Looking to start studying real mathematics.

In summary, these are some good books to start with if you want to learn rigorous mathematics applied to general relativity.
  • #1
Terilien
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0
hi physics forum. As many of you know, I've been studying general relatvity(its going quite well), but I'd like to delve into real rigorous mathematics. so essentially I'm looking for introductiosn to various topics, especially topology abstract algebra and differential geometry. Could someone tell me where to start and more importantly, how to go about doing it all?

thanks in advance.

Oh and my problems with certain concepts have vanished. It was due to notation.
 
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  • #2
There are some good books on differential geometry, for example Do Carmo's two books: "Differential Geometry of Curves and Surfaces" and "Riemannian Geometry".

If you want to see more rigorous maths applied to GR, you should (if you have not already) try Barrett O'Neill's texts on "Semi-Riemannian Geometry with Applications to Relativity", and "Geometry of Kerr Black Holes".

Another book you may want to try is "Symmetries and Curvature Structure in General Relativity" by G. S. Hall. It's mainly a math book (not physics).

I think it might be useful to start with GR-related setting (since you are already studying GR), but now focusing more on the mathematical structures, hence the recommendations.

I am sure other more experienced people can give better advise though.
 
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  • #3
Have you already done a rigorous version of (real) analysis?
 
  • #4
no but it would be nice!
 
  • #5
i do not know of any generally accepted intro to topology, which is a very large subject, but Michael Artin's Algebra is excellent for beginning abstract algebra.

topology has many sides, and one often begins with the most boring aspect, namely general, or point set topology. I myself read Kelley for that many decades ago. and many students have started with a book by munkres.

the more interesting aspects are differential and algebraic topology, or perhaps also geometric topology.

a nice little very elementary introduction, but substantial, is by chinn and steenrod. some other lovely and elementary but excellent books are by andrew wallace: intro to alg top, and diff top, first steps. thurston has a nice intro to geometry: three dimensional geometry and topology.

my compliments to you for seeking actual recommendations of good books, unlike some posters here who waste their time and ours simply railing against what they claim are all the bad books out there.

actually given that it is largely a labor of love, and requires a long apprenticeship and unrewarded effort to be able to produce one, i am impressed at the number of wonderful books available.
 
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1. What is real mathematics?

Real mathematics is the study of abstract concepts and logical reasoning to solve problems and understand the world around us. It covers a wide range of topics including algebra, geometry, calculus, and statistics.

2. How can I start studying real mathematics?

To start studying real mathematics, it is important to have a strong foundation in basic math concepts such as arithmetic, fractions, and algebra. You can then move on to more advanced topics by taking courses at a university or through self-study using textbooks and online resources.

3. Do I need to have a strong mathematical background to study real mathematics?

While having a strong mathematical background can be helpful, it is not necessary to begin studying real mathematics. With dedication and hard work, anyone can learn and excel in mathematics.

4. What are some resources for studying real mathematics?

There are many resources available for studying real mathematics, including textbooks, online courses, and video lectures. You can also join math clubs or participate in math competitions to enhance your understanding and skills.

5. What career opportunities are available for those who study real mathematics?

Studying real mathematics can open up various career opportunities in fields such as finance, computer science, engineering, and research. Many industries value individuals with strong mathematical skills, making it a versatile and valuable degree to have.

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