What Mathematical Concepts Underlie String Theory and Poincaré's Conjecture?

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SUMMARY

The discussion centers on the mathematical foundations necessary for understanding string theory and Poincaré's conjecture. Key areas identified include differential geometry and algebraic topology, which are essential for advanced mathematical proofs in theoretical physics. The discussion highlights that Grigori Perelman's proof of Poincaré's conjecture utilized these advanced mathematical techniques. Additionally, prerequisites for studying these fields include real analysis, multivariate real analysis, graduate algebra, and group theory.

PREREQUISITES
  • Differential geometry
  • Algebraic topology
  • Real analysis
  • Multivariate real analysis
NEXT STEPS
  • Study differential geometry techniques
  • Explore algebraic topology concepts
  • Learn real analysis and its applications
  • Investigate multivariate real analysis
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This discussion is beneficial for graduate engineering students, mathematicians, and physicists interested in the mathematical underpinnings of string theory and advanced proofs in topology.

GreenLRan
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Hi,
I'm a graduate engineering student, and as an undergrad I had all the typical engineering math courses: Cal I - III, linear algebra, and differential equations. I am curious. What else is out there?

You hear about the top theoretical physicists of the day trying to prove string theory or find "the theory of everything," also recently the problem involving poincare's conjecture that was solved (I'm guessing this was solved with differential equations and some complex geometry?).

What branch(s) of mathematics are involved with these kinds of proofs? This is not my field particularly as I am not a physicist (I do have a minor in physics however), but I very much want to improve my skills as a mathematician.

Can anyone provide any guidance, references, sources, or books that I may look into?

Thank you!
 
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I don't know about how physicists do it, but as far as mathematicians, this is what I have gathered about these topics (I am more familiar with the former than the latter, which isn't saying a lot, so I'll just stick to what I am pretty certain the prereqs are):

Perelman used techniques in differential geometry and algebraic topology, from what I gather. Both of these areas are quite advanced even at the introductory level, especially algebraic topology. In order to get to algebraic topology you need to understand proofs, then you need to do some courses in graduate algebra (maybe one algebra course and a group theory course), then some topology, then you probably will have picked up enough set theory etc. on the way and you can take the grad course in algebraic topology.

As for differential topology as far as I am aware you need real analysis and multivariate real analysis, then you can tackle it.
 

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