Associate Professor Nicolas Boumal will talk about: 'Smooth, globally Polyak-Łojasiewicz functions are nonlinear least-squares'

Polyak-Łojasiewicz (PŁ) functions abound in the literature, especially in nonconvex optimization. When they are also smooth, they become surprisingly simple---with an exotic twist. The plan is for us to discover the structure of those functions and of their sets of minimizers via gradient flow and fiber bundles.

Joint work with Christopher Criscitiello and Quentin Rebjock.

Professor Andrew Saxe will talk about; 'Demystifying depth: principles of learning in deep neural networks' 

Deep neural networks have revolutionized artificial intelligence, yet their inner workings remain poorly understood. This talk presents mathematical analyses of the nonlinear dynamics of learning in several solvable deep network models, offering theoretical insights into the role of depth. These models reveal how learning algorithms, data structure, initialization schemes, and architectural choices interact to produce hidden representations that afford complex generalization behaviours. A recurring theme across these analyses is a neural race: competing pathways within a deep network vie to explain the data, with an implicit bias toward shared representations. These shared representations in turn shape the network’s capacity for systematic generalization, multitasking, and transfer learning. I will show how such principles manifest across diverse architectures—including feedforward and linear attention networks. Together, these results provide analytic foundations for understanding how environmental statistics, network architecture, and learning dynamics jointly structure the emergence of neural representations and behaviour.