Hyperlinear profile and entanglement

An approximate representation of a finitely-presented group is an assignment of unitary matrices to the generators, such that the defining relations are close to the identity in the normalized Hilbert-Schmidt norm. A group is said to be hyperlinear if every non-trivial element can be bounded away from the identity in approximate representations of the group. Determining whether all groups are hyperlinear is a major open problem, as a non-hyperlinear group would provide a counterexample to the famous Connes embedding problem.

Computational Spectroscopy of Quantum Field Theories

Quantum field theories play an important role in many condensed matter systems for their description at low energies and long length scales. In 1+1 dimensional critical systems the energy spectrum and the spectrum of scaling dimensions are intimately related in the presence of conformal symmetry. In higher space-time dimensions this relation is more subtle and not well explored numerically. In this talk we motivate and review our recent effort to characterize 2+1 dimensional quantum field theories using computational techniques 2+targetting the energy spectrum on a spatial torus.

Decomposable Specht modules

I will give a brief survey of the study of decomposable Specht modules for the symmetric group and its Hecke algebra, which includes results of Murphy, Dodge and Fayers, and myself. I will then report on an ongoing project with Louise Sutton, in which we are studying decomposable Specht modules for the Hecke algebra of type $B$ indexed by `bihooks’.

The Quest for Solving Quantum Chromodynamics: the tensor network approach

The strong interaction of quarks and gluons is described theoretically within the framework of Quantum Chromodynamics (QCD). The most promising way to evaluate QCD for all energy ranges is to formulate the theory on a 4 dimensional Euclidean space-time grid, which allows for numerical simulations on state of the art supercomputers. We will review the status of lattice QCD calculations providing examples such as the hadron spectrum and the inner structure of nucleons. We will then point to problems that cannot be solved by conventional Monte Carlo simulation techniques, i.e.

Determining a local Hamiltonian from a single eigenstate

I'll ask whether the knowledge of a single eigenstate of a local lattice Hamiltonian is sufficient to uniquely determine the Hamiltonian. I’ll present evidence that the answer is yes for generic local Hamiltonians, given either the ground state or an excited state. In fact, knowing only the correlation functions of local observables with respect to the eigenstate appears generically sufficient to exactly recover both the eigenstate and the Hamiltonian, with efficient numerical algorithms.

Mapping the Universe with Euclid

In 2020 the European Space Agency (ESA) will launch the Euclid satellite mission. Euclid is an ESA medium class astronomy and astrophysics space mission, and will undertake a galaxy redshift survey over the redshift range 0.9 < z < 1.8, while simultaneously performing an imaging survey in both visible and near infrared bands. The complete survey will provide hundreds of thousands images and several tens of Petabytes of data.