Neural networks are undoubtedly successful in practical applications. However complete mathematical theory of why and when machine learning algorithms based on neural networks work has been elusive. Although various representation theorems ensures the existence of the ``perfect’’ parameters of the network, it has not been proved that these perfect parameters can be (efficiently) approximated by conventional algorithms, such as the stochastic gradient descent. This problem is well known, since the arising optimisation problem is non-convex. In this talk we show how the optimization problem becomes convex in the mean field limit for one-hidden layer networks and certain deep neural networks. Moreover we present optimality criteria for the distribution of the network parameters and show that the nonlinear Langevin dynamics converges to this optimal distribution. This is joint work with Kaitong Hu, Zhenjie Ren and Lukasz Szpruch. 

Berry's random wave conjecture is a heuristic that the eigenfunctions of a classically ergodic system ought to display Gaussian random behaviour, as though they were random waves, in the large eigenvalue limit. We discuss two manifestations of this conjecture for eigenfunctions of the Laplacian on the modular surface: Planck scale mass equidistribution, and an asymptotic for the fourth moment. We will highlight how the resolution of these two problems in this number-theoretic setting involves a delicate understanding of the behaviour of certain families of L-functions.

Quantum fields can sometimes have negative energy density.  In gravitational contexts, this threatens to permit both causality violations (such as traversable wormholes, warp drives, and time machines) and violations of the Second Law for black holes.  I will discuss the thermodynamic principles that rule out such pathological situations.  These principles have led us to an interesting lower bound on the energy flux, even for field theories in flat spacetime! This Quantum Null Energy Condition has now been proven for all relativistic field theories.  I will give an intuitive argument explaining why such ``quantum energy conditions'' ought to hold. 
 

  Our understanding of physical phenomena is intimately linked to the way we understand the relevant observables describing them. While a big deal of progress has been made for processes occurring in flat space-time, much less is known in cosmological settings. In this context, we have processes which happened in the past and which we can detect the remnants of at present time. Thus, the relevant observable is the late-time wavefunction of the universe. Questions such as "what properties they ought to satisfy in order to come from a consistent time evolution in cosmological space-times?", are still unanswered, and are compelling given that in these quantities time is effectively integrated out. In this talk I will report on some recent progress in this direction, aiming towards the idea of a formulation of cosmology "without time". Amazingly enough, a new mathematical structure, we called "cosmological polytope", which has its own first principle definition, encodes the singularity structure we ascribe to the perturbative wavefunction of the universe, and makes explicit its (surprising) relation to the flat-space S-matrix. I will stress how the cosmological polytopes allow us to: compute the wavefunction of the universe at arbitrary points and arbitrary loops (with novel representations for it); interpret the residues of its poles in terms of flat-space processes; provide a  general geometrical proof for the flat-space cutting rules; reconstruct the perturbative wavefunction from the knowledge of the flat-space S-matrix and a subset of symmetries enjoyed by the wavefunction.

The classical theory of integration concern integrals of differential forms over domains of integration. In geometric terms, this corresponds to a canonical pairing between de Rham cohomology and singular homology. For varieties defined over the reals, one can make use of complex conjugation to define a real-valued pairing between de Rham cohomology and its dual, de Rham homology. The corresponding theory of integration, that we call single-valued integration, pairs a differential form with a `dual differential form’. We will explain how single-valued periods are computed and give an application to superstring amplitudes in genus zero. This is joint work with Francis Brown.
 

We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue problems. Using information that is available or easy to estimate, our bounds improve the classical Davis-Kahan sin-theta theorem by a factor that can be arbitrarily large, and can give nontrivial information even when the sin-theta theorem suggests that a Ritz vector might have no accuracy at all. We also present extensions in three directions, deriving error bounds for invariant subspaces, singular vectors and subspaces computed by a (Petrov-Galerkin) projection SVD method, and eigenvectors of self-adjoint operators on a Hilbert space.