Machine Learning has emerged as one of the most transformative forces in science and technology. Beneath its powerful algorithms lie mathematical foundations deeply rooted in classical disciplines such as Applied Mathematics and Systems Control. This lecture adopts a mathematician’s perspective to examine why Machine Learning works so effectively and how its data-driven paradigms can be rigorously integrated into traditional analytical frameworks.
We will revisit the historical and conceptual links between Machine Learning and Systems Control—also known as Cybernetics—a field shaped by the pioneering ideas of Ampère and Wiener. Their parallel evolution reveals a deep mathematical unity and highlights the power of mathematics to model complex systems and drive innovation.
This dual perspective is mutually enriching. On one hand, Machine Learning raises fundamental mathematical questions that challenge and inspire the mathematical community. On the other, it offers opportunities to expand the scope of classical applied mathematics by developing hybrid methodologies that integrate data-driven insights.
The lecture will conclude by outlining promising directions for future research at the intersection of Machine Learning, Applied Mathematics, and Control Theory.

I will discuss the large-scale/ long-time behavior of a Brownian particle in a random, incompressible (divergence-free) vector field. If this vector field has correlations which decay sufficiently slowly, then the behavior of the particle is superdiffusive instead of diffusive. In particular, after time $t$, the particle will typically be a much larger distance than $\sqrt{t}$ from the origin. This phenomenon was explained in the physics literature in the late 1980s, using heuristic renormalization group arguments, as a divergence of effective diffusivity as we zoom out to large length scales. I will discuss recent mathematical innovations based on some analytic/PDE methods that allow us to make some of these renormalization group arguments rigorous.
​​​​​​​(This is joint work with A. Bou-Rabee and T. Kuusi.)

We discuss several models of opinon dynamics, where agents tend to align their opinions with their neighbours when they interact. We start with the classical voter model, where opinions can have only two values. Then we explain the Deffuant model and present several open conjectures. We give some results and conjectures about the compass model which is a variant of the Deffuant model where opinions take values on the unit circle. Finally we discuss the averaging process on infinite graphs.
The results were obtained in collaboration with Markus Heydenreich and Timo Vilkas.