We consider the quintic two-dimensional focusing nonlinear Schrödinger equation ${\mathrm{iu}}_t=-\Delta u-\left|u\right|{}^{4}u$ which is $L^2$-supercritical. Even though the existence of finite-time blow-up solutions in the energy space $H^1$ is known, very little is understood concerning the singularity formation. Numerics suggest the existence of a stable blow-up dynamic corresponding to a self-similar blowup at one point in space. We prove the existence of a different type of dynamic and exhibit an open set among the $H^1$-radial distributions of initial data for which the corresponding solution blows up in finite time on a sphere. This is the first result of an explicit description of a blow-up dynamic in the $L^2$-supercritical setting

We consider Kapranov's Chow quotient compactification of the moduli space of ordered $n$-tuples of hyperplanes in $P^{r-1}$ in linear general position. For $r=2$, this is canonically identified with the Grothendieck-Knudsen compactification of $M_{0,n}$ which has, among others, the following nice properties:

(1) modular meaning: stable pointed rational curves;

(2) canonical description of limits of one-parameter degenerations;

(3) natural Mori theoretic meaning: log-canonical compactification.

We generalize (1) and (2) to all $(r,n)$, but we show that (3), which we view as the deepest, fails except possibly in the cases $(2,n)$, $(3,6)$, $(3,7)$, $(3,8)$, where we conjecture that it holds

We determine which automorphic representations of the discrete spectrum of${\mathrm{GL}}_{2n}$ are distinguished by the symplectic group. This concludes a project initiated by Jacquet and Rallis [JR2]

We prove the existence of small amplitude, ($2\pi /\omega$)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency $\omega$ belonging to a Cantor-like set of asymptotically full measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser implicit function theorem. In spite of the complete resonance of the equation, we show that we can still reduce the problem to a finite-dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows us to deal also with nonlinearities that are not odd and with finite spatial regularity