We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We conclude with Higher order extension of Souriau model based on works of R..S Ingarden and W. Jaworski. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics.
Francesco Caravelli LANL, USANonequilibrium properties of memristive circuits: connection to Hopfield models
Memristors are nonlinear passive circuit elements which can be thought as time varying resistances. Circuits with memristors have been shown both experimentally and numerically to be of use for various problems of machine learning. In this talk we provide further theoretical background for this statement. We show that the dynamics for memristive circuits is such that a specific class of quadratic functional is being minimized. This shows that this class of optimization problems, in general hard to solve, can be thus approximately quickly solved using memristive circuits. We analyze this statement in the case of random circuits, showing in a certain approximation what is the behavior of the number of stationary points as a function of the topological parameters of the circuit. This provides a connection between spin glasses and the Hopfield model.
Domenico Felice Max Planck Institute for Mathematics in the Sciences, GermanyTowards a Canonical Divergence within Information Geometry
We propose a new canonical divergence and prove that it is a potential function for the dual structure of a statistical manifold. We recover the canonical divergence proposed by Ay and Amari [1] in the case of: (a) dual flatness [2], (b) conjugate symmetry [3], and (3) statistical symmetry as studied by Henmi and Kobayashi [4]. The new divergence coincides with the well-known canonical divergence of dually flat manifolds [2] and is proportional to the squared Riemannian distance in the self-dual case."References: [1] Ay, N., Amari, S.I.: A novel approach to canonical divergences within information geometry. Entropy 17, 8111-8129 (2015)[2] Amari, S.I., Nagaoka, H.: Methods of Informatin Geometry, Translations of Mathematical monographs, vol. 191. Oxford University Press (2000)[3] Lauritzen, S.L.: Differential geometry in statistical inference. Lecture Notes-Monograph Series 10, 163-218 (1987)[4] Henmi,M., Kobayashi, R.: Hooke's law in statistical manifolds. Nagoya Math. J. 159, 1-24 (2000)
Kirill Glavatskiy The University of Sydney, AustraliaAgent-based modeling of housing market: can methods of statistical physics predict human behavior?
Kirill Glavatskiy, Mikhail Prokopenko, Michael Harre.Following the crashes of global market in 2008 questions were asked of the ability of economics to address the vulnerability of such an important system. It became evident that conventional tools are not able to address systemic risks in markets with heterogeneous structure, where humans behave according to multiple behavioral patterns. This is the case, in particular, for housing market. Over the recent years there has been observed a steady increase of real estate prices in major metropolitan cities in Australia, while some other cities do not show these trends. This indicates that there might be a possibility of “housing bubble” which will eventually burst. In this work we employ methods of statistical physics to investigate the evolution of the housing market in Australia and possibilities for existence of the housing bubble and its burst. The ideological background for this approach is the fact that when it comes to macroscopic phenomena, individual human choices do not play a significant role. Rather, their collective behavior determines the overall evolution, and statistical interactions between individuals become decisive. Employing the analogy between humans in economic description and molecules in physical description, we investigate the conditions for human self-organization from the perspective of statistical physics. We address the problem using three approaches, which complement each other on different levels of description: agent-based numerical simulations, maximum entropy principle, and bifurcations at the phase transitions.Within agent based modeling we introduce heterogeneous society with agents who follow one of the several behavioral patterns. This corresponds to a multicomponent fluid, which self-organize in the mean-field fashion. Unlike a fluid, the agents are allowed to change their identity, switching between behavioral patterns. Depending on the particular state the current community (city), this evolution may or may not lead to a bifurcation and hence phase transition. Coming close to the bifurcation point would indicate that the particular community is close to a “bubble” state.Within maximum entropy principle we investigate the macroscopic structure of interacting communities. Introducing the analogue of the entropy for the system of our agents, we identify the set of economic constraints, within which they realize their individual strategies. Maximizing the entropy for each of the community allows us to identify the analogy of the spatial distribution of the temperature. Temperature distribution determines the fluxes in the system and allows us to understand the sources and sinks for the market.Having the recent census data from the Australian Bureau of Statistics we perform analysis for each community (city) within Australia. By doing this we build a map of Australia, which shows how “heated” different regions are with respect to each other. This information can be used by policy making agencies to properly react on market challenges.
J.Michael Herrmann The University of Edinburgh, United KingdomRobots can Understand Physics from Fisher Information
Petru Hlihor Romanian Institute of Science and Technology, Romania, and Max Planck Institute for Mathematics in the Sciences, GermanyA Defense Against Adversarial Examples based on Image Reconstruction by Variational Autoencoders
Calum Imrie University of Edinburgh, United KingdomSelf-Organised Transitions in Swarms with Turing Patterns
Vladimir Jaćimović University of Montenegro, MontenegroMean fields in networks of interacting particles
Dimitri Marinelli Romanian Institute of Science and Technology, RomaniaQuantum Information Geometry and Stochastic Reconfiguration
Michel Nguiffo Boyom Université des Sciences et Techniques de Languedoc, FranceComplex systems and Geometric structures
Loosely speaking a complex system is a measurable set $(\Xi,\Omega)$; $\Gamma(\Xi,\Omega)$ is the group of measurable isomorphisms of $\Xi$ (viz the group of efficient statistics.) An information Geometry of $(\Xi,\Omega)$ is a $\Gamma$-Geometry in a statistical model of $(\Xi,\Omega)$. Relevant informations are invariants of such a $\Gamma$-geometry. The relevancy of informations is linked with the existence of nice geometric structures in both $(\Xi,\Omega)$ and its chosen model. Among rich geometries in statistical models are the symplectic geometry, the geometry of Koszul, the bi-invariant Riemannian geometry in Lie groups, the left invariant symplectic geometry in Lie groups. The aim of the talk is to address those concerns, some related open geometric problems and a few recent contributions.A few references. [AN] Amari and Nagaoka: Methods of information geometry, AMS-Oxford monograph 91.[BF] Barbaresco F. GeometriC Theory of Heat from Souriau Lie group thermodynamics and Koszul geomtry; Entropy 2016.[NB1] Nguiffo Boyom M. Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and cohomology; Entropy 12 vol 18 2016.[NB2] Nguiffo Boyom M. Numerical properties of Koszul connections; arxiv.1708.01106. 3 august 2017.
Thomas Oikonomou Nazarbayev University, KazakhstanThe Failure of the MaxEnt Principle for the generalized entropies
Alexandra Peşte Romanian Institute of Science and Technology, Romania, and Max Planck Institute for Mathematics in the Sciences, GermanyOn the Geometry of the Latent Space of Variational AutoEncoders: An Explanatory Analysis
Sabin Roman Romanian Institute for Science and Technology, RomaniaOn the Use of Natural Gradient for Variational AutoEncoders
Nathaniel Virgo ELSI, Tokyo, Japan, JapanDecomposing multivariate information
We propose a decomposition of multivariate information which is based on a generalisation of Amari's hierarchy over a lattice imposed on combinations of primary random variables, so-called "structures". While related, our construction differs from the well-known lattice construction of Williams/Beer's in that the quantities to be interpreted as information terms sit on the edges and that no variable set is distinguished as a predictor of others; all variables are on the same level, similar to the approach by Rosas et al. (2016). We show that this construction can address some of the questions posed by James & Crutchfield (2017)
Riccardo Volpi Romanian Institute of Science and Technology, Romania (joint work with Luigi Malagò)Geometry of latent representations for word embeddings
There are universal features associated with a range of critical phenomena, such as thermal phase transitions or transitions associated with 'exceptional points' where eigenvectors of a matrix coalesce. These include, for example, the breakdown of adiabatic approximations, the curvature divergence of the parameter-space at critical points, or the loss of information concerning the initial state of the system as one passes through critical points. This talk will sketch some of these ideas that suggest that perhaps exceptional point physics is not all that different from critical phenomena in thermal physics.
Entropic Dynamics (ED) is a framework in which dynamical laws are derived as an application of entropic methods of inference. The dynamics of the probability distribution is driven by entropy subject to constraints that are eventually codified into the phase of a wave function. The central challenge is to identify the relevant physical constraints and, in particular, to specify how those constraints are themselves updated.
In this talk I describe how the information geometry of the space of probabilities is extended to the ensemble-phase space of probabilities and phases. The result is a highly symmetric Riemannian geometry that incorporates the symplectic and complex structures that characterize the geometry of quantum mechanics. The ED that preserves these structures is a Hamiltonian flow and the simplest Hamiltonian suggested by the extended metric leads to quantum mechanics. Thus, in the entropic dynamics framework, Hamiltonians and complex wave functions arise as the natural consequence of information geometry.
We construct an all-inclusive statistical-mechanical model for self-organization based on the hierarchical properties of the nonlinear dynamics towards the attractors that define the period-doubling route to chaos [1-3]. The aforementioned dynamics imprints a sequential assemblage of the model that privileges progressively lower entropies, while a new set of configurations emerges due to the collective partitioning of the original system into secluded portions. The initial canonical balance between numbers of configurations and Boltzmann-Gibbs (BG) statistical weights is drastically altered and ultimately eliminated by the sequential actions of the attractor. However the emerging set of configurations implies a different and novel entropy growth process that eventually upsets the original loss and has the capability of locking the system into a self-organized state with characteristics of criticality, therefore reminiscent in spirit to the so-called self-organized criticality [4,5].
Some specifics of the approach we develop are: We systematically eliminate access to configurations of an otherwise elementary thermal system model by progressively partitioning it into isolated portions until only remains a subset of configurations of vanishing measure. Each isolated portion becomes essentially a micro-canonical ensemble. The thermal system consists of a large number of (effective) degrees of freedom, each occupying entropy levels with the form of inverse powers of two. The sequential process replaces the original configurations by an emerging discrete scale invariant set of ensemble configurations with allowed entropies that are necessarily inverse powers of two. In doing this we achieve the following results:1) The constrained thermal system becomes a close analogue of the dynamics towards the multifractal attractor at the period-doubling onset of chaos.2) The statistical-mechanical properties of the thermal system depart from those of the ordinary Boltzmann-Gibbs form and acquire features from q-statistics.3) Redefinition of entropy levels as logarithms of the original ones recovers the BG scheme and the free energy Legendre transform property.
Furthermore, the sequences of actions on the entropies associated with the degrees of freedom have the following consequences:i) Confine degrees of freedom on very few configurations of ever decreasing entropies.ii) The reduction in numbers of configurations goes down from the initial exponential of the number of degrees of freedom to only one per micro-canonical ensemble, there being an equivalent exponential number of such ensembles for the entire system. As the thermodynamic limit is approached a set of initial configurations with nonzero measure reduces to a set of vanishing measure. But in the process a new set of numbers of configurations develops. These are given by the degeneracies of the micro-canonical ensembles.iii) The new emerging numbers of configurations grow more slowly than exponentially with the size of the system as they are binomial coefficients.
As with any partition function, the sum of the numbers of configurations times their probabilities cannot vanish nor diverge but be unity. Initially (the BG case) the numbers grow exponentially and the weights also decrease exponentially with system size. After the actions of the attractor the numbers of new ensemble degeneracies grow slower than exponentially and the new weights must now decrease accordingly. The new “canonical” partition function acquires q-exponential weights typical of q-statistics. The precise value of the tuning parameter q and its relationship with the inverse temperature is determined.
[1] Robledo A., Moyano, L.G., “q-deformed statistical-mechanical property in the dynamics of trajectories en route to the Feigenbaum attractor”, Physical Review E 77, 032613 (2008).[2] Robledo, A., “A dynamical model for hierarchy and modular organization: The trajectories en route to the attractor at the transition to chaos”, Journal of Physics: Conf. Ser. 394, 012007 (2012).[3] Diaz-Ruelas, A., Robledo, A., “Emergent statistical-mechanical structure in the dynamics along the period-doubling route to chaos”, Europhysics Letters 105, 40004 (2014).[4] Bak, P., “How Nature Works” (Copernicus, New York, 1996).[5] Watkins, N.W., Pruessner, G., Chapman, S.C., Crosby, N.B., Jensen, J.K., “25 Years of Self-organized Criticality: Concepts and Controversies”, Space Science Reviews 198, 3 (2016).
Francesco Caravelli LANL, USANonequilibrium properties of memristive circuits: connection to Hopfield models
Memristors are nonlinear passive circuit elements which can be thought as time varying resistances. Circuits with memristors have been shown both experimentally and numerically to be of use for various problems of machine learning. In this talk we provide further theoretical background for this statement. We show that the dynamics for memristive circuits is such that a specific class of quadratic functional is being minimized. This shows that this class of optimization problems, in general hard to solve, can be thus approximately quickly solved using memristive circuits. We analyze this statement in the case of random circuits, showing in a certain approximation what is the behavior of the number of stationary points as a function of the topological parameters of the circuit. This provides a connection between spin glasses and the Hopfield model.
Domenico Felice Max Planck Institute for Mathematics in the Sciences, GermanyTowards a Canonical Divergence within Information Geometry
We propose a new canonical divergence and prove that it is a potential function for the dual structure of a statistical manifold. We recover the canonical divergence proposed by Ay and Amari [1] in the case of: (a) dual flatness [2], (b) conjugate symmetry [3], and (3) statistical symmetry as studied by Henmi and Kobayashi [4]. The new divergence coincides with the well-known canonical divergence of dually flat manifolds [2] and is proportional to the squared Riemannian distance in the self-dual case."References: [1] Ay, N., Amari, S.I.: A novel approach to canonical divergences within information geometry. Entropy 17, 8111-8129 (2015)[2] Amari, S.I., Nagaoka, H.: Methods of Informatin Geometry, Translations of Mathematical monographs, vol. 191. Oxford University Press (2000)[3] Lauritzen, S.L.: Differential geometry in statistical inference. Lecture Notes-Monograph Series 10, 163-218 (1987)[4] Henmi,M., Kobayashi, R.: Hooke's law in statistical manifolds. Nagoya Math. J. 159, 1-24 (2000)
Kirill Glavatskiy The University of Sydney, AustraliaAgent-based modeling of housing market: can methods of statistical physics predict human behavior?
Kirill Glavatskiy, Mikhail Prokopenko, Michael Harre.Following the crashes of global market in 2008 questions were asked of the ability of economics to address the vulnerability of such an important system. It became evident that conventional tools are not able to address systemic risks in markets with heterogeneous structure, where humans behave according to multiple behavioral patterns. This is the case, in particular, for housing market. Over the recent years there has been observed a steady increase of real estate prices in major metropolitan cities in Australia, while some other cities do not show these trends. This indicates that there might be a possibility of “housing bubble” which will eventually burst. In this work we employ methods of statistical physics to investigate the evolution of the housing market in Australia and possibilities for existence of the housing bubble and its burst. The ideological background for this approach is the fact that when it comes to macroscopic phenomena, individual human choices do not play a significant role. Rather, their collective behavior determines the overall evolution, and statistical interactions between individuals become decisive. Employing the analogy between humans in economic description and molecules in physical description, we investigate the conditions for human self-organization from the perspective of statistical physics. We address the problem using three approaches, which complement each other on different levels of description: agent-based numerical simulations, maximum entropy principle, and bifurcations at the phase transitions.Within agent based modeling we introduce heterogeneous society with agents who follow one of the several behavioral patterns. This corresponds to a multicomponent fluid, which self-organize in the mean-field fashion. Unlike a fluid, the agents are allowed to change their identity, switching between behavioral patterns. Depending on the particular state the current community (city), this evolution may or may not lead to a bifurcation and hence phase transition. Coming close to the bifurcation point would indicate that the particular community is close to a “bubble” state.Within maximum entropy principle we investigate the macroscopic structure of interacting communities. Introducing the analogue of the entropy for the system of our agents, we identify the set of economic constraints, within which they realize their individual strategies. Maximizing the entropy for each of the community allows us to identify the analogy of the spatial distribution of the temperature. Temperature distribution determines the fluxes in the system and allows us to understand the sources and sinks for the market.Having the recent census data from the Australian Bureau of Statistics we perform analysis for each community (city) within Australia. By doing this we build a map of Australia, which shows how “heated” different regions are with respect to each other. This information can be used by policy making agencies to properly react on market challenges.
J.Michael Herrmann The University of Edinburgh, United KingdomRobots can Understand Physics from Fisher Information
Petru Hlihor Romanian Institute of Science and Technology, Romania, and Max Planck Institute for Mathematics in the Sciences, GermanyA Defense Against Adversarial Examples based on Image Reconstruction by Variational Autoencoders
Calum Imrie University of Edinburgh, United KingdomSelf-Organised Transitions in Swarms with Turing Patterns
Vladimir Jaćimović University of Montenegro, MontenegroMean fields in networks of interacting particles
Dimitri Marinelli Romanian Institute of Science and Technology, RomaniaQuantum Information Geometry and Stochastic Reconfiguration
Michel Nguiffo Boyom Université des Sciences et Techniques de Languedoc, FranceComplex systems and Geometric structures
Loosely speaking a complex system is a measurable set $(\Xi,\Omega)$; $\Gamma(\Xi,\Omega)$ is the group of measurable isomorphisms of $\Xi$ (viz the group of efficient statistics.) An information Geometry of $(\Xi,\Omega)$ is a $\Gamma$-Geometry in a statistical model of $(\Xi,\Omega)$. Relevant informations are invariants of such a $\Gamma$-geometry. The relevancy of informations is linked with the existence of nice geometric structures in both $(\Xi,\Omega)$ and its chosen model. Among rich geometries in statistical models are the symplectic geometry, the geometry of Koszul, the bi-invariant Riemannian geometry in Lie groups, the left invariant symplectic geometry in Lie groups. The aim of the talk is to address those concerns, some related open geometric problems and a few recent contributions.A few references. [AN] Amari and Nagaoka: Methods of information geometry, AMS-Oxford monograph 91.[BF] Barbaresco F. GeometriC Theory of Heat from Souriau Lie group thermodynamics and Koszul geomtry; Entropy 2016.[NB1] Nguiffo Boyom M. Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and cohomology; Entropy 12 vol 18 2016.[NB2] Nguiffo Boyom M. Numerical properties of Koszul connections; arxiv.1708.01106. 3 august 2017.
Thomas Oikonomou Nazarbayev University, KazakhstanThe Failure of the MaxEnt Principle for the generalized entropies
Alexandra Peşte Romanian Institute of Science and Technology, Romania, and Max Planck Institute for Mathematics in the Sciences, GermanyOn the Geometry of the Latent Space of Variational AutoEncoders: An Explanatory Analysis
Sabin Roman Romanian Institute for Science and Technology, RomaniaOn the Use of Natural Gradient for Variational AutoEncoders
Nathaniel Virgo ELSI, Tokyo, Japan, JapanDecomposing multivariate information
We propose a decomposition of multivariate information which is based on a generalisation of Amari's hierarchy over a lattice imposed on combinations of primary random variables, so-called "structures". While related, our construction differs from the well-known lattice construction of Williams/Beer's in that the quantities to be interpreted as information terms sit on the edges and that no variable set is distinguished as a predictor of others; all variables are on the same level, similar to the approach by Rosas et al. (2016). We show that this construction can address some of the questions posed by James & Crutchfield (2017)
Riccardo Volpi Romanian Institute of Science and Technology, Romania (joint work with Luigi Malagò)Geometry of latent representations for word embeddings
The thermodynamics of computation specifies the minimum amount that entropy must increase in the environment of any physical system that implements a given computation, when there are no constraints on how the system operates (the so-called “Landauer limit”). However common engineered computers use digital circuits that physically connect separate gates in a specific topology. Each gate in the circuit performs its own “local” computation, with no a priori constraints on how it operates. In contrast, the circuit’s topology introduces constraints on how the aggregate physical system implementing the overall “global” computation can operate. These constraints cause additional minimal entropy increase in the environment of the overall circuit, beyond that caused by the individual gates.
Here we analyze the relationship between a circuit’s topology and this additional entropy increase, which we call the “circuit Landauer cost”. We also compute a second kind of circuit cost, the “circuit mismatch cost”. This is the extra entropy that is generated if a physical system is designed to achieve minimal entropy production for a particular distribution q over its inputs, but is instead used with an input distribution p that differs from q.
We show that whereas the circuit Landauer cost cannot be negative, circuits can have positive or negative mismatch cost. In fact the total circuit cost (given by summing the two types of cost) can be positive or negative. Thus, the total amount of entropy increase in the environment can be either larger or smaller when a particular computation is implemented with a circuit.
Furthermore, in general different circuits computing the same Boolean function have both different Landauer costs and different mismatch costs. This provides a new set of challenges, never before considered, for how to design a circuit to implement a given computation with minimal thermodynamic cost. As a first step in addressing these challenges, we use tools from the computer science field of circuit complexity to analyze the scaling of thermodynamic costs for different computational tasks.
The natural gradient method is one of the most prominent information-geometric methods within the field of machine learning.
It was proposed by Amari in 1998 and uses the Fisher-Rao metric as Riemannian metric for the definition of a gradient within optimisation tasks. Since then it proved to be extremely efficient in the context of neural networks, reinforcement learning, and robotics. In recent years, attempts have been made to apply the natural gradient method for training deep neural networks. However, due to the huge number of parameters of such networks, the method is currently not directly applicable in this context. In my presentation, I outline ways to simplify the natural gradient for deep learning. Corresponding simplifications are related to the locality of learning associated with the underlying network structure.
The scope of IG has been clearly defined in S. Amari and H. Nagaoka monograph by emphasizing the crucial importance of the dual affine structure of regular parametric statistical models. Amari’s research program is today still in progress e.g., the papers presented in this conference. One direction of research is the extension of the dual affine structure to non-parametric models. While the parametric assumption can fit most of the needs of Statististics and Machine Learning, this assumption is too restrictive in such fields as Statistical Phisics, Stochastics, or Evolution Equations. It is an easy but useful exercise to retell the geometry of finite dimensional models—-such as the probability simplex or the Gaussian distribution—-in a non-parametric language. In the first part of the talk, we give a non-parametric presentation of the IG of the finite probability simplex based on the idea of statistical bundle. After that, we discuss some of the available proposals for generalizing the same set-up to technically more complex situation.
Participants
Eleonora Andreotti
University of L'Aquila, Italy
Nihat Ay
Max Planck Institute for Mathematics in the Sciences, Germany
Pradeep Kumar Banerjee
Max Planck Institute for Mathematics in the Sciences, Germany
Frederic Barbaresco
THALES, France
Martin Biehl
Araya Inc., Tokyo, Japan
Dorje Brody
Imperial College London, United Kingdom
Francesco Caravelli
LANL, USA
Ariel Caticha
University of Albany, USA
Berlin Chen
Swarthmore College, USA
Emanuele Crosato
The University of Sydney, Australia
Sarah de Nigris
ENS de Lyon, France
Domenico Felice
Max Planck Institute for Mathematics in the Sciences, Germany
Keyan Ghazi-Zahedi
Max Planck Institute for Mathematics in the Sciences, Germany
Kirill Glavatskiy
The University of Sydney, Australia
Claudius Gros
Goethe University Frankfurt, Germany
Masayuki Henmi
The Institute of Statistical Mathematics, Japan
J.Michael Herrmann
The University of Edinburgh, United Kingdom
Petru Hlihor
Romanian Institute of Science and Technology, Romania
Calum Imrie
University of Edinburgh, United Kingdom
Vladimir Jaćimović
University of Montenegro, Montenegro
Jürgen Jost
Max Planck Institute for Mathematics in the Sciences, Germany
Markus Junginger
MHP - A Porsche Company, Germany
Harsha Kallumadatil Velluva
Indian Institute of Technology Bombay, Mumbai, India, India
Mark Kirstein
TU Dresden, Germany
Luigi Malagò
Romanian Institute of Science and Technology, Romania
Dimitri Marinelli
Romanian Institute of Science and Technology, Romania
Georg Martius
Max Planck Institute for Intelligent Systems, Germany
Hiroshi Matsuzoe
Nagoya Institute of Technology, Japan
Rostislav Matveev
Max Planck Institute for Mathematics in the Sciences, Germany
Matteo Moroni
ENS Lyon, France
Jan Naudts
Universiteit Antwerpen, Belgium
Michel Nguiffo Boyom
Université des Sciences et Techniques de Languedoc, France
Ramil Nigmatullin
University of Sydney, Australia
Atsumi Ohara
University of Fukui, Japan
Thomas Oikonomou
Nazarbayev University, Kazakhstan
Eckehard Olbrich
Max Planck Institute for Mathematics in the Sciences, Germany
Alexandra Peşte
Romanian Institute of Science and Technology, Romania
Cristina Pinneri
Max Planck Institute for Intelligent Systems, Deutschland
Giovanni Pistone
Collegio Carlo Alberto, Italy
Daniel Polani
University of Hertfordshire, United Kingdom
Mikhail Prokopenko
The University of Sydney, Australia
Sharwin Rezagholi
Max Planck Institute for Mathematics in the Sciences, Germany
Alberto Robledo
Universidad Nacional Autonoma de Mexico (UNAM), Mexico
Sabin Roman
Romanian Institute for Science and Technology, Romania
Antonio Maria Scarfone
Istituto dei Sistemi Complessi (ISC - CNR), Italy
Lorenz Schwachhöfer
TU Dortmund, Germany
Elham Shamsara
Medical University of Mashhad, Iran
Richard Spinney
University of Sydney, Australia
Hiroki Suyari
Chiba University, Japan
Omri Tal
Max Planck Institute for Mathematics in the Sciences, Germany
Pau Vilimelis Aceituno
Max Planck Institute for Mathematics in the Sciences, Germany
Nathaniel Virgo
ELSI, Tokyo, Japan, Japan
Riccardo Volpi
Romanian Institute of Science and Technology, Romania
Tatsuaki Wada
Ibaraki University, Japan
Galen Wilkerson
Istituto Nazionale di Documentazione, Innovazione e Ricerca Educativa, Italy
David Wolpert
Santa Fe Institute, USA
G. Çiğdem Yalçın
Istanbul University, Turkey
Marius Yamakou
Max Planck Institute for Mathematics in the Sciences, Germany
Organizing Committee
Nihat Ay
Max Planck Institute for Mathematics in the Sciences (Leipzig), Germany
Mikhail Prokopenko
University of Sydney, Australia
Program Committee
Nihat Ay
Max Planck Institute for Mathematics in the Sciences (Leipzig), Germany
Domenico Felice
Max Planck Institute for Mathematics in the Sciences (Leipzig), Germany
Carlos Gershenson
Universidad Nacional Autónoma de México (Mexico City), Mexico
Paolo Gibilisco
Università degli Studi di Roma "Tor Vergata", Italy
Daniel Polani
University of Hertfordshire (Hatfield), United Kingdom