1. A brief history of nonvanishing theorems. The nonvanishing of a Dirichlet series 2 a(n)n~\ or the existence of a pole, at a particular value of s often has applications to arithmetic. Euler gave the first example of this, showing that the infinitude of the primes follows from the pole of Ç(s) at s = 1. A deep refinement was given by Dirichlet, whose theorem on primes in an arithmetic progres...

<p style='text-indent:20px;'>We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension <inline-formula><tex-math id="M1">$ n $</tex-math></inline-formula>, assuming smooth varieties in id="M2">$ n-1 $</tex-math></inline-formula>. We also show that existence good minimal models non-uniruled klt id="M3">$ $</te...

In this paper we will address the question of how many nonvanishing, linearly independent tangent vector fields can exist on a sphere Sn−1 ⊆ R. By this we mean the following, a tangent vector field on Sn−1 = {x ∈ R : ‖x‖ = 1} is a map v : Sn−1 → R such that v(x) ⊥ x for all x ∈ Sn−1. However, by assumption v is nonvanishing, so we can normalize such that ‖v(x)‖ = 1 and we obtain a map v : Sn−1 ...

We discuss the nonvanishing of the family of central values L( 1 2 , f ⊗ χ), where f is a fixed automorphic form on GL(2) and χ varies through class group characters of an imaginary quadratic field K = Q( √ −D), as D varies; we prove results of the nature that at least D1/5000 such twists are nonvanishing. We also discuss the related question of the rank of a fixed elliptic curve E/Q over the H...