In this work, we establish new nonvanishing results for automorphic $L$-functions on $\mathrm{GL}\left(n,{\mathbf{A}}_Q\right)$. In particular, we show that, given a cuspidal automorphic form $\pi$ on $\mathrm{GL}\left(3,{\mathbf{A}}_Q\right)$ and an arbitrary point $s_0\in \mathbf{C}$, there exist infinitely many Dirichlet characters $\chi$ with prescribed ramification such that the twisted completed $L$-functions do not vanish at $s_0$: $\Lambda \left(s_0,\pi \otimes \chi \right)\ne 0$.

We define a Hopf algebra of motivic iterated integrals on the line and prove an explicit formula for the coproduct $\Delta$ in this Hopf algebra. We show that this formula encodes the group law of the automorphism group of a certain noncommutative variety. We relate the coproduct $\Delta$ to the coproduct in the Hopf algebra of decorated rooted plane trivalent trees, which is a plane decorated version of the one defined by Connes and Kreimer [CK]. As an application, we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. These formulas play a key role in the mysterious correspondence between the structure of the motivic fundamental group of ${\mathbb{P}}^1-\left(\left\{0,\infty \right\}\cup {\mu }_N\right)$, where ${\mu }_N$ is the group of all $N$th roots of unity, and modular varieties for ${\mathrm{GL}}_m$ (see [G6], [G7]). In Section 7 we discuss some general principles relating Feynman integrals and mixed motives. They are suggested by Section 4 and the Feynman integral approach for multiple polylogarithms on curves given in [G7]. The appendix contains background material.

We study limits of quasi-Fuchsian groups for which the bending measures on the convex hull boundary tend to zero, giving necessary and sufficient conditions for the limit group to exist and be Fuchsian. As an application, we complete the proof of a conjecture made in [24, Conjecture 6.5] that the closures of pleating varieties for quasi-Fuchsian groups meet Fuchsian space exactly in Kerckhoff's lines of minima of length functions. Doubling our examples gives rise to a large class of cone manifolds which degenerate to hyperbolic surfaces as the cone angles approach $2\pi$.

Given a birational self-map of a compact complex surface, it is useful to find an invariant measure that relates the dynamics of the map to its action on cohomology. Under a very weak hypothesis on the map, we show how to construct such a measure. The main point in the construction is to make sense of the wedge product of two positive, closed (1, 1)-currents. We are able to do this in our case because local potentials for each current have ``finite energy'' with respect to the other. Our methods also suffice to show that the resulting measure is mixing, does not charge curves, and has nonzero Lyapunov exponents.

We obtain $L^p$ eigenfunction bounds for the harmonic oscillator $H = -\Delta + x^2$$H=-\Delta +x^2$ in $\mathbb{R}^n$${ℝ}^n$ and for other related operators, improving earlier results of Thangavelu and of Karadzhov. We also construct suitable counterexamples that show that our estimates are sharp.