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In this talk, I will discuss how to advance the mathematical understanding
of multi-loop, multi-leg, and multi-scale Feynman integrals and to bridge
the gap between the mathematical framework of scattering amplitudes and
their phenomenological applications. I will highlight their broad relevance,
not only in achieving groundbreaking predictions for LHC observations but
also in extending their impact beyond elementary particle physics. For
instance, these computations play a crucial role in the precise calculation
of bound-state observables and have contributed to breakthroughs in
gravitational wave research.