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Modeling with Measures: from Structured Populations to Crowd Dynamics
Four different communities met at the Lorentz Center: (1) Measure-valued balance laws; (2) Structured-population dynamics and selection-mutation models using measure formulation; (3) Abstract functional analysis and dynamical systems in spaces of measures on metric spaces; and (4) Modeling, analysis and simulation of collective dynamics applied to nonlocal transport of mass measures capturing the dynamics of human crowds in heterogeneous domains.
The aim was threefold: (1) Identify measure-valued balance scenarios (relevant for practical applications like population balances in heterogeneous environments) which are really open mathematically; (2) Establish the mathematical/modeling source of difficulties; (3) Start discussions and collaborations to solve some of the identified open problems.
As a direct consequence of the open problems presented by senior scientists, the 23 participants have formed five working groups:
(i) Boundary conditions for measures-valued balance laws;
(ii) Conservation laws with stochastic initial data, coefficients, productions;
(iii) Hyperbolic-elliptic systems;
(iv) Stability for measures-valued structured-population models;
(v) Operator splitting techniques for equations in metric spaces.
N.B. Participants came from 7 countries. There were 5 female scientists. The junior-to-senior researchers ratio was 12/11.
We have succeeded to identify a couple of interesting open problems, which were able to generate a couple of basic questions triggering our attention. The titles of the working groups indicate the directions which were taken. Within the working groups contacts have evolved and are now followed-up by bilateral visits. The junior researchers have benefitted much from the tutorial lectures and the discussions afterwards.
There will be a special issue of the AIMS journal Mathematical Biosciences and Engineering (MBE) devoted to the topic of the workshop 'Modeling with Measures: from Structured Populations to Crowd Dynamics'. Any participant or group of participants is invited to submit a paper to this special issue.
From the feedback of the participants, we feel strongly that the workshop was a success. The participants have agreed that a follow-up workshop in the same research direction (and organization spirit) would be very much welcome. Potential candidate locations for this are CIRM/Marseille (France) and the Banach Center (Poland).