Bachelor Thesis

This thesis studies the classical Sewing Lemma and Young’s integration theory, which provides a framework for computing integrals of the form $\int fdg$ for Hölder continuous functions $f$ and $g$ with exponents $\alpha +\beta >1$. Then we present a non-commutative extension of the Sewing Lemma, using it to construct solutions to differential equations involving matrix-valued functions.

The thesis was conducted under the supervision of Professor Dario Trevisan.

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Selected Bibliography:

• L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Mathematica 67, 1936.
• M. Gubinelli, Controlling rough paths, Journal of Functional Analysis 216, 2004.
• Denis Feyel, Arnaud de La Pradelle, and Gabriel Mokobodzki, A non-commutative sewing lemma, Electronic Communications in Probability 13, 2008.
• Wolfgang Scherer, Mathematics of quantum computing. An introduction, Springer, 2019.
• Eugene Stepanov and Dario Trevisan, On exterior differential systems involving differentials of Hölder functions, Journal of Differential Equations 337, 2022.