We prove the famous Faber intersection number conjecture and other more general results by using a recursion formula of n-point functions for intersection numbers on moduli spaces of curves. We also present some vanishing properties of Gromov-Witten invariants.

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We study the solvability problem for the multidimensional Riccati equation −∇$u$=|∇$u$|^{q}+ω, where$q$>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation −Δ$u$−ω$u$=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions on$R$^{n}in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type$−Lu=f(x, u, ∇u)$+ω where, and$L$is a uniformly elliptic operator.

Let$d$be a positive integer, $A={\mathbb{C}} [t_{1}^{\pm1},\ldots ,t_{d}^{\pm1}]$ be the Laurent polynomial algebra, and $W=\operatorname{Der} (A)$ be the derivation Lie algebra of$A$. Then we have the semidirect product Lie algebra$W$⋉$A$which we call the extended Witt algebra of rank$d$. In this paper, we classify all irreducible Harish-Chandra modules over$W$⋉$A$with nontrivial action of$A$.

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