- Unit Coordinator: Marco Di Francesco, Antonio Esposito
- Programme: InterMaths
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of L'Aquila
- Language: English
- Delivery: In-class
- Aims:
At the end of the course, the student will be familiar with multi-agent systems models of discrete type (both deterministic and stochastic), with their meso-scopic formulation, with their continuum formulation, both of first and second order, and with their formulation on a graph. The students will acquire the mathematical techniques to solve those models suitably and will be able to use those models in various interdisciplinary applications and to adapt them to specific problems in contexts of interest in health-care systems such as diagnostics and imaging, neural networks, genetics, epidemiology, dynamic data management, and biological aggregation phenomena in physiology.
- Content:
- Discrete particle systems for interacting agents. Models with external field. Models with nonlocal aggregation/repulsion forces. Models with alignment, self-propulsion and friction. Swarms models (Vicsek) . Opinion models (Sznajd, Krause). Examples of asymptotic behaviour. The stochastic case.
- Control for discrete models. Mean-field games. Application to optimisation problems. Many species models and models with species transitions. Applications in genetics, imaging, and data science.
- Complementary topics of abstract measure theory. Measure topologies. Transport of measures.
- Mesoscopic models of Vlasov type. Derivation as mean field limits from discrete particle models. Formal derivation of continuum second order models. Derivation of first order models in friction dominated regimes.
- Derivation of linear and nonlinear diffusion models from particle systems. Existence of solutions to the nonlinear diffusion equation. Asymptotic self-similar behavior.
- Aggregation-diffusion equations. Existence of solutions with linear diffusion. Existence in the diffusion-less case and formation of clusters in finite time. Stationary states with quadratic diffusion. The case of many species. Application to epidemiology.
- Introduction to graph modelling of nonlocal type. Applications to neural networks.
- Probabilistic label-switching models and applications.
- Pre-requisites:
Ordinary differential equations, real and functional analysis
- Reading list:
Lecture notes will be provided.
- Unit Coordinator: Debora Amadori
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of L'Aquila
- Language: English
- Aims:
Aim of the course is to present some mathematical models currently used in the analysis of collective phenomena, such as vehicular and pedestrian trauc, and locking phenomena. Emphasis will be given to the mathematical treatment of specific problems coming from real world applications.
- Content:
- Macroscopic trauc models. LWR model, its derivation. Fundamental diagrams. The Riemann problem, examples. Second order models for trauc low: Payne-Whitham model, description, drawbacks; Aw-Rascle model, shocks description, domains of invariance, instabilities near vacuum.
- Theory: systems of conservation laws, strict hyperbolicity, Rankine-Hugoniot conditions; Lax admissibility condition. The Riemann problem for systems: the linear case; GNL and LD fields; rarefactions and contact discontinuities. BV functions, examples and properties. A compactness theorem.
- Wave front tracking algorithm: approximate rarefactions, possible types of interactions. Bounds on number of waves and on total variation. Compactness of approximate solutions. The initial-boundary value problem on the half line: boundary Riemann problem, interactions with the boundary, control of the total variation by means of a Lyapunov-type functional. The Toll gate problem.
- Networks, basic definitions, conservation of the lux. Examples. Distributions along the roads, maximization of the lux. Riemann problem on a junction composed by 2 incoming roads and 2 outgoing roads. The case of 2 incoming roads and 1 outgoing road: the "right of way" rule. Junction between one incoming and one outgoing road, different luxes.
- Pedestrian low: normal and panic situation. Macroscopic models for evacuation, conservation of "mass", eikonal equation. The Hughes model for pedestrian low. The eikonal equation: non uniqueness, viscosity solutions, relation with vanishing viscosity approximation. The Hughes model in one space dimension. Curve of turning points, Rankine-Hugoniot conditions. The case of constant initial density and of symmetric initial data; conservation of the left and right mass; an example with mass exchange across the turning point. Macroscopic models for pedestrian low that include: knowledge of a preferred path, discomfort from walking along walls, tendency of avoiding high densities of pedestrian in a neighborhood (nonlocal term of convolution type), angle of vision, obstacle in the domain. Linearized stability around a smooth solution.
- Introduction to the theory of locking. Examples: Krause model for opinion dynamics, Cucker-Smale model, model for attraction-repulsion phenomena. The Cucker-Smale locking model: basic properties, estimates on the kinetic energy. A "locking theorem": proof by bootstrapping method (Ha and Tadmor). Some drawbacks of the model. Introduction to the kinetic limit for locking: the N-particle distribution function, Liouville equation, marginal distribution, continuity equation. The formal mean-Seld limit: a Vlasov- type equation.
- Reading list:
M.D. Rosini, Macroscopic models for vehicular lows and crowd dynamics: theory and applications. Springer. 2013.
M. Garavello, B. Piccoli, Trauc low on networks. Conservation laws models. AIMS Series on Applied Mathematics. 2006.
- Unit Coordinator: Josef Šlapal
- ECTS Credits: 4
- Semester: 2
- Year: 2
- Campus: Brno University of Technology
- Aims:
The aim of the course is to show the students possibility of a unified perspective on seemingly different mathematical subjects.
- Content:
1. Sets and classes
2. Mathematical structures
3. Isomorphisms
4. Fibres
5. Subobjects
6. Quotient objects
7. Free objects
8. Initial structures
9. Final structures
10. Cartesian product
11. Cartesian completeness
12. Functors
13. Relection and corelection
- Pre-requisites:
Students are expected to know the mathematics taught within the bachelor's study programme and the graph theory taught in the master's study
- Reading list:
- Jiří Adámek, Theory of Mathematical Structures, D. Reidel Publ. Company, Dordrecht, 1983.
- A.Adámek, H.Herrlich. G.E.Strecker: Abstract and Concrete Categories, John Willey & Sons, New York, 1990
- Additional info:
The course will familiarise students with basic concepts and results of the theory of mathematical structures. A number of examples of concrete structures will be used to demonstrate the exposition.
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of Aveiro
- Language: English
