It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. We will prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we will exhibit various examples of Cayley graphs of finitely presented groups (e.g. PGL(5, Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgakopoulos. This is a joint work with Mikael de la Salle.

Quantum computers derive their computational power from the ability to manipulate superposition states of quantum registers. The generic quantum attack against a symmetric encryption scheme with key length n using Grover's algorithm has O(2^(n/2)) time complexity. For this kind of attack, an adversary only needs classical access to an encryption oracle. In this talk I discuss adversaries with quantum superposition access to encryption and decryption oracles. First I review and extend work by Kuwakado and Morii showing that a quantum computer with superposition access to an encryption oracle can break the Even-Mansour block cipher with key length n using only O(n) queries. Then, improving on recent work by Boneh and Zhandry, I discuss indistinguishability notions in chosen plaintext and chosen ciphertext attacks by a quantum adversary with superposition oracle access and give constructions that achieve these security notions.

Wave scattering problems arise in numerous applications in acoustics, electromagnetics and linear elasticity. In the boundary element method (BEM) one reformulates the scattering problem as an integral equation on the scatterer boundary, e.g. using Green’s identities, and then seeks an approximate solution of the boundary integral equation (BIE) from some finite-dimensional approximation space. The conventional choice is a space of piecewise polynomials; however, in the “high frequency” regime when the wavelength is small compared to the size of the scatterer, it is computationally expensive to resolve the highly oscillatory wave solution in this way. The hybrid numerical-asymptotic (HNA) approach aims to reduce the computational cost by enriching the BEM approximation space with oscillatory functions, carefully chosen to capture the high frequency asymptotic solution behaviour. To date, the HNA methodology has been implemented almost exclusively in a Galerkin variational framework. This has many attractive features, not least the possibility of proving rigorous convergence results, but has the disadvantage of requiring numerical evaluation of high dimensional oscillatory integrals. In this talk I will present the results of some investigations carried out with my MSc student Emile Parolin into collocation-based implementations, which involve lower-dimensional integrals, but appear harder to analyse in terms of convergence and stability.

Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution.  This result is based on work of Crawley-Bouvey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.