# European Institute for Statistics, Probability, Stochastic Operations Research and their Applications

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April 18-22, 2016

METASTABILITY

in statistical mechanics and stochastic processes

 SUMMARY REGISTRATION SPEAKERS PROGRAMME ABSTRACTS

SUMMARY

The mathematical study of the phenomenon of metastability has been a standing issue since the foundation of Statistical Mechanics. The development of a rigorous mathematical theory, however, started mostly in the 70’s with the contribution of researchers of at least three different areas: rigorous statistical mechanics, reliability theory and dynamical systems. At present, the subject has achieved a sizeable degree of mathematical maturity, but researchers face several pressing issues. On the one hand, the diversity of scenarios leading to the phenomenon has resulted in a multitude of different approaches with not clearly
determined degree of equivalence. An effort is needed to contrast these different approaches and determine comparative advantages and levels of overlapping of their ranges of applicability. On the other hand, progress in recent years has brought solid mathematical foundations to more challenging setups, such as non-reversible evolutions, dynamics in continuous space and processes in graphs. These advances, however, remain by and large known only by a small number of specialists who deserve an opportunity to expose them to larger audiences.
The workshop is an attempt to address all these issues.

## ORGANISERS

 Roberto Fernandez Utrecht University Francesca Nardi TU Eindhoven Julien Sohier University of Créteil

INVITED SPEAKERS

 F. Barret University of Nanterre M. Baudel University of Orléans N. Berglund University of Orleans A. Bianchi University of Padova H. V. Can Aix-Marseille University S. Dommers Ruhr University Bochum S. Dutercq University of Orleans A. Gaudillière University of Marseille F. den Hollander University of Leiden S. Jansen University of Bochum O. Jovanovski University of Leiden R. Kotecký University of Warwick/Charles University, Prague C. Landim IMPA C. Maes University of Leuven P. Müller Bonn University F.R. Nardi TU Eindhoven E. Pulvirenti University of Leiden F. Redig TU Delft B. Schapira University of Marseille S. Shlosman University of Marseille B. Scoppola University Roma La Sapienza E. Scoppola University Roma Tre I. Seo Courant Institute S. Shlosman CPT - Luminy M. Slowik TU Berlin C. Spitoni University of Utrecht S. Taati University of Leiden D. Valesin University of Groningen A. van Enter University of Groningen A. Zocca CWI

PROGRAMME

MONDAY April 18

 09.00 - 09.25 Registration 09.25 - 09.30 Welcome Onno Boxma 09.30 - 10.20 Elisabetta Scoppola Metastability: old problem and new results 10.25 - 11.15 Francesca R. Nardi Metastability for General Dynamics with Rare Transitions: Escape Time and Critical Configurations 11.20 - 11.40 Coffee/tea break 11.40 - 12.30 Alex Gaudillière 12.30 - 15.00 Lunch 15.00 - 15.50 Martin Slowik Metastability in stochastic dynamics: Low-lying eigenvalues of the generator via two-scale decomposition 15.55 - 16.10 Coffee/tea break 16.10 - 17.00 Cristian Spitoni Sum of exit times in a series of metastable states 17.05 - 17.55 Patrick Muller Hydrodynamic limits, propagation of chaos and large deviations in local mean-field models with unbounded spins

TUESDAY April 19 - programme related to NETWORKS

 09.30 - 10.20 Frank den Hollander Metastability on networks 10.25 - 11.15 Sander Dommers Metastability of the Ising model on random regular graphs at zero temperature 11.20 - 11.40 Coffee/tea break 11.40 - 12.30 Oliver Jovanovski Metastability for Glauber dynamics on random graphs 12.30 - 15.00 Lunch 15.00 - 15.50 Senya Schlosman Metastable behavior of communication networks with moving nodes 15.55 - 16.10 Coffee/tea break 16.10 - 17.00 Alessandro Zocca 17.05 - 17.55 Siamak Taati Metastability of the hard-core process on bipartite graphs 18.30 - Conference dinner

WEDNESDAY April 20

 09.30 - 10.20 Claudio Landim Markov chains model reduction 10.25 - 11.15 Insuk Seo Metastability and Eyring-Kramers Formula for Non-reversible Markov Processes 11.20 - 11.40 Coffee/tea break 11.40 - 12.30 Aernout van Enter Metastability, from Ideal to Real to Super 12.30 - 14.00 Lunch 14.00 - 14.50 Nils Berglund Metastability of stochastic Allen-Cahn PDEs 14.55 - 15.45 Sebastien Dutercq Interface dynamics of a metastable mass-conserving diffusion 15.50 - 16.10 Coffee/tea break 16.10 - 17.00 Manon Baudel Spectral theory for random Poincare maps 17.05 - 17.55 Frank Redig Splitting and exchange models with SU(1,1) symmetry

THURSDAY April 21

 09.30 - 10.20 Roman Kotecký Metastability for the Widom-Rowlinson model 10.25 - 11.15 Elena Pulvirenti Metastability for Widom-Rowlinson model: II 11.20 - 11.40 Coffee/tea break 11.40 - 12.30 Sabine Jansen Low-temperature Lennard-Jones chains 12.30 - 14.00 Lunch 14.00 - 14.50 Christian Maes Driving-induced stability with long-range effects 14.55 - 15.45 Bruno Shapira A metastability result for the contact process on finite graphs 15.50 - 16.10 Coffee/tea break 16.10 - 17.00 Daniel Valesin 17.05 - 17.55 Hao Van Can t.b.a.

FRIDAY April 22

 09.00 - 09.50 Alessandra Bianchi Limiting dynamics of the condensate in the reversible inclusion process on a finite set 09.55 - 10.45 10.50 - 11.10 Coffee/tea break 11.10 - 12.00 Florent Barret Transition times for a class of parabolic SPDEs in one dimension 12.00 - 12.50 Benedetto Scoppola Metastability for irreversible PCA dynamics 12.50 - 14.00 Lunch

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ABSTRACTS

Florent Barret

Transition times for a class of parabolic SPDEs in one dimension

We consider a class of parabolic semi-linear stochastic partial differential equations driven by space-time white noise on a compact space interval. We obtain precise asymptotics of the transition times between metastable states. A version of the so-called Eyring-Kramers formula is proven in an infinite dimensional setting. The proof is based on a spatial finite difference discretization of the stochastic partial differential equation. The expected transition time is computed for the finite dimensional approximation and controlled uniformly in the dimension.

Manon Baudel

Spectral theory for random Poincare maps

We consider an ordinary differential equation with N stable periodic orbits perturbed by weak noise, for which we want to quantify the rare transitions between periodic orbits. The behaviour of the stochastic system is analysed through the spectrum of the kernel of a Markov process describing successive returns to a Poincare section. Because of the N stable periodic orbits, N eigenvalues of the Markov process are expected to be close to 1.
Using Laplace transforms of first-hitting times and a Dirichlet boundary value problem, we show that the eigenvalue problem on the Poincare section can be reduced to an eigenvalue problem on a union of small balls surrounding the stable orbits. In this way we obtain a homogeneous integral equation, whose eigenvalues are the roots of a Fredholm determinant and are directly linked to the eigenvalues of our Markov process.
Using perturbative arguments and Harnack inequalities, we approximate the Markov process by an N-state Markov chain. Under a metastable hierarchy assumption, we can prove that the N eigenvalues of the discrete-space Markov chain are close to 1.
(jointwork in progress with Nils Berglund)

Nils Berglund

Metastability of stochastic Allen-Cahn PDEs

Consider an Allen-Cahn SPDE on the d-dimensional torus, for d=1 or 2, driven by weak space-time white noise. We provide Eyring-Kramers-type asymptotics for the mean transition time between stable stationary solutions, going beyond large-deviation (Arrhenius) estimates. In the case d=1, such asymptotics were obtained independently by the speaker and Barbara Gentz (Bielefeld), using a sprectral Galerkin approximation, and by Florent Barret, using finite-difference discretizations. The mean transition time is known for all finite domain sizes, and contains a prefactor expressible in terms of ratios of spectral determinants. In the case d=2, the SPDE is only well-defined after a suitable renormalization is carried out. I will report on joint work with Hendrik Weber (Warwick) and Giacomo di Gesu (CERMICS), yielding an expression for the prefactor when the domain is sufficiently small.

Alessandra Bianchi

Limiting dynamics of the condensate in the reversible inclusion process on a finite set

The inclusion process is a stochastic lattice gas where particles perform random walks subjected to mutual attraction, thus providing the natural bosonic counterpart of the well-studied exclusion process.
Due to attractive interaction between the particles, the inclusion process can exhibit a condensation transition where a finite fraction of all particles concentrates on a single site. In this talk we characterize the dynamics of the condensate for the reversible inclusion process on a finite set S, in the limit of total number of particles going to infinity.
By potential theoretic techniques, we determine the time scales associated to the transitions of the condensate from one site to another, and we show that the limiting dynamics of the condensate is a suitable continuous time random walk on S.
(joint work with S. Dommers and C.
Giardinà)

Sander Dommers

Metastability of the Ising model on random regular graphs at zero temperature

We study the metastability of the ferromagnetic Ising model on a random $r$-regular graph in the zero temperature limit. We prove that in the presence of a small positive external field the time that it takes to go from the all minus state to the all plus state behaves like $\exp(\beta n (r/2+O(\sqrt{r})))$ when the inverse temperature $\beta\rightarrow\infty$ and the number of vertices $n$ is large enough but fixed. The proof is based on the so-called pathwise approach and bounds on the isoperimetric number of random regular graphs.

Sebastien Dutercq

Interface dynamics of a metastable mass-conserving diffusion

We will consider a diffusion process defined by the stochastic differential equation $dx_t=-\nabla V_\gamma(x_t)dt+\sqrt(2\epsilon)dW_t$
where $V_\gamma$ is a potential with a conservation law and invariant under a group of symmetries. First we will describe the metastable states of the system, and then we will define a hierarchy on these metastable states.
We will see how we can interpret the dynamics of this system in terms of the motion of its interfaces, and give sharp results on expected first-hitting times and its spectral gap.

Frank den Hollander

Metastability on networks

This talk presents an overview on how potential theory for reversible Markov processes can be used to analyse the metastable behaviour of Metropolis dynamics on random graphs at low temperature. The analysis starts from a number of general hypotheses about the energy landscape and leads to three metastability theorems: (1) the average crossover time from the metastable state to the stable state satisfies the classical Arrhenius law, with an exponent and a prefactor that are controlled by the energy and the entropy of the critical droplet; (2) the critical droplet is the gate for the metastable crossover; (3) the crossover time divided by its average is exponentially distributed.
The three theorems are model-independent and amplify the universal nature of metastability (in the low-temperature setting considered here). However, they involve a number of key geometric quantities that are model-dependent, the identification of which is typically hard. Yet, these quantities capture the underlying richness of the metastability phenomenon.
The talk is a prelude to the talks by Oliver Jovanovski and Siamak Taati,
who will discuss specific models.

Sabine Jansen

Low-temperature Lennard-Jones chains

We investigate the low-temperature behavior of a chain of atoms with Lennard-Jones type interaction. We prove a path large deviations principle for the infinite chain and analyze the principal eigenvector and spectral gap of the transfer operator. There are two interesting regimes: at positive pressure, the spectral gap stays bounded away from zero and correlations decay exponentially fast, uniformly in the temperature. When the pressure goes to zero sufficiently fast, the chain breaks and the spectral gap is exponentially small; this matches known approaches that model fracture as a double-well problem with one well at infinity.

Oliver Jovanovski

Metastability for Glauber dynamics on random graphs

We give a description of metastability arising in the low-temperature Ising model when the setting is a random graph obtained from the "Configuration model" algorithm.
(joint work with S. Dommers, F. Nardi and F. den Hollander)

Roman Kotecký

Metastability for the Widom-Rowlinson model

The Widom-Rowlinson model is one of the few models of interacting particles in the continuum for which a proof of ”liquid-gas” coexistence has been provided. An important feature of this model that facilitates its investigation is the fact that it can be rewritten as a model of two species of particles with mutual hardcore interaction. Reviewing first this feature and its use for the proof of phase coexistence, we will proceed to a discussion of transition from a metastable supercooled gas phase to the liquid phase in this model. In particular, the theorem about Arrhenius law with a nonstandard entropic correction will be formulated.
Based on a work in progress.
(joint with F. den Hollander, S. Jansen, and E. Pulvirenti; Elena Pulvirenti will be discussing main ideas of the proof in her talk.)

Claudio Landim

Markov chains model reduction

We review in this lecture the martingale approach to metastability. This method has been developed recently to solve the following problem. Let $E_N$ be a sequence of countable state
spaces. Given a sequence $\eta^N(t)$ of $E_N$-valued, continuous-time Markov chains, we identify slow variables [functions $\Psi_N : E_N \to S$, where $S$ is a fixed, usually finite, set],
and time scales $\theta_N$ for which the reduced model, or coarse-grained process, $\Psi_N(\eta^N(t\theta_N))$ converges to a $S$-valued Markov chain. We introduce the problem and the main tools in a simple context, and we present some examples where this approach has been successfully applied.

Christian Maes

Driving-induced stability with long-range effects

We give a sufficient condition under which an applied nonequilibrium driving in a medium always has a stabilizing effect on an attached quasi-static probe. We show that the resulting Lamb shift in the symmetric part of the stiffness matrix with respect to equilibrium is positive and depends strongly on the nonequilibrium density far away from the probe. For illustration we take the example of diffusive medium particles with a self-potential in the shape of a Mexican hat, repulsive at the origin and attractive near the border of a disk where the potential energy gets very large. They undergo a rotational force around the origin. For no or small rotation in the medium, the origin is an unstable fixed point for the probe and the precise shape of the self-potential at large distances from the origin is irrelevant for the statistical force there. Above a certain threshold of the rotation amplitude the origin becomes stable for the probe and more details of the medium start to matter. The stabilization is robust under different types of rotation, and the temperature dependence gets reversed in the transition from equilibrium (where lower temperature destabilizes more) to nonequilibrium (where lower temperature stabilizes more).
(joint work with U.Basu, P. de Buyl, K. Netocny)

Patrick Müller

Hydrodynamic limits, propagation of chaos and large deviations in local mean-field models with unbounded spins

We consider systems of stochastic differential equations that describe the dynamics of unbounded spin-systems where spins interact via long-range spatially variable interactions. We prove the convergence of certain empirical mesures under proper rescaling to solution of a pde and we show that propagation of chaos holds. Finally, we derive a large deviation principle for on the space of paths of measures.

Francesca R. Nardi

Metastability for General Dynamics with Rare Transitions: Escape Time and Critical Configurations

Metastability is an ubiquitous physical phenomenon in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions for Markov chains. For Metropolis chains associated with statistical mechanical systems, this phenomenon has been described in an elegant way through a pathwise approach in terms of the energy landscape associated to the Hamiltonian of the system. In the seminar we will first explain the main results and ideas of this approach and compare it with other existing ones. Then we will provide a similar description in the general rare transitions setup that can be applied to irreversible systems as well.
Besides their theoretical content, we believe that our results are a useful tool to approach metastability for non-Metropolis systems such as Probabilistic Cellular Automata. Moreover, we will describe results pertaining to exponential hitting times which range of applicability includes irreversible systems, systems with exponentially growing volumes and systems with a general starting measure.
(joint work with Emilio N. M. Cirillo, R. Fernandez, F. Manzo, E. Scoppola and J. Sohier)

Elena Pulvirenti

Metastability for Widom-Rowlinson model: II

In this talk I will recall the Widom-Rowlinson model (introduced by Roman Kotecký in a prior talk). I will review the main metastability results when choosing a stochastic dynamics in which particles are randomly created and annihilated on a finite two-dimensional torus, according to an infinite reservoir with a given chemical potential. I will explain some of the ideas needed in the proof, which uses the potential theoretic approach to metastability, and show how to obtain asymptotic behavior of the capacity in the limit of low temperature via large and moderate deviations.
(joint work in progress with F. den Hollander, S. Jansen, R. Kotecký)

Frank Redig

Splitting and exchange models with SU(1,1) symmetry

Inspired by a recent model of wealth distribution in econophysics, the so-called immediate exchange model, we study a general class of models where at random times a continuous quantity is exchanged among two agents according to the following mechanism (X,Y) goes to (X(1-U)+ YV, XU + Y(1-V)) where U,V are independent random variables on [0,1]. We show the following
1. If U, V are Beta(t,s) distributed, then this model has a discrete dual process, which is in a natural way related to the symmetric inclusion process.
2. The discrete dual process is self-dual , and converges in a scaling limit to the original continuous process.
3. The generator of the discrete process has full SU(1,1) symmetry.
We then generalize the construction to a broader class of splitting and redistribution models, including models with SU(2) symmetry and asymmetric redistributions.
(joint work with Federico Sau (Delft))

Bruno Schapira

A metastability result for the contact process on finite graphs

We show that the contact process on any sequence of growing finite graphs exhibits a metastable behavior, when the infection parameter is larger than the critical rate on $\mathbb Z$. Based on a joint work with Daniel Valesin

Benedetto Scoppola

Metastability for irreversible PCA dynamics

In this talk we present a class of parallel irreversible Markov chains, that are probabilistic cellular automata, and we show various models in which is possible to describe in details metastable behaviours.
In some cases we are able to show rigorously a dramatic speeding of the dynamics, due to the combined effect of parallelism and irreversibility of the dynamics. We shall discuss also applications of this dynamics to the discrete optimization.
(joint work(s) with E. Scoppola, P.Dai Pra', A Troiani, C. Lancia)

Elisabetta Scoppola

Metastability: old problem and new results

We recall problems and results in metastability, in particular within the so called "pathwise approach". The exponential law of the decay time will be discussed in some detail with new techniques.

Insuk Seo

Metastability and Eyring-Kramers Formula for Non-reversible Markov Processes

In this talk, we present a metastability result for a class of non-reversible Markov processes. Our framework is based on the recent developments of the Dirichlet and the Thomson principles for the non-reversible Markov chains, and on the martingale approach for the metastability. In particular, we provide an argument which enables us to compute the sharp estimates for the mean transition times between the metastable sets, and as a consequence the Eyring-Kramers formula for the non-reversible process is obtained. Finally, we discuss the metastability of the so-called planar Potts model, which is a generalization of the Curie-Weiss model, as an application of our result.
(joint work with C. Landim)

Senya Shlosman

Metastable behavior of communication networks with moving nodes

I will explain the results of the study of queuing networks, which consist of servers which can move. Each client in the network has its destination, but the servers of the network move on their own. The interplay of the two factors make the overall behavior of the system quite intricate.
(joint work with François Baccelli, Alexandre Rybko and Alexandre Vladimirov)

Martin Slowik

Metastability in stochastic dynamics: Low-lying eigenvalues of the generator via two-scale decomposition

Metastability is a phenomenon that occurs in the dynamics of a multi-stable non-linear system subject to noise.  It is characterized by the existence of multiple, well separated time scales. The talk will be focus on the metastable behavior of reversible Markov chains on countable state spaces.  In the talk, I will first review the potential theoretic approach to metastability that has been proven to be a powerful tool to derive sharp estimates on quantities characterizing the metastable behavior, e.g. metastable exit times.  In particular, I will discuss systems in which the entropy of microscopic paths plays a crucial role, and precise computations of microscopic point to point capacities are not possible.  Second, I will present an approach to derive matching upper and lower bounds on the small eigenvalues of the generator.

Cristian Spitoni

Sum of exit times in a series of metastable states

In this talk we consider the problem of not degenerate in energy metastable states forming a series in the framework of reversible fi?nite state space Markov chains. We assume that starting from the state at higher energy the system necessarily visits the second one before touching the stable state. In this framework we give a sharp estimate of the exit time from the metastable state at higher energy and we prove an addition rule. Furthermore, we discuss the application of this theory to the Blume-Capel model in the zero chemical potential case, and to a reversible Probabilistic Cellular Automaton without self-interaction.

Siamak Taati

Metastability of the hard-core process on bipartite graphs

The hard-core gas model is widely studied in statistical mechanics, but also appears in connection with communication networks.  In the dynamic version, particles appear and disappear with constant rates on the sites of a graph subject to the "hard-core constraint":  two neighbouring sites can hold at most one particle at a time.  When the rate of appearance is large compared to the disappearance rate, the particles tend to remain close to efficient maximal packing arrangements.  Transition from one locally optimal arrangement to another requires the formation of a "critical droplet", which is a configuration solving a particular isoperimetric problem.  I will talk about a description of such transitions in case the underlying graph is bipartite.  We use potential-theoretic methods and give sharp asymptotics for the expected transition time for few choices of the underlying graph.
(joint work with Frank den Hollander and Francesca Nardi)

Daniel Valesin

The contact process on finite trees and random regular graphs

The contact process is a class of interacting particle systems that models the spread of an infection in a population. Vertices of a graph are individuals that can either be in state 0 (healthy) or 1 (infected). Infected individuals heal with rate 1 and transmit the infection to each neighbor with rate $\lambda > 0$. We study the contact process on trees or locally tree-like graphs. Since seminal works by Pemantle (1992) and Liggett (1996), it has been established that the contact process on the infinite $d$-regular tree ($T_d$) exhibits two phase transitions. Specifically, depending on the value of the infection parameter $\lambda$, the process started from finite occupancy can (1) die out; (2) survive globally but not locally; (3) survive locally. In this talk, we examine the counterparts of these distinct regimes in the context of two classes of finite graphs that locally approximate the tree $T_d$: truncated trees and random regular graphs. We show that the process exhibits very different behaviors in these two contexts.
(joint work with Jean-Christophe Mourrat, Michael Cranston and Thomas Mountford)

Aernout van Enter

Metastability, from Ideal to Real to Super

To describe metastable phases in infinite volume has been problematical for different reasons. We point out some examples of short-range models which satisfy the criteria of " Ideal Metastability" due to Sewell, and comment why these criteria still are physically dubious. Sewell's criterion of "Real Metastability" implies exponentially diverging metastability times. I discuss some examples of bootstrap percolation models, in which the metastability times become  superexponential, rather than exponential.

Alessandro Zocca

Hitting time asymptotics for hard-core interaction on grid graphs

Motivated by the study of random-access networks performance, we consider the hard-core model with Metropolis transition probabilities on finite grid graphs and investigate the asymptotic behavior of the first hitting time between its stable states in the low-temperature regime. In particular, we develop a novel combinatorial method to show how the order-of-magnitude of this first hitting time depends on the grid sizes and on the boundary conditions for various types of grid graphs. Our analysis also proves the asymptotic exponentiality of the scaled hitting time and yields the mixing time of the process in the low-temperature limit as side-result. In order to derive these results, we extended the model-independent framework for first hitting times known as "pathwise approach" to allow for more general initial and target states.

PRACTICAL INFORMATION

### ●      Venue

Eurandom, Mathematics and Computer Science Dept, TU Eindhoven,

Den Dolech 2, 5612 AZ  EINDHOVEN,  The Netherlands

Eurandom is located on the campus of Eindhoven University of Technology, in the (4th floor) (about the building). The university is located at 10 minutes walking distance from Eindhoven main railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).
Accessibility TU/e campus and map.

### ●      Accommodation

For invited participants, we will take care of accommodation. Other attendees will have to make their own arrangements.

We have a preferred hotel, which can be booked at special rates. Please email Patty Koorn for instructions on how to make use of this special offer.

For other hotels around the university, please see: Hotels (please note: prices listed are "best available").

More hotel options can be found on the webpages of the , Postbus 7, 5600 AA Eindhoven.

### ●      Travel

For those arriving by plane, there is a convenient direct train connection between Amsterdam Schiphol airport and Eindhoven. This trip will take about one and a half hour. For more detailed information, please consult the NS travel information pages or see Eurandom web page location.

Many low cost carriers also fly to Eindhoven Airport. There is a bus connection to the Eindhoven central railway station from the airport. (Bus route number 401) For details on departure times consult http://www.9292ov.nl

The University  can be reached easily by car from the highways leading to Eindhoven (for details, see our route descriptions or consult our map with highway connections.

### ●      Conference facilities : Conference room, Metaforum Building  MF11&12

The meeting-room is equipped with a data projector, an overhead projector, a projection screen and a blackboard. Please note that speakers and participants making an oral presentation are kindly requested to bring their own laptop or their presentation on a memory stick.

### ●      Conference Secretariat

Upon arrival, participants should register with the workshop officer, and collect their name badges. The workshop officer will be present for the duration of the conference, taking care of the administrative aspects and the day-to-day running of the conference: registration, issuing certificates and receipts, etc.

### ●      Contact

Mrs. Patty Koorn, Workshop Officer, Eurandom/TU Eindhoven, koorn@eurandom.tue.nl

The organisers acknowledge the financial support/sponsorship of:

#### Last updated 13-04-16, by PK

P.O. Box 513, 5600 MB Eindhoven, The Netherlands
tel. +31 40 2478100
e-mail: info@eurandom.tue.nl