08:30 to 08:50 Registration 08:50 to 09:00 Welcome by John Toland, Director of the Institute 09:00 to 09:35 M Pollicott (University of Warwick)Geodesic flows: Mixing, zeta functions and resonances Historically important examples of chaotic'' dynamical systems are Anosov flows, in particular, and geodesic flows on negatively curved manifolds. In particular, they provide a concrete setting to explore a wealth of interesting topics: (i) mixing rates (which can be studied using zeta function and resonances); (ii) large deviations and fluctuation theorems (Gallavotti-Cohen theorem in non-equilibrium statistical mechanics); and (iii) escape rates (the rate at which mass escapes from an open system) and extremes. INI 1 09:35 to 10:10 Rigorous computation of invariant measures and Lyapunov exponents Co-author: Isaia Nisoli (Universitade Federal Rio de Janeiro) We will consider the problem of computation of invariant measures and other aspects related to the statistical behavior of the dynamics up to certified errors. In this way the output of a computation represent some rigorous quantitative estimation on the behavior of the dynamics under study, going towards more reliable tools for the simulation of dynamical models. We will show some general approach which can be applied in several cases of systems having some hyperbolic behavior, including maps with indifferent fixed points. Time permitting we will also consider a class piecewise hyperbolic maps related to the Lorenz attractor. INI 1 10:10 to 10:45 Paleo-climatic time series: statistics and dynamics Co-authors: Arnaud Debussche (ENS Cachan), Jan Gairing (HU Berlin), Claudia Hein (HU Berlin), Michael Högele (U Potsdam), Ilya Pavlyukevich (U Jena) Dynamical systems of the reaction-diffusion type with small noise have been instrumental to explain basic features of the dynamics of paleo-climate data. For instance, a spectral analysis of Greenland ice time series performed at the end of the 1990s representing average temperatures during the last ice age suggest an $\alpha-$stable noise component with an $\alpha\sim 1.75.$ We model the time series as a dynamical system perturbed by $\alpha$-stable noise, and develop an efficient testing method for the best fitting $\alpha$. The method is based on the observed $p$-variation of the residuals of the time series, and their asymptotic $\frac{\alpha}{p}$-stability established in local limit theorems.\par\smallskip Generalizing the solution of this model selection problem, we are led to a class of reaction-diffusion equations with additive $\alpha$-stable L\'evy noise, a stochastic perturbation of the Chafee-Infante equation. We study exit and transition between meta-stable states of their solutions. Due to the heavy-tail nature of an $\alpha$-stable noise component, the results differ strongly from the well known case of purely Gaussian perturbations. INI 1 10:45 to 11:10 Morning Coffee 11:10 to 11:45 T Kuna (University of Reading)Typical behaviour of extremes of chaotic dynamical systems for general observables In this talk we discuss the distribution of extreme events for a chaotic dynamical system for a general class of observables. More precisely, we link directly the distribution of events over threshold to the local geometrical structure on the surface of the attractor. It is shown how this can provide us with information about the local stable and unstable dimensions. Using Ruelle's response theory, we discuss the sensitivity of the parameters of the distribution under perturbations. This is a joint work with Vlaerio Lucarini, Davide Faranda and Jeroen Wouters. INI 1 11:45 to 12:20 Interplay between Mathematics and Physics Co-author: Tian Ma (Sichuan University) In this talk, we shall present three first principles and a few examples, demonstrating the symbiotic interplay between theoretical physics and advanced mathematics. We start with a general principle that dynamic transitions of all dissipative systems can be classified into three categories: continuous, catastrophic and random. We shall illustrate this principle with a few examples in both equilibrium and non-equilibrium phase transitions, including the metastable oscillation mechanism of the El Nino Southern Oscillation (ENSO) and the existence of 3rd-order transitions beyond the Andrews critical point. Then we present two basic principles: the principle of interaction dynamics (PID) and the principle of representation invariance (PRI), to study the nature's fundamental interactions/forces. Intuitively, PID takes the variation of the action functional under energy-momentum conservation constraint. PRI requires that physical laws be independent of representations of the gauge groups. These two principles give rise to a unified field model for four interactions, which can be naturally decoupled to study individual interactions. With PID, for example, we derive new gravitational field equations with a vector field $\Phi_\mu$, which can be considered as a spin-1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin-2 massless bosonic particle) and this spin-1 massless bosonic particle, leading to a unified theory for dark matter and dark energy. In addition, the PID offers a completely different and much simpler way of introducing Higgs fields. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:40 to 14:15 Quantifying uncertainty and improving reduced-order predictions of partially observed turbulent dynamical systems Co-author: A. J. Majda (Courant Institute, NYU) The issue of mitigating model error in reduced-order prediction of high-dimensional dynamics is particularly important when dealing with turbulent geophysical systems with rough energy spectra and intermittency near the resolution cut-off of the corresponding numerical models. I will discuss a new framework which allows for information-theoretic quantification of uncertainty and mitigation of model error in imperfect stochastic/statistical predictions of non-Gaussian, multi-scale dynamics. In particular, I will outline the utility of this framework in derivation of a sufficient condition for improving imperfect predictions via a popular but heuristic Multi Model Ensemble approach. Time permitting, the role and validity of 'fluctuation-dissipation' arguments for improving imperfect predictions of externally perturbed non-autonomous turbulent systems will also be addressed. INI 1 14:15 to 14:50 Extreme value theory for randomly perturbed systems: getting the local dimensions We present some new results for extreme values distributions in dynamical systems perturbed "via" random transformations and with observational noise. In both cases the linear scaling parameters of the Gumbel law will allow to get informations on the local behavior respectively of the stationary measure (random transformations), and of the invariant measure (observational noise). This collects work done with Aytac, Faranda, Freitas, Lucarini and Turchetti. INI 1 14:50 to 15:25 A new recurrences based technique for detecting robust extrema in long temperature records Co-author: Sandro Vaienti (University of Marseille) By using new techniques originally developed for the analysis of extreme values of dynamical systems, several long records of temperatures at different locations are analysed by showing that they have the same recurrence time statistics of a chaotic dynamical system perturbed with dynamical noise and by instrument errors. The technique provides a criterion to discriminate whether the recurrence of a certain temperature belongs to the natural climate variability or can be considered as a real extreme event with respect to a specific time scale fixed as parameter. The method gives a self-consistent estimation of the convergence. INI 1 15:25 to 15:50 Afternoon Tea 15:50 to 16:25 Extreme sea waves in the coastal zone INI 1 16:25 to 17:00 J Vollmer (Max-Planck-Institut fur Dynamics and Self-Organisation)Dew droplets and cloud droplets: droplet growth, size distributions, and corrections to scaling I present the results of comprehensive laboratory experiments and numerical studies addressing droplet growth and droplet size distributions in systems where droplets grow due to sustained supersaturation of their environment. Both, for droplets condensing on a substrate (like dew) and droplets entrained in an external flow (like in clouds), we observe remarkable shortcomings of classical scaling theories addressing these growth processes. The origins of the discrepancies are identified, and appropriate extensions of the theories are discussed. INI 1 17:00 to 18:00 Welcome Wine Reception
 09:00 to 09:35 Hydrodynamic turbulence as a problem in non-equilibrium statistical mechanics The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems which describe units of fluid of smaller and smaller spatial extent. These units are macroscopic but have few degrees of freedom, and can be studied by the methods of (microscopic) non-equilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Specifically, we obtain the formula $$\zeta_p={p\over3}-{1\over\ln\kappa}\ln\Gamma({p\over3}+1)$$ for the exponents of the structure functions ($\langle|\Delta_rv|^p\rangle\sim r^{\zeta_p}$). The meaning of the adjustable parameter $\kappa$ is that when an eddy of size $r$ has decayed to eddies of size $r/\kappa$ their energies have a thermal distribution. The above formula, with $(\ln\kappa)^{-1}=.32\pm.01$ is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture which can thus also be used in related problems. INI 1 09:35 to 10:10 GM Buttazzo (Università di Pisa)Optimal location problems with routing cost Co-authors: Serena Guarino (University of Pisa (Italy)), Fabrizio Oliviero (University of Pisa (Italy)) A model problem for the location of a given number $N$ of points in a given region $\Omega$ and with a given resources density $\rho(x)$ is considered. The main difference between the usual location problems and the present one is that in addition to the location cost an extra {\it routing cost} is considered, that takes into account the fact that the resources have to travel between the locations on a point-to-point basis. The limit problem as $N\to\infty$ is characterized and some applications to airfreight systems are shown. INI 1 10:10 to 10:45 Phase transitions and large deviations in geophysical fluid dynamics Geophysical turbulent flows (atmosphere and climate dynamics, the Earth core dynamics) often undergo very rapid transitions. Those abrupt transitions change drastically the nature of the flow and are of paramount importance, for instance in climate. By contrast with most theoretical models of phase transitions, for turbulent flows it is difficult to characterize clearly the attractors (they are not simple fixed points of a deterministic dynamics or statistical equilibrium states) and the trajectories that lead to transitions from one attractor to the others. I will review recent researches in this subject, including experimental and numerical studies of turbulent flows. Most of the talk will focus on theoretical works in the framework of the 2D stochastic quasi-geostrophic Navier-Stokes equations, the quasi-geostrophic equations, and the stochastic Vlasov equations. We will discuss predictions of phase transitions, validity of large deviation results of the Freidlin-Wentzell type, or more involved approaches when the Freidlin-Wentzell approach is not valid. The results involve several works that have been done in collaborations with J. Laurie, M. Mathur, C. Nardini, E. Simonnet, J. Sommeria, T. Tangarife, H. Touchette, and O. Zaboronski. INI 1 10:45 to 11:10 Morning Coffee 11:10 to 11:45 Turbulence transition in shear flows: coherent structures, edge states and all that Pipe flow, plane Couette flow and several other shear flows show a transition to turbulence for flow rates where the linear profile is still stable. The turbulent dynamics is transient, so that the transition is related to the formation of a chaotic saddle in the state space of the system. The saddle is supported by exact coherent states and their heteroclinic connections. I will summarize the common features that appear across all these shear flows, sketch the numerical techniques used to identify and track the relevant structures in the state space of the system and point out possible applications beyond fluid mechanics. INI 1 11:45 to 12:20 V Lucarini ([Universität Hamburg/University of Reading])Noise, Fluctuation, and Response in Geophysical Fluid Dynamics Response theory provides formidable methods for addressing many problems in statistical mechanics. Recently, it has been proposed as a gateway for various challenges in geophysical fluid dynamics, such as the provision of a rigorous conceptual framework for computing climate response to a variety of forcings and for deriving effective equations for coarse-grained variables, thus paving the way for constructing accurate parametrization of unresolved processes in numerical models. In this contribution, we first would like to present some new results showing how one can use response theory to compute the impact of adding stochastic forcing to deterministic chaotic systems. Then, we will discuss the applicability of the fluctuation-dissipation theorem in the context of non-equilibrium systems, focusing on the role played by the choice of observable. Finally, we will present some applications of response theory in geophysical fluid dynamical systems, ranging from low-order models such as the Lorenz 63 and Lorenz 96 models to General Circulation Models of the atmosphere. INI 1 12:20 to 12:55 Nonequilibrium statistical mechanics of climate variability: modelling issues and applications to data assimilation techniques Stochastic models and computational tools for the study of transitions between different metastable states (or regimes) in climate system are discussed using the barotropic quasi-geostrophic (QG) equation as a test case. Specifically, a stochastic partial differential equation (SPDE) is obtained by adding appropriate forcing and damping terms to the QG equation to make this equation dynamically consistent with the predictions of equilibrium statistical mechanics, while allowing to study nonequilibrium phenomena such as transitions between different regimes. In the small noise regime, the most likely states of the invariant measure for this SPDE coincide with the selective decay states and we establish conditions under which these states are not unique, implying the existence of different climate regimes. We also analyze the mechanism and rate of the dynamical transitions between these regimes by computing the most likely paths connecting them. Finally we will discuss how the se results can be used in the context of data assimilation procedure based on Kalman or ensemble filters to improve the efficiency of these methods in the presence of regime shifts. INI 1 12:55 to 13:30 Lunch at Wolfson Court