probability 1, and we give applications to the scaling of harmonic measure at the tip.

We give a complete proof of the Bers–Sullivan–Thurston density conjecture. In the light of the ending lamination theorem, it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds. The ending invariants are either Riemann surfaces or filling laminations supporting Masur domain measured laminations and satisfying some mild additional conditions. With any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence. We derive the necessary compactness theorem combining the Rips machine with non-existence results for certain small actions on real trees of free products of surface groups and free groups. We prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface. In order to be able to apply the ending lamination theorem, we have to prove that this algebraic limit has the desired topological type and that these non-realized laminations are ending laminations. That this is the case is the main novel technical result of this paper. Loosely speaking, we show that a filling Masur domain lamination which is not realized is an ending lamination.