Foundational Course

This module is designed to teach you a lot of the fundamentals of analysis. The arguments that we will learn here will come up time and time again in future modules, so be sure to understand them well, and remember them.

This module is designed to teach you a lot of the fundamentals of analysis. The arguments that we will learn here will come up time and time again in future modules, so be sure to understand them well, and remember them.

This course covers the basics of ordinary differential equations. These are differential equations where the derivatives are taken in one variable only. They come in many forms and so we need a lot of strategies to tackle them.

Differential equations are an important topic in a huge range of applied topics, but they are especially vital to understand in physics. We hope that what you learn in this course will be of practical use in your future endeavours. 

For each of the lectures, we have a multi-choice quiz for you to attempt. Do give them a good try - this is a particular module where practicing solutions is going to help much more than writing or studying proofs.

It will also serve as a prerequisite for the upcoming summer course on the Mathematics of Flow and Diffusion Models.

This course aims to cover all of the basics you'll need for studying measure theory and functional analysis. The goal of this module is to work our way up to the definition of the Lebesgue integral and its properties. To justify it, we begin with the Riemann integral, and show some cases where it is incapable of providing answers, thus motivating the need for a stronger theory. 

It will also serve as a prerequisite for the upcoming summer course on the Mathematics of Flow and Diffusion Models.

This course provides a concise yet thorough introduction to the foundational concepts of group theory, topology, and manifolds, tailored for learners who seek a robust understanding of these mathematical structures. It emphasises the core principles, essential techniques, and key results that form the backbone of modern applications in fields such as artificial intelligence (AI), machine learning (ML), and quantum computing. The course strikes a balance between theoretical rigour and practical relevance, ensuring that participants gain both deep insights and the ability to connect abstract ideas to real-world problems.

In addition, this course serves as an essential prerequisite for anyone planning to explore advanced topics in Lie Groups. It introduces and develops the critical foundational notions that are indispensable for a meaningful engagement with Lie theory, including group actions, topological spaces, and smooth manifolds. Mastery of these concepts will provide a seamless transition into the study of continuous symmetries and their applications across physics, geometry, and computational sciences.

We're excited to have you join us on this journey to master the mathematical foundations and practical applications of linear algebra in machine learning. This course is designed to provide a rigorous understanding of advanced linear algebra concepts while connecting them directly to real-world machine learning algorithms and use cases.

Welcome to Finite-Dimensional Hilbert Spaces: Your Essential Step into Quantum Computing and Quantum Mechanics!

Whatever your background, as long as you bring a foundational understanding to build on, this course is designed to make advanced mathematical concepts accessible and engaging. Finite-dimensional Hilbert spaces are not just a mathematical curiosity; they’re the initial step for understanding the mathematics behind quantum mechanics and computing. If you've been curious about the structure behind quantum systems or want to lay a solid foundation for future studies in infinite-dimensional spaces, you're in the right place!

In each of the sections of this course, you will find a lecture explaining the content, quizzes that will guide you in applying the concepts, and additional pen-and-paper exercises encourage you to deepen your understanding at your own pace. We strongly recommend to tackle all of them, as they will help you build intuition and rigor!!

Knowing about methods for constructing proofs is a vital, but often overlooked part of any researcher's journey. In fact, if you didn't study mathematics at university level, it is likely that you were never formally introduced to proof-writing strategies! We aim to remedy this fact with this mini-course, which will give you the tools to be able to write mathematical arguments clearly and precisely, and teach you some of the common methods that you will encounter throughout your time in the QF academy.

As these methods are so ubiquitous in graduate-level mathematical education, this course should be seen as a prerequisite to the other mathematics courses in the academy.