Explore three key problems in the mathematics of deep learning through this 44-minute lecture by Andrew Dudzik from Google DeepMind, presented at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the challenges of aligning neural network architectures with classical computer programs, particularly for algorithmic tasks like sorting, shortest path calculations, and basic arithmetic. Discover how these alignment problems relate to fundamental mathematical concepts such as polynomial functors, cohomology, and higher categories. Gain insights into the limitations of large language models (LLMs) in performing algorithmic tasks and understand the mathematical approaches being explored to address these challenges.