• Talk: An algebraic geometry of paths via the iterated-integral signature

    I gave this talk on March 20, 2024 at the Nonlinear Algebra Seminar at MPI MiS Leipzig. Abstract: Contrary to previous approaches bringing together algebraic geometry and signatures of paths, we introduce a Zariski topology on the space of paths itself, and study path varieties consisting of all paths whose signature satisfies certain polynomial equations. Particular emphasis lies on the role of the non-associative halfshuffle, which makes it possible to describe varieties of paths that satisfy certain relations all along (...)

  • Lecture: Stieltjes integration and iterated integrals

    Semester schedule: Lecture Monday 12:15-13:45 (starting April 17) Exercise Class Wednesday 12:15-13:45 (starting April 26) Potsdam-Golm Mathematics Institute (Haus 9), Room 1.22 or Zoom The exercise class is of course optional for researchers joining us, students wishing to take the exam are expected to regularly present and discuss their exercise solutions in the class on Wednesday. As the first lecture will be devoted to an overview of the contents and to a repetition of integration against a one-dimensional variable, it (...)

  • Talk: Orthogonal and special linear invariants of paths via the iterated-integral signature: Overview and new developments

    I gave this blackboard talk on May 13, 2022 at the 15th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis. The talk was largely based on joint work with Joscha Diehl, Micheal Ruddy, Jeremy Reizenstein and Nikolas Tapia. Abstract: Looking at the action of the orthogonal group, we apply Fels-Olver’s moving frame method paired with the log-signature transform to construct a set of integral invariants for curves in R^d from the iterated-integrals signature. In particular we show that (...)

  • Research idea: An algebraic geometry of paths via the iterated-integral signature

    I gave this talk on March 31, 2022 (which also happens to be Trans Day of Visibility) at Technische Universität Berlin for the Rough Algebra Day organised by Yannic Vargas, Sylvie Paycha, Bernd Sturmfels and Peter Friz. Abstract: In this presentation we discuss research opportunities starting from the observation that the iterated-integral signature constructs a correspondence between finite or infinite dimensional varieties of paths and finite or infinitely generated ideals in the shuffle algebra. This leads to a notion of (...)

  • Online talk: In how far is the path characterized by the signature?

    I gave this talk on February 04, 2022 in the analysis seminar of Sylvie Paycha of University of Potsdam, which happened online on zoom. Abstract: The aim of our project is to give a definite answer to the question posed (in its earliest form) by Chen some 65 years ago, whether or not the signature, i.e. the full time increment footprint in the group-like elements, characterizes the path. We intend to show that the signature uniquely characterizes all reduced rough (...)