Explore a 47-minute conference talk by Yves André on the direct summand conjecture and its implications in commutative algebra. Delve into the history of this elementary-looking statement proposed by M. Hochster 55 years ago, which has become a central piece in the realm of homological conjectures. Learn about André's proof of the conjecture using p-adic (perfectoid) methods and discover how the arguments have been simplified over time. Gain insights into the application of these techniques to birational geometry in mixed characteristic. This talk provides a comprehensive overview of recent developments in this field, offering mathematicians and researchers a deeper understanding of the direct summand conjecture and its far-reaching consequences in modern algebra and geometry.