We present a structure theorem for the subsurface projections of train-track splitting sequences. For the proof we introduce induced tracks, efficient position, and wide curves. As a consequence of the structure theorem, we prove that train-track sliding and splitting sequences give quasi-geodesics in the train-track graph; this generalizes a result of Hamenstädt.

Following the work of Kashiwara and Rouquier and of Gan and Ginzburg, we define a family of exact functors from category $\mathcal{O}$ for the rational Cherednik algebra in type $A$ to representations of certain colored braid groups and we calculate the dimensions of the representations thus obtained from standard modules. To show that our constructions make sense in a more general context, we also briefly study the case of the rational Cherednik algebra corresponding to complex reflection group $\mathbb{Z}/l\mathbb{Z}$.

The Hall algebra $U_E^{+}$ of the category of coherent sheaves on an elliptic curve $E$ defined over a finite field has been explicitly described and shown to be a 2-parameter deformation of the ring of diagonal invariants $R^{+}=C[x_1^{\pm 1},\dots ,y_1,\dots {]}^{S_{\infty }}$ (in infinitely many variables). We study a geometric version of this Hall algebra, by considering a convolution algebra of perverse sheaves on the moduli spaces of coherent sheaves on $E$. This allows us to define a canonical basis $B$ of $U_E^{+}$ in terms of intersection cohomology complexes. We also give a characterization of this basis in terms of an involution, a lattice, and a certain PBW-type basis.

For a given $\ell$-adic sheaf $\mathcal{F}$ on a commutative algebraic group $G$ over a finite field $k$ and an integer $r\ge 1$, we define the $r$th local norm $L$-function of $\mathcal{F}$ at a point $t\in G\left(k\right)$ and prove its rationality. This function gives information on the sum of the local Frobenius traces of $\mathcal{F}$ over the points of $G\left(k_r\right)$ (where $k_r$ is the extension of degree $r$ of $k$) with norm $t$. For $G$ the $1$-dimensional affine line or the torus, these sums can in turn be used to estimate the number of rational points on curves or the absolute value of exponential sums which are invariant under a large group of translations or homotheties.

The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext-groups between the two ideal sheaves in question, twisted by a line bundle. We express the Chern classes of these virtual bundles in terms of Nakajima operators.