Complex oxides are fascinating systems which host a vast array of unique phenomena, such as high-temperature (and unconventional) superconductivity, "colossal" magnetoresistance, all forms of magnetism and ferroelectricity, as well as (quantum) phase transitions and couplings between these states.
Developing new materials via a better understanding of the mechanisms underlying macroscopic properties has been the Graal of materials science. The most prominent effort in Materials Science has been toward a better control of the microstructure, toward smaller and smaller scales, and in recent years, toward nanomaterials.
Electron interference in the solid enables to determine the electron coherence time, the phase electron gains during transport, the statistics of quasi-particles in strongly interacting systems, and the processes of dephasing.
The advent of means to prepare well-controlled nanoscale building blocks has opened up many new opportunities to understand difficult problems which lie at the core of materials science. As an example, nanometer-size inorganic nanocrystals can be transformed from one state to another with remarkably simplified kinetics compared to extended or bulk solids.
Nanowires are of both fundamental and technological interest. They represent the critical components in the potential nanoscale electronic and photonic device applications.
Beginning with a seminal paper of Diaconis (1988), the aim of so-called "probabilistic numerics" is to compute probabilistic solutions to deterministic problems arising in numerical analysis by casting them as statistical inference problems. For example, numerical integration of a deterministic function can be seen as the integration of an unknown/random function, with evaluations of the integrand at the integration nodes proving partial information about the integrand. Advantages offered by this viewpoint include: access to the Bayesian representation of prior and posterior uncertainties; better propagation of uncertainty through hierarchical systems than simple worst-case error bounds; and appropriate accounting for numerical truncation and round-off error in inverse problems, so that the replicability of deterministic simulations is not confused with their accuracy, thereby yielding an inappropriately concentrated Bayesian posterior. This talk will describe recent work on probabilistic numerical solvers for ordinary and partial differential equations, including their theoretical construction, convergence rates, and applications to forward and inverse problems. Joint work with Andrew Stuart (Warwick).
Organizers: Philipp Hennig