ArpiLYSM3: VERSUS, oppositions in Mathematics and more

The aim of the school is to consider multidisciplinary aspects inside Mathematics. In many situations, mathematical concepts appear in different paradigms.

These offers different point of views which deserve to be compared with rispect to their analogies and differences. These comparisons involve usually different extra-mathematics fields such as Physics, Biology, Economics, Social Sciences, etc. that will be handled in this school. 

 

The main streams  considered will include: 

- Typical vs. Untypical: a journey through large deviations, by Raphaël Chétrite

 

- ODEs vs. PDEs: the large number agent systems, by Stefano Rossi

 

This course discusses the rigorous derivation of effective kinetic PDEs starting from the dynamics of N particles described by a system of ODEs and  the comparison between the behavior of finite and  infinite-particle sytems,  with regard to long-time behavior and the emergence of irreversibility.

In this course, we show how most concepts from (classical) optimal transport find their quantum counterparts in quantum mechanics through a “canonical” quantization of the Wasserstein distance. These concepts include Brenier’s theorem, bipartite matching, and Kantorovich duality.

    - Order vs. Disorder in neural networks: a journey through memories and forgetting, by Alberto Fachechi.

This course introduces the fundamental principles of associative neural networks through the lens of the equilibrium statistical mechanics of spin glasses. The course highlights how  analytical techniques  provide a quantitative understanding of memory retrieval or not, storage capacity, and phase transitions.

    - Finite vs. Infinite dimension integrability, between geometry and analysis of dynamical systems, by Lorenzo Baroni.

This course aims to extend ideas and techniques from finite to infinite dimensional Hamiltonian dynamics, focusing on Liouville–Arnold integrability. By a proper  choice of topology, one provides a decomposition of the phase space as rigid as in the classical case,. A task that other approaches  miss.