Emilio Rossi Ferrucci (Imperial College, London)
I will begin by introducing geometric (controlled) rough paths from the algebraic/combinatorial point of view, as done in recent work with Cass, Litterer and Driver. Geometric rough paths can be viewed as theories of integration against irregular multidimensional paths, which satisfy an integration by parts identity, expressed algebraically through the shuffle relation. Gubinelli's controlled paths represent Taylor-type expansions performed w.r.t. to a reference rough path. A cornerstone of our treatment of the theory is the use of the ordered shuffle for the definition of a rough path "above" a controlled path. I will end by speaking about some ongoing work of my own, that explores which aspects of the theory change in the setting of branched rough paths, which, unlike the geometric ones, are not required to satisfy the laws of ordinary calculus.
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