For a Weyl group $W$, we investigate simple closed formulas (valid on elliptic conjugacy classes) for the character of the representation of $W$ in the homology of a Springer fiber. We also give a formula (valid again on elliptic conjugacy classes) of the $W$-character of an irreducible discrete series representation with real central character of a graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin double cover of $W$ and the Dirac operator for graded affine Hecke algebras play key roles.

We study syzygies of the Segre embedding of $\mathbf{P}\left(V_1\right)\times \cdots \times \mathbf{P}\left(V_n\right)$, and prove two finiteness results. First, for fixed $p$ but varying $n$ and $V_i$, there is a finite list of master $p$-syzygies from which all other $p$-syzygies can be derived by simple substitutions. Second, we define a power series $f_p$ with coefficients in something like the Schur algebra, which contains essentially all the information of $p$-syzygies of Segre embeddings (for all $n$ and $V_i$), and show that it is a rational function. The list of master $p$-syzygies and the numerator and denominator of $f_p$ can be computed algorithmically (in theory). The central observation of this paper is that by considering all Segre embeddings at once (i.e., letting $n$ and the $V_i$ vary) certain structure on the space of $p$-syzygies emerges. We formalize this structure in the concept of a $\Delta$-module. Many of our results on syzygies are specializations of general results on $\Delta$-modules that we establish. Our theory also applies to certain other families of varieties, such as tangent and secant varieties of Segre embeddings.

In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of ${\mathbb{A}}^2$. We show that it is isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of ${\mathrm{GL}}_{\infty }$. We explain this coincidence via the geometric Langlands correspondence for elliptic curves, by relating it also to the convolution algebra in the equivariant K-theory of the commuting variety. We also obtain a few other related results (action of the elliptic Hall algebra on the K-theory of the moduli space of framed torsion free sheaves over ${\mathbb{P}}^2$, virtual fundamental classes, shuffle algebras, …).

We give a criterion under which a solution $g\left(t\right)$ of the Kähler–Ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. As $t$ tends to the singular time $T$ from each direction, we prove the convergence of $g\left(t\right)$ in the sense of Gromov–Hausdorff and smooth convergence away from the exceptional divisors. We call this behavior for the Kähler–Ricci flow a canonical surgical contraction. In particular, our results show that the Kähler–Ricci flow on a projective algebraic surface will perform a sequence of canonical surgical contractions until, in finite time, either the minimal model is obtained, or the volume of the manifold tends to zero.

For $1\le p\le 2$, we prove sharp weak-type $(p,p)$-estimates for the Beurling–Ahlfors operator acting on the radial function subspaces of $L^p\left(\mathbb{C}\right)$. A similar sharp $L^p$-result is proved for $1<p\le 2$. The results are derived from martingale inequalities which are of independent interest.