A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory (GIT) quotients come with a natural ample line bundle and hence often a natural projective embedding. This question translates to determining the equations of the moduli space under this embedding. This article deals with one of the most classical quotients, the space of ordered points on the projective line. We show that under any weighting of the points, this quotient is cut out (scheme-theoretically) by a particularly simple set of quadric relations, with the single exception of the Segre cubic threefold, the space of six points with equal weight. We also show that the ideal of relations is generated in degree at most four, and we give an explicit description of the generators. If all the weights are even (e.g., in the case of equal weight for odd $n$), then we show that the ideal of relations is generated by quadrics

In this article, we show that if $\mathbb{G}$ is a simply connected Chevalley group of either classical type of rank bigger than $1$ or type $E_6$ and if $q>9$ is a power of a prime number $p>5$, then $G=\mathbb{G}({\mathbb{F}}_q((t^{-1})))$, up to an automorphism, has a unique lattice of minimum covolume, which is $\mathbb{G}({\mathbb{F}}_q[t])$

We consider continuous $\mathrm{SL}(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism that fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle that is not uniformly hyperbolic can be approximated by one that is conjugate to an $\mathrm{SO}(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes, and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schrödinger operator is a Cantor set

Convexity properties of Weil-Petersson (WP) geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson–Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considerations are combined to establish convexity along Weil-Petersson geodesics of the functions, the distance between horocycles for a hyperbolic metric

We prove a general result on equality of the weak limits of the zero counting measure, $d{\nu }_n$, of orthogonal polynomials (defined by a measure $d\mu$) and $(1/n)K_n(x,x)d\mu (x)$. By combining this with the asymptotic upper bounds of Máté and Nevai [16] and Totik [33] on $n{\lambda }_n(x)$, we prove some general results on ${\int }_I(1/n)K_n(x,x)d{\mu }_s\to 0$ for the singular part of $d\mu$ and ${\int }_I|{\rho }_E(x)-(w(x)/n)K_n(x,x)|\mathrm{dx}\to 0$, where ${\rho }_E$ is the density of the equilibrium measure and $w(x)$ the density of $d\mu$

Consider the three-dimensional Anderson model with a zero mean and bounded independent, identically distributed random potential. Let $\lambda$ be the coupling constant measuring the strength of the disorder, and let $\sigma (E)$ be the self-energy of the model at energy $E$. For any $\epsilon >0$ and sufficiently small $\lambda$, we derive almost-sure localization in the band $E\le -\sigma (0)-{\lambda }^{4-\epsilon }$. In this energy region, we show that the typical correlation length ${\xi }_E$ behaves roughly as $O((|E|-\sigma (E){)}^{-1/2})$, completing the argument outlined in the preprint of T. Spencer [18]