Title: “An Introduction to the Theory of Rough Paths”

Abstract: The theory of rough paths, developed by Terry Lyons, provides a robust mathematical framework to analyze controlled differential equations driven by irregular signals, including those that are highly oscillatory or exhibit non-smooth behavior. Traditional differential equation methods rely on smoothness assumptions, which break down in the presence of rough signals, such as those encountered in stochastic processes like Brownian motion. In this talk, I will introduce the fundamental concepts of rough path theory, focusing on intuition, motivation, and basic constructions, without assuming prior familiarity with the subject. We will explore how rough paths extend classical integration concepts and how these ideas open doors to studying complex stochastic and deterministic systems. Applications in stochastic differential equations and stochastic analysis will also be briefly discussed to illustrate the broad impact of the theory.