13-2 Additional Pure Mathematics For Entertainment

Theoretical Physics with Mark Weitzman
18 Mar 202307:03
EducationalLearning
32 Likes 10 Comments

TLDRIn this YouTube video, the host recommends several textbooks for physicists interested in pure mathematics for entertainment. The books cover a range of topics including number theory, geometry, analysis, combinatorics, graph theory, algebra, geometry, group theory, and real analysis. Notable titles include 'Proofs from the Book,' 'Glimpses of Algebra and Geometry,' and 'Algebraic Geometry: A Computational Introduction.' The host highlights the readability and educational value of these books, with some available for free online.

Takeaways
  • πŸ“š The video recommends 'Proofs from the Book' for its collection of ingenious proofs in various areas of mathematics, including number theory, geometry, analysis, combinatorics, and graph theory.
  • 🌟 'Proofs from the Book' has received high praise with 69 five-star ratings on Amazon and has gone through six editions, indicating its popularity and quality.
  • πŸ“˜ 'Glimpses of Algebra and Geometry' is another recommended book that combines algebra and geometry, covering topics like regular solids, elliptic curves, and Riemann surfaces.
  • πŸ” The book by John Stillwell on group theory is highlighted for its approach to the subject from a quaternion perspective, making it accessible and enjoyable for undergraduates.
  • πŸ€” The speaker admits difficulty with combinatorics but recommends a book that covers basic and infinite combinatorics, including topics like incompleteness theorems and cardinal numbers.
  • πŸ“ˆ Sheldon Axler's book on measure, integration, and real analysis is praised for its approachability and is available as an open-source book online for free.
  • πŸ“ An undergraduate book on algebraic geometry is mentioned, which introduces concepts like varieties and schemes, and explains computational approaches like GrΓΆbner bases.
  • πŸ’‘ The video suggests that these books are used for entertainment and learning in between studying physics, indicating their accessibility and relevance to a broader audience.
  • πŸ“Š The speaker shares personal anecdotes about taking breaks from physics to explore these mathematical topics, showing the value of interdisciplinary learning.
  • πŸ‘‹ The video concludes with a promise to continue adding more book recommendations, suggesting an ongoing series or interest in sharing knowledge.
  • πŸš€ The mention of 'sweat all these bank failures' and market fluctuations provides a glimpse into the speaker's personal life and current events that may influence their study habits.
Q & A

  • What is the main purpose of the video?

    -The main purpose of the video is to recommend several textbooks related to pure mathematics for the entertainment and enrichment of physicists.

  • What is the first book mentioned in the video and what does it cover?

    -The first book mentioned is 'Proofs from the Book,' which covers five areas: number theory, geometry, analysis, combinatorics, and graph theory.

  • What does the term 'ingenious' refer to in the context of the book 'Proofs from the Book'?

    -In the context of 'Proofs from the Book,' 'ingenious' refers to the fantastic and creative proofs that are considered to belong in the book in God's book, a term attributed to Erdos.

  • What is the second book recommended in the video and what topics does it mix together?

    -The second book recommended is 'Glimpses of Algebra and Geometry,' which mixes together algebra and geometry, covering topics such as regular solids, elliptic curves, stereographic projections, regular polygons, and Riemann surfaces.

  • Who is the author of the book on group theory mentioned in the video?

    -The author of the book on group theory mentioned in the video is John Stillwell.

  • What is unique about the approach taken in John Stillwell's book on group theory?

    -John Stillwell's book on group theory is unique because it develops a lot of concepts from a quaternion perspective, covering generalized rotation groups, the exponential map, and the structures of Lie algebras.

  • What is the third book mentioned in the video and what is its main focus?

    -The third book mentioned is a book on combinatorics and graph theory, which focuses on basic elements of combinatorics, graph theory, and infinite combinatorics, including topics like incompleteness theorems and cardinals.

  • What is the author's opinion on measure theory and real analysis?

    -The author is not a big fan of measure theory and real analysis, but recommends a book by Sheldon Axler as the least painful approach to these subjects.

  • Which book is available as an open-source book and can be found online for free?

    -The book by Sheldon Axler on measure, integration, and real analysis is available as an open-source book and can be found online for free.

  • What is the final book mentioned in the video and what topics does it introduce at an undergraduate level?

    -The final book mentioned is an undergraduate level book that begins to introduce topics in algebraic geometry, including computational approaches and concepts like GrΓΆbner bases.

  • What does the author suggest doing with the recommended books during breaks from physics?

    -The author suggests reading these recommended books during breaks from physics, such as during a week or two off, to provide entertainment and learning.

Outlines
00:00
πŸ“š Essential Textbook Recommendations for Pure Mathematics

The speaker introduces a series of textbook recommendations for those interested in pure mathematics, particularly for physicists seeking entertainment. They highlight 'Proofs from the Book,' which is highly rated and covers five mathematical areas: number theory, geometry, analysis, combinatorics, and graph theory. The book is praised for its ingenious proofs and is suggested for readers to choose topics of interest. The speaker also recommends 'Glimpses of Algebra and Geometry,' an undergraduate-level book that combines algebra and geometry, covering a wide range of topics including regular solids, elliptic curves, stereographic projections, and more. The book is noted for its readability and enjoyable content.

05:02
πŸ” Exploring Group Theory and Advanced Mathematics

The speaker continues with further textbook recommendations, focusing on group theory and other advanced mathematical topics. They mention a book by John Stillwell, which is an undergraduate text that explores league groups and Lie algebras from a mathematical perspective, using quaternions to develop generalized rotation groups and the structures of Lie algebras. Another book discussed is on combinatorics and graph theory, which covers basic graph theory, Ramsey theory, colorings, and delves into infinite combinatorics and related topics such as incompleteness theorems and cardinal numbers. The speaker also touches on a book by Sheldon Axler on measure, integration, and real analysis, which is noted for its accessible approach and availability as an open-source book. Lastly, they introduce an undergraduate book on algebraic geometry that introduces concepts like varieties and schemes and includes computational methods like Grobner bases, important for symbolic algebra programs.

Mindmap
Keywords
πŸ’‘Pure Mathematics
Pure Mathematics refers to the study of mathematical concepts independently of any application to real-world problems. It is the branch of mathematics that focuses on the intrinsic properties of numbers, shapes, and spaces without concern for practical applications. In the video, the speaker is recommending books that are suitable for physicists who are interested in the entertainment aspect of pure mathematics, emphasizing the beauty and intellectual challenge of the subject.
πŸ’‘Proofs from the Book
This is a reference to a specific book mentioned in the script, which is a collection of elegant and ingenious proofs in mathematics. The book is well-regarded, having gone through multiple editions and receiving numerous five-star ratings on Amazon. It covers various areas of mathematics, and the term is used to highlight the book's reputation and the quality of its content, which the speaker recommends for entertainment and learning.
πŸ’‘Number Theory
Number Theory is a branch of pure mathematics that deals with the properties and relationships of numbers, particularly integers. It is one of the five areas covered in the book 'Proofs from the Book'. The speaker mentions it as one of the topics that readers can explore according to their interests, indicating the breadth of mathematical concepts that can be found in the book.
πŸ’‘Combinatorics
Combinatorics is the study of finite or countable discrete structures, particularly the counting of the possible arrangements of a set of items. The speaker mentions a book that covers this area, indicating that it includes both basic and advanced topics, such as infinite combinatorics and graph theory. The term is used to illustrate the diversity of mathematical disciplines that can be explored for entertainment and intellectual stimulation.
πŸ’‘Graph Theory
Graph Theory is a branch of mathematics that studies graphs, which are mathematical structures used to model pairwise relations between objects. The speaker discusses a book that covers the basics of graph theory, including topics like Ramsey Theory and colorings, as well as more advanced concepts like incompleteness theorems and cardinals. This term is used to show the depth and variety of topics within the field of combinatorics.
πŸ’‘Algebraic Geometry
Algebraic Geometry is a branch of mathematics that combines techniques from algebra and geometry to study problems from a geometric perspective. The speaker mentions a book that introduces this field at an undergraduate level, which is particularly interesting because it begins to delve into complex concepts like varieties and schemes. The term is used to highlight the book's approach to a higher-level area of mathematics in an accessible way.
πŸ’‘Grobner Bases
Grobner Bases are a set of techniques used in computational algebra to solve systems of polynomial equations. The speaker mentions this concept in the context of a book that covers it, indicating its importance in symbolic algebra programs like Maple and Mathematica. The term is used to illustrate the practical applications of theoretical concepts in mathematics and computer science.
πŸ’‘Measure Theory
Measure Theory is a branch of mathematical analysis that studies the theory of integration, providing a rigorous foundation for calculus. The speaker mentions a book by Sheldon Axler that covers this topic, noting that it is a less painful approach to learning about measures, integration, and product measures. The term is used to describe a foundational area of mathematics that is essential for understanding more advanced concepts.
πŸ’‘Quaternion
Quaternions are a system of hypercomplex numbers that extend the complex numbers. They are used in various areas of mathematics and physics, including the study of rotations in three dimensions. The speaker recommends a book that develops many concepts from a quaternion perspective, indicating the unique approach the book takes to explain generalized rotation groups and Lie algebras. The term is used to highlight an alternative perspective in the study of mathematical structures.
πŸ’‘Unsolvability of the Quintic
The Unsolvability of the Quintic refers to the fact that there is no general formula for solving polynomial equations of degree five or higher using only radicals. The speaker mentions this in the context of a book that covers a range of algebraic and geometric topics, including Galois Theory, which is directly related to this concept.
Highlights

The book 'Proofs from the Book' is highly recommended for its ingenious proofs and has gone through six editions with 69 five-star ratings on Amazon.

The book covers five areas: number theory, geometry, analysis, combinatorics, and graph theory.

The term 'proofs from the book' was coined by ErdΕ‘s to describe exceptionally brilliant proofs.

The book 'Glimpses of Algebra and Geometry' mixes algebra and geometry, covering a range of topics including regular solids, elliptic curves, and stereographic projections.

The book 'Glimpses of Algebra and Geometry' is a readable and enjoyable undergraduate text.

John Stillwell's book on group theory is an undergraduate text that develops concepts from a quaternion perspective.

Stillwell's book covers generalized rotation groups, the exponential map, and the structure of Lie algebras.

The book on combinatorics and graph theory is a short text covering basic graph theory, combinatorics, and infinite combinatorics.

The combinatorics book includes topics like incompleteness theorems and cardinal numbers.

Sheldon Axler's book on measure, integration, and real analysis is a less painful approach to the subject.

Axler's book is available as an open-source book, free to download as a PDF.

An undergraduate book on algebraic geometry introduces concepts like varieties and schemes with a computational approach.

The algebraic geometry book covers Grobner bases, important for symbolic algebra programs like Maple and Mathematica.

The algebraic geometry book is a comprehensive 600-page text with a companion website offering additional resources.

The speaker takes breaks from physics to read these mathematics books, finding them entertaining and informative.

The speaker plans to continue adding books to the list as they discover new ones.

Transcripts

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