Complex Systems

Complex systems provide a good description for many natural and social phenomena. Ecosystems, social networks, financial markets, quantum many body systems (e.g., superfluid helium 3), self-organizing artificial intelligence (such as neural systems) are all characterized by having many elements (species, members, traders, atoms), which interact locally, but give rise to phenomena, which cannot be explained by the properties of the single elements themselves (the emergence of fads and trends, financial market booms and crashes, superfluidity). Essential characteristics of a complex system include: feedbacks between processes occurring at different scales, amplification of minor variations in initial conditions, and the emergence of patterns in the absence of a global controller. The research at Labex MME-DII  focuses both on the study of abstract mathematical models of such systems (PDEs, Markov Processes, the Schroedinger problem, mean field analysis) and on the application of such models to the  analysis of specific complex systems (financial markets, quantum many body systems, networks). Some of the topics of interest are:

Theoretical Foundations: Applications:
  • Markov processes:
    • with scale-dependent limit properties;
    • with variable memory
  • The Schroedinger problem
  • Non-linear PDE:
    • hyperbolic;
    • parabolic;
    • elliptic
  • Reaction-diffusion systems
  • Control theory for non-linear PDE
  • Perturbations, robustness and speed of convergence to equilibrium
  • Blow-up and extinction phenomena
  • Evolutionary and adaptive dynamics
  • Mean-field games
  • Interactive epistemology
  • Application of stochastic processes to neuroscience
  • Interacting particle systems
  • Complex liquids
  • Financial markets
  • Complex networks
  • Opinion formation
  • Sustainable supply chain management
  • Cultural dynamics

Researchers with interest in Complex Systems:

Julien Barral  (LAGA)

I am interested in understanding and classifying fine geometric properties of measures and processes arising in statistical mechanics, dynamical systems and ergodic theory and metric number theory. Also I have an interest in applications of multi fractals in modeling social or natural phenomena, from signals to systems.

Florent Barret (MODAL’X)

I am interested in modelling and understanding Markov processes which could have very different behaviors depending on the time scale they are considered. Metastability and averaging are such situations. In these cases, the process exhibits different asymptotic behaviors depending on the time scale, and could converge to simplified processes. Metastable diffusions have numerous stable attractors and could jump randomly from one to another. Averaging principle leads to distinguish between slow and fast components of some process and tries to keep only the slow component. These are some of the situations I consider.

Elena Veronica Belmega (ETIS)

My research is on resource allocation problems in inter-connected networks of distributed nodes using tools from game theory, information theory, online and convex optimization and learning with applications mainly in wireless networks, multiple antenna communications, Internet of Things, and smart grids.

Pierre Berger  (LAGA)

I am interested in the study of dynamical systems (i.e. systems defined by the iteration of a map or of a flow), which have robust properties for perturbation of the systems. This can be their chaotic behavior (existence of a unique statistical behavior, or infinitely many simultaneous statistical behaviors), or reduction to systems of lower dimension.

Sacha Bourgeois-Gironde (LEMMA)

Sacha Bourgeois-Gironde’s research develops at the interface between economics and cognitive sciences. It mainly consists in investigating some biological bases of economic behaviors and cognition. The central theme is a temporal and functional lag between the evolution of neurobiological mechanisms that supported the adaptation to modern economic environments and the nature and demands of these environments. The study of rationality and irrationality, which transformed dramatically economic sciences in the recent decades, is understood within that particular scope. How institutions solve problem of individual rationality is at the core of his research. The human institutions on which his work focusses are money and language.

Jérémie Cabessa  (LEMMA)

L’épistémologie interactive fournit un cadre formel permettant de modéliser les concepts de connaissance et de croyance des agents impliqués dans diverses situations interactionnelles. L’étude des fondements  épistémiques de la théorie des jeux consiste alors à comprendre les relations qui existent entre les hypothèses épistémiques liées à certaines situations interactionnelles et les concepts de solutions qu’elles induisent en théorie des jeux.

Delphine David (CEPN)

We apply mean field games theory to model agents’ interaction on financial markets. We mainly study HJB equations and/or Kolmogorov equations to draw a general behavior of the system.

Hung T. Diep (LPTM)

I use statistical physics to study collective behaviors of systems of interacting particles. Applications are found in the field of phase transitions and critical phenomena as well as in condensed matter and theory of magnetism. I combine in my works theories and methods of simulation which are complementary. These methods can be also applied in socio-economic problems. I have started studying some problems in sociophysics.

Vladimir Georgescu (AGM)

Spectral and scattering theory of quantum many body systems and quantum fields.

Alain Grigis (LAGA)

My recent center of interest is about resonances. I have worked on localization of resonances for a system of Schrödinger operators coming from molecular predissociation. It was in collaboration with André Martinez professor in Bologna. I have also worked with Vladimir Buslaev on Stark-Wannier resonances. We have treated the case when the corresponding Hill equation has only finitely many gaps open. I have a project to extend this work to the general case of infinitely many open gaps. I am also finishing a work with Alexey Pozharskii from Saint-Petersburg, on resonances for a Schrödinger operator with fast oscillating potential.

Laura Hernandez (LPTM)

During the last years my research work evolved to the study of Complex Systems in Physics, Ecology and Social Sciences. Concerning the study of  ”non-physical problems” from a physicist’s point of view, the motivation is two-fold: on one hand, the hope that a non-standard approach, based on the application of the tools of Statistical Physics and of Dynamical Systems to problems that are studied in a very different way in  the corresponding disciplinary fields, could open the perspective for new findings. On the other hand, the fact that, if common properties happen to characterize the processes taking place in different systems, it is essential to study these processes in order to propose suitable models. I’m interested in the study of opinion formation and cultural dynamics models both in terms of the theoretical study of social inspired models, and also in terms of the research of stylized facts on social data in order to give some insight on the properties of a pertinent model for the studied system. I am also working on the application of complex networks formalism in order to study the organization of mutualist ecosystems and the application of these tools to the study of economic complexity.

Pierre Hodara (AGM)

Modeling and estimation of interacting particle systems.

Andreas Honecker (LPTM)

I work on theoretical physics with a focus on solid state physics, but I also have some interest in statistical physics, complex systems, and related fields. Much of my work builds on computer simulations.

Thierry Huillet  (LPTM)

Stochastic processes with applications to physics and biology.

Flora Koukiou (LPTM)

Modeling of random and complex systems using probability theory. Random measures and martingale theory are used for the study of the phase transitions.

Maël Le Treust (ETIS)

My research is at the interface between information theory and game theory. The initial problem is to characterize the coordination that can be implemented by autonomous devices with different objectives. The mathematical tools required for this study come from information theory, including Shannon’s entropy, which determines the maximum level of compression of an information source, as well as mutual information, which characterizes the maximum level of transmission through a noisy channel.

Christian Léonard (MODAL’X)

The Schrödinger problem is an entropy minimization problem for stochastic processes subject to initial and final marginal constraints. It resembles the dynamical version of the Monge-Kantorovich optimal transport problem. Indeed, the latter can be obtained as a limit of a sequence of Schrödinger problems when a slowing down procedure is applied. The optimal transport and Schrödinger problems are useful tools for developing an economic theory of marriage. The solutions of these problems look like geodesics on the set of probability measures on the state space. As on a Riemannian manifold where the acceleration of a geodesic is tightly connected to the curvature of the space, the acceleration of the solutions to the optimal transport and  Schrödinger problems  is a key for obtaining results about the curvature of the set of probability measures on the (possibly discrete) state space. As corollaries of these new notions of curvature, one obtains estimates on the rate of relaxation of complex interacting systems in terms of functional inequalities (spectral gap and logarithmic Sobolev inequalities, for instance).

Eva Loecherbach (AGM)

My research focuses on the study of stochastic processes, their applications in biology (in particular in neuroscience), interacting particle systems, variable memory and variable range processes, statistics of stochastic processes, limit theorems and the study of the longtime behavior of Markov processes.

Elisabeth Logak (AGM)

I work on the mathematical analysis of nonlinear partial differential equations arising in physical, chemical or biological models. I have studied the singular limit of reaction-diffusion systems that leads to interface propagation involving the mean curvature as well as nonlocal terms. I also worked on special solutions of Fisher-type equations, such as travelling fronts or stationary periodic patterns. My recent interests focus on differential systems with nonlocal diffusion related to the spread of epidemics on networks.

Laurent Menard  (MODAL’X)

My main research interests lie in the probabilistic study of large random combinatorial objects. Strictly speaking, I am a probabilist although a good portion of my work relies on combinatorics. More precisely, my main contributions are in the study of large random planar maps (namely planar graphs embedded in the sphere), and random trees. My research focuses on the metric properties of these objects and on the study of random processes living on these objects, such as percolation or the Ising model. In addition to my work on random maps, I have recently started to study sparse random graphs and random matrices. The main idea is to study the empirical distribution of the spectra of adjacency matrices of large random networks. I already have had results for locally tree-like graphs, and I am still investigating some surprising connections with free probability theory. I also stared to work on the related subject of singular values of sparse matrices which could lead to interesting applications in the estimation of covariance matrices.

Luc Miller (MODAL’X)

• Analyse des équations aux dérivées partielles. • Théorie du contrôle, de l’observation et de la stabilisation des semigroupes d’opérateurs et des systèmes distribués linéaires. • Analyse microlocale semi-classique d’équations de la physique mathématique  (ondes, élasticité, plaques, Schrödinger, chaleur, diffusions anormales fractionnaires). • Propagation des ondes hautes-fréquences (singularités, énergie, problèmes aux bords).

Marius Ochea (THEMA)

My research area is defined by the field of evolutionary game dynamics which sets out to investigate dynamical systems arising from evolutionary game theory. Evolutionary game theory studies the behavior of homo heuristicus, an agent whose decision-making toolbox contains simple, adaptive rules to deal with interactive problems. Among the heuristics of interest are imitation, perturbed best-reply and reinforcement learning. More recently, I am working on specific classes of adaptive heuristics – such as regret-matching and correlated fictitious play – that may induce rich dynamics on players’ joint distribution of actions.

Christophe Oguey (LPTM)

Complex liquids: we developed a geometrical method to analyze complex liquid structures by imbedded entangled graphs. We set the basis of the morphology of 3-star polyphile self-assemblies, and of the morphological transitions in these types of mesophases. By maximum entropy arguments, we got estimates of the topological correlations in foams and large cellular aggregates. We also studied in details the static and dynamics of films and film junctions in soap froths. For protein complexes, we set up a computer program, VLDP, analyzing the topology of macromolecules or complexes. It has been applied to get large scale contact distributions in globular proteins, to decipher the porosity a membrane protein and to analyze the stability and dynamics of the nucleosome, the fundamental unit of DNA compaction in eukaryotic cells.

Alex Pitti (ETIS)

I work on artificial intelligence and neural networks. I study neural architecture for cognitive systems and bio-inspired robotics.


Mathias Quoy (ETIS)

I am concerned with modelling the Prefrontal Cortex (PFC), Hippocampus (HS) and Basal Banglia (BG). In parallel with the neurobiological models, I also study Neural Networks in a dynamical system perspective. In particular, I study Random Recurrent Neural Networks (RRNN) both with rate coding neurons and spiking neurons. Of particular interest is the chaotic dynamics exhibited by both networks. This enables to store spatio-temporal inputs through hebbian learning in RRNN for instance. During a sabbatical at the Brain Lab at UNR (Reno, NV, USA). I studied RAIN (Random Asynchronous Irregular firing Networks). These studies were supported by ANR project ASTICO, PEPS ST2I MARTINE, DARPA project Synapse.

Lourdes Rojas Rubio (THEMA)

My research aims to study the dynamic of political change, theoretically and empirically. Two theoretical models are proposed. First, a probabilistic choice model of political cultural transmission will be developed to study the dynamics of population and to determine in which case the transmission mechanisms of political culture induce heterogeneous or homogeneous preferences for a specific political system. A second theoretical model is proposed, to study the role of political culture and its interaction with income inequality and institutional factors to determine the endogenous dynamics of political regimes and policy outcomes. Finally, an empirical study will be done to estimate the impact of political culture and its interaction with income on the consolidation of a democratic system. Also the effect of political culture on the probability of entry in democracy and on the probability of remaining in democracy will be estimated.

Armen Shirikyan (AGM)

I have worked in the following three domains of PDEs (in a broad sense): 1. Qualitative theory of nonlinear hyperbolic PDEs; 2. Large-time asymptotics of stochastic PDEs; 3. Control theory for nonlinear PDEs. The results obtained in my works (which are purely theoretical) have applications in statistical hydrodynamics and mathematical theory of turbulence.

Pascal Seppecher (CEPN)

My research focuses on the understanding of the dynamics of complex monetary economies. These economies combine radical decentralization together with general interdependence. The study of their dynamics requires precise reconstruction, through the development of macroeconomic models that integrate agent-based techniques (to take the decentralized aspect into account) and the stock-flow consistent approach (to model monetary interdependences in a precise way). In the models I develop, uncertainty is radical and thus agents cannot form « rational expectations ». Therefore, my areas of interest include modelling of individual economic decisions in situations of uncertainty; processes of collective adaptation; and emergences of norms and conventions.

Philippe Souplet (LAGA)

Problems under study: parabolic and elliptic PDE’s. Reaction-diffusion equations and systems, diffusive Hamilton-Jacobi equations, chemotaxis models, population dynamics, nonlocal equations, free-boundary problems, equations on manifolds, linear semigroups. Questions under study for these problems: – Asymptotic behavior; – Blow-up and extinction phenomena; -Analysis of singularities; – A priori estimates and universal bounds of solutions; – Local-in-time theory (existence, uniqueness, regularity); – Stationary solutions: existence, regularity, singularities, a priori estimates; – Elliptic and  parabolic Liouville-type theorems

Daniel Thiel (CEPN)

System dynamics models applied to management problems and sustainable supply chain management.

Alessandro Torcini (LPTM)

My research activity has been related to research fields ranging from Theory and Simulation of Liquids to Nonlinear Dynamics of Complex Systems, from Out-of-equilibrium Statistical Mechanics to Biological Physics. In particular, I have studied biophysically inspired mesoscopic models for the folding of proteins and more recently models for the dynamics of single neurons as well as of neural networks. In the last year my activity has been focused along two main axes: 1) synchronization in complex networks with application to neural dynamics as well as to Josephson Junctions arrays and power grids; 2) analysis of the emergence of collective dynamics in neural networks.

Denis Ullmo (LPTMS)

We analyse the various regime characterizing the behavior of the solutions of simple mean field games.

Yijun Xiao (MODAL’X)

-Optimal diaphony and L2-discrepancy of some low dsicrepancy sequences, – Some algebraic and geometric property of (t,m,s)-net, – Applications of Quasi-Monte Carlo methods, – Numerical probability