How to self study pure math - a step-by-step guide

Summary

This video provides a comprehensive collection of resources for self-studying pure mathematics. It covers essential subjects like linear algebra, real analysis, topology, complex analysis, group theory, Galois theory, differential geometry, and algebraic topology. For each topic, the video recommends the best books and videos, many of which are freely available online. Notable resources include 'Linear Algebra Done Right' by Sheldon Axler, 'Understanding Analysis' by Stephen Abbott, and visual guides for complex analysis. The video also highlights various university lecture series and problem sets, making it an invaluable guide for independent learners.

Takeaways

• 😀 This video provides a comprehensive collection of resources for self-studying pure mathematics, including books, videos, and online materials.
• 😀 The video covers a range of topics that one might encounter in a pure math degree, from first-year linear algebra to advanced topics like algebraic topology and differential geometry.
• 😀 For linear algebra, 'Linear Algebra Done Right' by Sheldon Axler is recommended due to its emphasis on examples and avoidance of determinants for learning the subject.
• 😀 'Understanding Analysis' by Stephen Abbott is the recommended book for real analysis, with a focus on rigorous proofs and a deep dive into calculus concepts from a proof's perspective.
• 😀 The University of Toronto provides excellent online notes for point set topology, which are filled with examples and a list of problems that reinforce the material.
• 😀 Complex analysis is introduced in a visual and intuitive way with 'Visual Complex Functions' by J. C. Polking, which uses color and phase portraits to illustrate key ideas.
• 😀 For group theory, the lectures by Professor Benedict Gross are recommended for their clarity and engaging teaching style. The book 'Topics in Algebra' by Herstein is also suggested for further study.
• 😀 Galois theory is introduced as a way to show that polynomials of degree 5 or higher do not have a general solution in radicals. Professor Tom Lanester’s notes are recommended for further learning.
• 😀 Differential geometry is the study of curved surfaces, and 'Introduction to Differentiable Manifolds and Riemannian Geometry' is suggested for learning this topic with both rigorous content and intuitive discussions.
• 😀 'Algebraic Topology' by Allen Hatcher is recommended for its clear illustrations and exercises, providing a solid foundation in homotopy, homology, and co-homology.
• 😀 If you prefer video learning, many of the mentioned resources come with accompanying video lectures or playlists, such as those by Sheldon Axler and Pierre Albin, which are great for reinforcing concepts visually.

Q & A

What is the main focus of this video?

The video provides a collection of resources for self-studying pure mathematics, covering various branches of math typically encountered during a pure math degree, from introductory topics to advanced subjects.

Which topics are typically covered in the first year of a pure math degree?

In the first year, students typically study linear algebra and real analysis.

What makes 'Linear Algebra Done Right' by Sheldon Axler suitable for self-study?

'Linear Algebra Done Right' is ideal for self-study due to its emphasis on concrete examples and a clear progression of concepts. It avoids using determinants and introduces the subject from a more abstract perspective, with exercises to reinforce learning.

What is the key feature of the book 'Understanding Analysis' by Stephen Abbott?

'Understanding Analysis' is known for its rigorous yet accessible approach to real analysis, covering topics like sequences, series, limits, derivatives, and integrals with a strong focus on proof-based learning.

Why is point set topology considered more abstract than calculus and linear algebra?

Point set topology takes concepts from calculus and linear algebra and applies them in a more general and abstract context, helping students explore the foundations of topology and its applications in higher mathematics.

What is a unique feature of 'Visual Complex Functions' in learning complex analysis?

'Visual Complex Functions' uses colorful phase portraits and illustrations to visually demonstrate the core ideas of complex analysis, making it easier to develop an intuitive understanding before diving into more technical details.

What role does group theory play in mathematics, and where can one find good resources to learn it?

Group theory is the study of symmetry, and it's foundational to many areas of mathematics. A recommended resource for learning group theory is Professor Benedict Gross’s lecture series, which is clear and engaging. For book learners, 'Topics in Algebra' by Herstein is highly recommended.

What is the main subject of Galois theory, and why is it significant?

Galois theory applies group theory to study the solvability of polynomial equations. It demonstrates that polynomials of degree 5 and higher do not have a general formula in radicals, which is a key result in abstract algebra.

How does the book 'Introduction to Differentiable Manifolds and Riemannian Geometry' approach the study of differential geometry?

This book provides a rigorous treatment of differential geometry, introducing manifolds and calculus in higher-dimensional space before exploring vector and tensor fields, integration on manifolds, and the main theorems of differential geometry such as Stokes's Theorem and Gauss-Bonnet Theorem.

What is algebraic topology, and what resources are recommended for studying it?

Algebraic topology uses tools from group theory to study topological spaces. The book 'Algebraic Topology' by Allen Hatcher is highly recommended for its clear explanations and numerous exercises. Professor Pierre Albin’s lecture series on YouTube, following Hatcher’s book, is another valuable resource.