School of Mathematics

The theorem of the title is that if the L-function L(E,s) of an elliptic curve E over the rationals vanishes to order r=0 or 1 at s=1 then the rank of the group of rational rational points of E equals r and the Tate-Shafarevich group of E is finite...

Resonances are complex analogs of eigenvalues for Laplacians on noncompact manifolds, arising in long time resonance expansions of linear waves. We prove a Weyl type asymptotic formula for the number of resonances in a strip, provided that the set...

We give an arithmetic version of the recent proof of the improved triangle removal lemma by Fox [Fox11], for the group $F_2^n$.
A triangle in $F_2^n$ is a tuple (x,y,z) such that $x+y+z = 0$. The triangle removal lemma for $F_2^n$ states that for...

A trusted source of independent and uniform random bits is a basic resource in many computational tasks, such as cryptography, game theoretic protocols, algorithms and physical simulations. Implementing such a source presents an immediate challenge...

We review the well known microscopic correspondence between random zeros of the Riemann zeta-function and the eigenvalues of random matrices, and discuss evidence that this correspondence
extends to larger mesoscopic collections of zeros or...

Thanks to the work of Katz and Sarnak on L-functions over function fields, we know that the Frobenius classes associated to L-functions of hyperelliptic curves over a finite field with $q$ elements, $F_{q}$, becomes equidistributed in the unitary...

A number is said to be $y$-smooth if all of its prime factors are less than $y$. Such numbers appear in many places throughout analytic and combinatorial number theory, and much work has been done to investigate their distribution.
I will try to...