Rachel Ward, From bits to bots: a mathematical perspective on generative AI
Data science and machine learning have undergone profound transformations in recent years, driven by the exponential growth of computational power and available data. In this talk, we will discuss the evolution from signal processing over half a century ago to the rise of machine learning and generative AI, highlighting mathematical foundations such as information theory, probability, linear algebra, and optimization. While modern AI research is becoming more empirical in recent years, we finish by highlighting open questions and directions where mathematicians and scientists are crucial for making foundational advancements.
Martin Hairer, (In)stability of stochastic systems
Motivated by natural questions arising in fluid dynamics, we consider random evolutions that exhibit an invariant linear subspace. A natural question is to quantify how (un)stable such a space is under the dynamic. In particular, we show that, for a class of random velocity fields that cover solutions to the stochastic Navier-Stokes equations, the Eulerian Lyapunov exponent for passive scalar advection / diffusion is finite. The proof relies on constructive bounds on the projective process that allow us to keep track of its dependence on the diffusion coefficient.
