What makes a neural network better, or worse, at fitting certain tasks? This question is arguably at the heart of neural network architecture design, and it is remarkably hard to answer rigorously. Over the past few years, there have been a plethora of attempts, using various facets of advanced mathematics, to answer this question under various assumptions. One of the most successful directions — algorithmic alignment — assumes that the target function, and a mechanism for computing it, are completely well-defined and known (i.e. the target is to learn to execute an algorithm). In this setting, fitting a task is equated to capturing the computations of an algorithm, inviting analyses from diverse branches of mathematics and computer science. I will present some of my personal favourite works in algorithmic alignment, along with their implications for building intelligent systems of the future Part of the LMS/IMA Joint Meeting on 'The Mathematical Foundations' of AI, which took place on Friday 13 October 2023 at De Morgan House, London and online via Zoom. ========== The London Mathematical Society has, since 1865, been the UK's learned society for the advancement, dissemination and promotion of mathematical knowledge. Our mission is to advance mathematics through our members and the broader scientific community worldwide. For further information: ► Website: https://www.lms.ac.uk/ ► Events: https://www.lms.ac.uk/events ► Grants and Prizes: https://www.lms.ac.uk/grants-prizes ► Publications: https://www.lms.ac.uk/publications ► Membership: https://www.lms.ac.uk/membership Follow us: ► Twitter: https://twitter.com/LondMathSoc ► Facebook: https://www.facebook.com/londonmathematicalsociety ► LinkedIn: https://www.linkedin.com/company/the-london-mathematical-society/ ► Youtube: @LondonMathematicalSociety