Among the challenges of contemporary physics research, many use powerful and effective mathematical tools, leading to a high level of sophistication and specialisation. Among these mathematical tools, the ISA project focuses on three precise, highly interconnected techniques: integrability, scattering theory, and spectral methods. These cross-cutting methods are used in two distinct but methodologically complementary physical contexts. The first context is that of integrable quantum models, where the scattering properties of flat waves are at the origin of powerful mathematical properties, but whose origin sometimes resists rigorous analysis. This is particularly the case in quantum field theories, for example in gauge-gravity dualities. The second context concerns integrated optical components developed using nanomaterials. It gives rise to technological applications as part of an IMB/ICB collaboration aimed at the development of optical sources and ultrasensitive sensors, especially for medical diagnosis. The interdisciplinary approach that is adopted combines theoretical study (IMB) with laboratory experiments (ICB), to create synergy that goes beyond the methods currently used independently by each community.