Applied Mathematics Research Centre

The AMRC hosts a series of seminars on subjects in statistical physics and fluid dynamics.

**Finite-Size Corrections in Simulations of Optimization Problems and Spin Glass Ground-States***Wednesday May 21, 2014, 15:00 h, DH Seminar Room*

Except for the rare cases that exact solutions exist for complex disordered systems, much of our understanding relies on their simulation. These are conducted often at rather limited system sizes due to disorder averages and the sheer combinatorial effort. Then, anomalously strong finite-size effects commonly cloud concrete conclusion. To aid in the discovery of rigorous relations to control those effects, I present a survey of a trove of data obtained with the Extremal Optimization heuristic with an emphasis on finite size corrections. In particular, studies of combinatorial optimization problems and spin glass ground states will be discussed, in sparse or dense mean-field models and on lattices in up to 7 dimensions, using bimodal, Gaussian, or a family of Levy-distributed bond-weights.

**Aspects of Lanchester theory***Wednesday May 7, 2014, 15:00 h, DH Seminar Room*

TBA

**Robots under Neural Control - Learning and Memory in Artificial Agents**

Conventional robot control relies on methods from electrical engineering and control theory. This often leads to unwanted properties, because these controllers are stiff, inflexible, and cannot easily adapt (“learn”) to new situations. Here we will show how to implement modular neural control thereby creating increasingly complex behavior in an artificial agent. Different from conventional methods, neural control directly links signals from sensors via a set of network modules to the motors of the machine without explicit “rules”. The machine thereby develops fast responses and flexible behavior and can adapt by ways of simulated neuronal plasticity to new situations. Specifically we will demonstrate, how a small CPG, which operates initially in a chaotic domain can – via neural chaos control – be driven into repetitive patterns, which are being used to determine the walking gait of an 18DOF hexapod robot (Steingrube et al. Nature Physics, 2010). As a second, potentially interesting aspect, we will show how such machines can be endowed with a neural working memory, which allows them to react to sequences of events also in an appropriately delayed and sequential way.

**Lattice paths and their application to polymers, percolation and the ASEP**

The use of recurrence relations, transfer matrices anf the constant term method will be used to determine a range of lattice path generating functions. The simplest application is to the adsorption of polymers onto a surface. As the surface fugacity increases there comes a point where the polymer sticks to the surface. This appears as a singularity in the generating function. Similar singularities occur in compact percolation and the ASEP model. The unifying feature is that all the generating functions are for directed lattice paths near a surface with various weighting parameters.

**Economic theory at Coventry University**

TBA

**Entanglement and magnetic properties of spin-1/2 Ising-Heisenberg lattice models**

The results on magnetic and entanglement properties of frustrated spin-1/2 Ising-Heisenberg models on a triangulated Kagome lattice and a symmetrical diamond chain will be presented. Various phases of the systems, possessing qualitatively different quantum behavior (characterized by means of concurrence, i.e. an entanglement measure) will be discussed. Additionally, strong correlations between magnetic and entanglement features of the models will be demonstrated. Note that the above models can be used for the description of properties of real solid state materials, such as the azurite and copper based polymeric coordination compounds.

**Ground-state, magnetic and partition function zeros properties of the spin-1 Ising-Heisenberg model on a diamond chain**

TBA

**Dynamics near criticality**

Heuristically, one can give arguments why the fluctuations of classical models of statistical mechanics near criticality are typically expected to be described by nonlinear stochastic PDEs. Unfortunately, in most examples of interest, these equations seem to make no sense whatsoever due to the appearance of infinities or of terms that are simply ill-posed.

I will give an overview of a new theory of "regularity structures" that allows to treat such equations in a unified way, which in turn leads to a number of natural conjectures. One interesting byproduct of the theory is a new (and rigorous) interpretation of "renormalisation group techniques" in this context.

At the technical level, the main novel idea involves a complete rethinking of the notion of "Taylor expansion" at a point for a function or even a distribution. The resulting structure is useful for encoding "recipes" allowing to multiply distributions that could not normally be multiplied. This provides a robust analytical framework to encode renormalisation procedures.

**Large-deviations and quantum non-equilibrium***Thursday January 30, 2014, 14:00 h, DH Seminar Room*

TBA

**Inflation after the Planck satellite**

Inflation, a description of rapid expansion during the early universe, is the leading paradigm of the big bang. Even quantum fluctuations are inflated by the expansion and they may become the initial seeds of all perturbations. So inflation links the very smallest (quantum) and largest (galactic) scales. I will summarise our knowledge of inflation in light of the first major data release from the Planck satellite this year. This satellite has made the most accurate ever observations of the cosmic microwave background temperature (CMB), which is likely to remain the “state-of-the-art” for perhaps a decade. The inflationary paradigm has been subjected to its sharpest test yet and remains an excellent fit to the data. However certain classes of models are now either under pressure or ruled out. Particular attention is paid to the constraints on non-linear perturbations, which would give rise to non-Gaussian perturbations of temperature perturbations. In the second part of the talk I will focus on how the very small scale perturbations could be measured. They are too small to be seen in the CMB and currently largely unconstrained. Possibilities include searching for primordial black holes and gravitational waves.