This research school focuses on cutting-edge techniques in harmonic analysis and their applications to differential operators with rough coefficients. It will be the 27th edition of the internationally recognized series International Internet Seminar on Evolution Equations. The famous Kato problem and its solution at the beginning of the 2000s has initiated a number of developments in the interplay of at least three mathematical fields – harmonic analysis, elliptic PDEs, functional calculus – leading to a surge of spectacular new results over the last two decades. A key idea behind most of the developments can roughly be described as replacing the Laplace operator, that plays a fundamental role in each of the three fields, by a more general differential operator with rough coefficients. The new analytic concepts that emerged from pursuing this program have become a basis for modern research in analysis. With the proposed research school we aim to pass these techniques and tools over to the next generation of mathematicians. The research school aims to • encourage young, promising scientists to get deeply involved into the field, • provide a platform to young scientists in a creative and benevolent atmosphere, • foster an intense dialogue and cooperation between young scientists and leading experts, • bring together experts from different directions within the broader field of evolution equations and spread the topic in our community. The International Internet Seminar on Evolution Equations is a series of seminars that first took place in 1997/98. For participating junior scientists the proposed research school is the culmination of eight months of intense acitivity around the topic.