Mathematics

General objectives: Many models in mathematical physics and natural sciences in general have variational principles (the principle of minimal energy, minimal action, ...) which describe their equilibrium configurations and dynamic evolutions. The aim of the course is to make students aware of the variety of problems that can be addressed with variational techniques and to provide them with the basic tools and mathematical language for analyzing the models arising in natural sciences. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results on the direct method of calculus variations, conditions for semicontinuity, asymptotic analysis via Gamma convergence, and she/he will be able to apply this methods in various contexts about which they will be provided the functional bases at least in dimension 1 (integral functionals and Sobolev spaces, geometric functionals and elements of geometric measure theory). Apply knowledge and understanding: at the end of the course the student will be able to begin the study of advanced calculus of variations. She/he will also be able to formulate a simple variational model (for example linked to a specific application) and analyze its asymptotic behavior or identify the characteristics that make it a robust model. Critical and judgmental skills: The student will have the basics to connect and use tools covered in various moments of his studies ranging from analysis, mathematical physics and to probability. She/he will therefore be able to appreciate the interest of a mathematical question in relation also to its use to answer a question coming from an applied problem. Communication skills: ability to rigorously expose the theoretical contents of the course and also ability to formulate the problem under consideration by understanding the role of deriving the right model and its analysis. Ability to explain moreover the results in the language related to the application under consideration, potentially understandable by non expert in calculating variations. Learning ability: the acquired knowledge will allow to face a possible master's thesis work in the field of applied mathematics in natural sciences both with a more theoretical approach and in connection with the analysis of a specific model of interest for applications.