Mr Oluwadamilola (Dami) Fasina will talk about; 'Learning with tensor paraproducts'

We discuss computational (Neural FIM) and analytical (tensor paraproducts) tools for learning structure of sets. In the first situation we focus on learning the metric amongst elements of a statistical manifold. To do so, we design a neural network which enables one to compute the Fisher information metric (FIM), so long the Jensen-Shannon divergences amongst probability distributions on the statistical manifold are preserved during training. In the second situation we focus on analyzing the structure of function compositions through separation of its low and high frequency components. This is accomplished by elaborating on J.M. Bony’s celebrated work on paraproducts by discretizing and allocating distinct scaling parameters along each dimension of the support of a function composition (with a prescribed regularity), permitting finer analytical control. A consequence of this extension is highlighted with a discussion of the regularity gains of kernels of integral operators. 

The Kervaire conjecture was formulated around 1963 after a conversation between Kervaire and Baumslag. It states that adding a generator and then a relator to a non-trivial group always yields a non-trivial group. To this day, the conjecture remains unproven in its most general form; however, it has been shown under certain additional hypotheses, either on the new relator or on the original group. For instance, the result holds for locally indicable groups and for locally residually finite groups. In this talk, I will explain Klyachko’s proof of the conjecture for torsion-free groups, which uses a funny property of the sphere known as the Car Crash Theorem, and van Kampen pictures. I will also discuss how these techniques were generalised by Fenn and Rourke to study equations over torsion-free groups defined by a large class of words (amenable words).