Cramer conjectured a random model for the distribution of the primes, which would suggest that, on the scale of the average prime gap, the primes can be modelled by a Poisson process. In particular, the set of limit points of normalized prime gaps would be the whole interval $[0,\infty)$. I will describe joint work with Banks and Freiberg which shows that at least 1/8 of the positive reals are in the set of limit points. 

Given a commutative ring A with ideal m, we consider the formal completion of A at m, and we ask when algebraic structures over the completion can be approximated by algebraic structures over the ring A itself. As we will see, Artin's approximation theorem tells us for which types of algebraic structures and which pairs (A,m) we can expect an affirmative answer. We will introduce some local notions from algebraic geometry, including formal and etale neighbourhoods. Then we will discuss some algebraic structures and rings arising in algebraic geometry and satisfying the conditions of the theorem, and show as a corollary how we can lift isomorphisms from formal neighbourhoods to etale neighbourhoods of varieties.

We will discuss the relation between perturbative gauge theory and
perturbative gravity, and look at how this relation extends to some exact
classical solutions. First, we will review the double copy prescription that
takes gauge theory amplitudes into gravity amplitudes, which has been
crucial to progress in perturbative studies of supergravity. Then, we will
see how the relation between the two theories can be made manifest when we
restrict to the self-dual sector, in four dimensions. A key role is played
by a kinematic algebraic structure mirroring the colour structure, which can
be extended from the self-dual sector to the full theory, in any number of
dimensions. Finally, we will see how these ideas can be applied also to some
exact classical solutions, namely black holes and plane waves. This leads to
a relation of the type Schwarzschild as (Coulomb charge)^2.

Dynamics of small particles in fluids have fascinated scientists for centuries. Phenomena such as Brownian motion, sedimentation, and electrophoresis continue to inspire cutting-edge research and innovations. The fluid in which the particles move is typically isotropic, such as water or a polymer solution. Recently, we started to explore what would happen if particles are placed in an anisotropic fluid: a liquid crystal. The study reveals that the liquid crystal changes dramatically both the statics and dynamics, leading to levitation of the particles, their anomalous Brownian motion and new mechanisms of electrokinetics. The new phenomena are rooted in anisotropy of the liquid crystal properties, such as different electric conductivity in the directions parallel and perpendicular to the average molecular orientation.

I will describe the computation of the supersymmetric Renyi entropy across an entangling 3-sphere for five-dimensional superconformal field theories. For a class of USp(2N) gauge theories I’ll also construct a holographic dual 1/2 BPS black hole solution of Euclidean Romans F(4) supergravity. The large N limit of the gauge theory results will be shown to agree perfectly with the supergravity computations.