Henri Berestycki (directeur d'études à l'EHESS)
Jean-Pierre Nadal (directeur d’études à l'EHESS et directeur de recherche au CNRS)
Pierre Rosenstiehl (directeur d'études à l'EHESS)
Masayasu Mimura (Meiji University)
Traveling wave approach to competitor-mediated coexistence in competition-diffusion systems
From the viewpoint of competitor-mediated coexistence, it is important to study the influence of an exotic species on other native species in terms of exogenous effects; in general, exotic species are weaker than native ones because they have evolved in a different environment. Even if an exotic species is weaker, however, it might cooperate with a native species, after which the competitive relations may have reversed among the native species. This motivates us to consider the ecological situation whereby one exotic competing species invades the native system of two strongly competing species. In this talk, we will discuss the problem of competitive exclusion or competitor-mediated coexistence using a three-species competition–diffusion system from traveling waves point of views.
Marc Barthélémy (CEA, Institut de Physique Théorique, CAMS)
Towards a Science of Cities
There are always more data about cities and urban systems. This is unprecedented in our history and opens the exciting possibility of a 'Science of Cities', with the aim of understanding and modeling phenomena taking place in the City. Urban morphology and morphogenesis, activity and residence location choice, urban sprawling and the evolution of urban networks, are just a few of the important processes that are discussed for a long time but that we now hope to understand quantitatively. Now is the time to participate to the first steps towards quantitative urbanism, and in this talk I will present some efforts in this direction. In particular, I will present some recent results on (i) how to understand and model scaling properties of cities and (ii) what are the main properties of evolving urban road networks. These two examples will illustrate the fact that understanding an object as complex as a city is necessarily interdisciplinary: we will need to build up on early studies in quantitative geography and spatial economics, on the knowledge of architects, urbanists and urban sociologists, and on the tools of geomatics together with modeling approaches coming from applied mathematics and statistical physics.
Jose A. Scheinkman (Edwin W. Rickert Professor of Economics at Columbia University)
Speculation and bubbles: The case of finitely lived assets
The history of ﬁnancial markets is strewn with periods in which asset prices seem to vastly exceed fundamentals—events commonly called bubbles. Nonetheless, there is very little agreement among economists on the economic forces that generate such occurrences. Part of the difficulty stems from the fact that economists’ discussions of bubbles often concentrate solely on the behavior of asset prices. The most common deﬁnition of a bubble is “a period in which prices exceed fundamental valuation.” Valuation, however, depends on a view of fundamentals, and efficient-market advocates correctly point out that valuations are almost always, ex post, wrong.
In this lecture I will discuss some other stylized facts that are associated with bubble episodes, in particular the increase in trading volume that often accompany bubbles. I will argue that a model that combines differences in beliefs with asymmetries between the cost of acquiring an asset and the cost of shorting that same asset can explain the correlation between bubbles and trading.
I will present a mathematical model developed with H. Berestycki and R. Monneau that allows for assets with a finite life, such as many credit instruments. In the model, the buyer of the asset today acquires also an option to resell that asset to other more optimistic traders in the future. In equilibrium, buyers would be part of the most optimistic group but would be willing to pay in excess of their optimistic views, because they value the option to resell. The value of this option can be legitimately called a bubble. The resale option is American - that is it can be exercised at any time before the expiration of the asset. Thus the value of the resale option is given by an associated optimal stopping time but the value of stopping in turn is given by a stopping time problem faced by the new buyer. Because of this recursive aspect in the option valuation, the value of the option is characterized by a nonlocal obstacle problem. The equilibrium value of the option to resell corresponds to the viscosity solution of the obstacle problem and we establish several properties of this solution in the paper.
I will discuss the effects of parameters such as interest rates or transaction costs on the magnitude of the bubble and trading volume. In particular, I will argue that a small Tobin tax while effective in lowering trading volume is unlikely to have much of an effect on the size of the bubble.
Michael Batty, (Centre for Advanced Spatial Analysis (CASA) University College London)
Defining Cities: Using Networks to Measure and Predict City Size
In this talk, I will begin with the notion that as cities get bigger, they get more than proportionately richer. This is an old idea in economics and one of its progenitors. Alfred Marshall, at the end of the nineteenth century defined these notions as ‘economies of urban agglomeration’. More recently the group of researchers at Santa Fe, in particular Luis Bettencourt and Geoff West, have argued that within the confines of comparable entities – cities – that define urban systems, as they grow, their income increases at a rate which is more than proportionate to their size, that is ,if their size increases by 100%, their income increases by some 112%. This is positive allometry or superlinear scaling as it is referred to and it has been demonstrated quite categorically for the US urban system comprising some 366 Metropolitan Statistical Areas. However in this talk, I will report the work of our group which has demonstrated equally conclusively that no such positive allometry exists for the UK urban system. Much of our analysis rests on the fact that it is extremely difficulty to define cities categorically with respect to their physical extent over which we need to measure their attributes – population, income etc. – and to this end we explore many thousands of realisations of UK cities, demonstrating that in general for most reasonable city sizes, there is no superlinearity – that is defining economies of agglomeration is problematic, while London is a massive outlier. This suggests that the world of the UK cities at least is much more complex than the US, and that one explanation is that the UK is one large, relatively integrated urban system – city even – while the US is still composed of distinct cities that have not yet become integrated in quite the same way that has happened in the UK. We develop these ideas using various definitions of cities, particularly as networks of streets that we explore using percolation theory.
Programme à venir
Jean-Philippe Bouchaud (Ecole Polytechnique, co-fondateur de Capital Fund Management)
Impact anormal et fragilité intrinsèque des marchés financiers
Comment le fait même d'acheter ou de vendre une action (ou tout autre actif) affecte-t-il son prix ? C'est une question fondamentale, à la fois pour comprendre le fonctionnement des marchés financiers et leur stabilité, mais aussi le débat très actuel (mis en exergue par le prix Nobel d'economie 2013) sur l'"efficience" des marchés. La surprise est que l'impact moyen d'une transaction de volume total Q n'est pas linéaire en Q, meme à très petit volume, mais se comporte plutot comme la racine de Q. Ceci signifie que la réponse linéaire est formellement divergente, comme pour un système "critique". Nous tenterons de développer un modèle théorique qui permet d'expliquer un tel effet, confirmé par des simulations numériques. Notre scenario suggère que les marchés financiers sont intrinsèquement fragiles et turbulents.
Peter Markowich (Cambridge)
Price formation modeling with PDE: from Boltzmann to free boundaries
We present a derivation of the Lasry-Lions free boundary price formation model through a scaling limit of a Boltzmann type system for the buyer and vendor densities. This Boltzmann system is obtained by simple and very heuristic kinetic modeling, describing an economic market where large numbers of buyers and vendors trade one good continuously.
Mark Lewis (Canada Research Chair in Mathematical Biology / University of Alberta)
Mathematical Models for Territorial Interactions
Mathematical models can help us understand the formation of complex spatial patterns, including the territories of wolves and coyotes. Here scent marks provide important clues regarding the use of space. In this talk I will show how biologically-based mechanistic rules can be put into a mathematical model which predicts the process of territorial formation as individuals create and respond to scent marks. The model predicts complex spatial patterns which are seen in nature, such stable "buff er zones" between territories which act as refuges for prey such as deer. The mathematical work is supported by detailed radio-tracking studies of animals. I will also employ the approach of game theory, where each pack attempts to maximize its fitness by increasing intake of prey (deer) and while decreasing interactions with hostile neighboring packs. Here the predictions are compared with radio-tracking data for wolves and coyotes. Finally I will show how a version of the territorial model has been applied to human populations in understanding spatial patterns arising from conflict between urban gangs.
Amandine Aftalion (Directrice de recherche au CNRS, Université de Versailles Saint-Quentin)
Questions de modélisation dans le sport : existe-t-il une stratégie de course optimale?
Résumé : Le but de cet exposé qui rapporte un travail en collaboration avec F.Bonnans (Inria) est de présenter un système d'équations différentielles décrivant l'évolution de l'énergie anaérobie et de la vitesse d'un coureur et d'en déduire une stratégie de course optimale. Ces modèles s'appliquent en particulier à la course à pied et au cyclisme. Notre modèle repose sur le principe fondamental de la dynamique et utilise des bilans d'énergie, une analogie hydraulique et des liens de contrôle entre les différentes variables. Notre résultat principal montre que varier sa vitesse (plutôt que courir à vitesse constante) permet de courir plus longtemps. Ce résultat théorique rejoint des observations physiologiques. En utilisant la théorie du contrôle optimal, nous obtenons des preuves sur la structure de la course optimale et relions le problème à une formulation relaxée où la force de propulsion représente une probabilité de distribution plutôt qu'une fonction du temps. L'analyse mathématique conduit à mettre une borne sur les variations de la force de propulsion afin d'avoir un modèle plus réaliste, donnant lieu à des variations de vitesse. Nous présenterons également des simulations numériques de ce système. Celles-ci reproduisent fidèlement des mesures physiologiques réalisées sur des coureurs de haut niveau. L'exposé s'achèvera par des extensions possibles des modèles.