We establish formulae for the Iwasawa invariants of Mazur-Tate elements of cuspidal eigenforms, generalizing known results in weight $2$. Our first theorem deals with forms of “medium” weight, and our second deals with forms of small slope. We give examples illustrating the strange behavior which can occur in the high-weight, high-slope case.

We show that the space of Kleinian punctured torus groups is not locally connected.

We show that a random walk on the mapping class group of an orientable surface gives rise to a pseudo-Anosov element with asymptotic probability one. Our methods apply to many subgroups of the mapping class group, including the Torelli group.

This paper introduces a class of smooth projective varieties that generalize and share many properties with partial flag varieties of type $A$. The quiver flag variety $M_{\vartheta }(Q,\underline{r})$ of a finite acyclic quiver $Q$ (with a unique source) and a dimension vector $\underline{r}$ is a fine moduli space of stable representations of $Q$. Quiver flag varieties are Mori dream spaces, they are obtained via a tower of Grassmann bundles, and their bounded derived category of coherent sheaves is generated by a tilting bundle. We define the multigraded linear series of a weakly exceptional sequence of locally free sheaves $\underline{E}=(O_X,E_1,\dots ,E_{\rho })$ on a projective scheme $X$ to be the quiver flag variety $\left|\underline{E}\right|:=M_{\vartheta }(Q,\underline{r})$ of a pair $(Q,\underline{r})$ encoded by $\underline{E}$. When each $E_i$ is globally generated, we obtain a morphism ${\varphi }_{\left|\underline{E}\right|}:X\to \left|\underline{E}\right|$, realizing each $E_i$ as the pullback of a tautological bundle. As an application, we introduce the multigraded Plücker embedding of a quiver flag variety.

We show that (equivariant) $K$-theoretic $3$-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the (equivariant) $K$-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through three general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum $K$-theory ring of a Grassmannian, which determine the multiplication in this ring. We also compute the dual Schubert basis for this ring and show that its structure constants satisfy $S_3$-symmetry. Our formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.