What are the necessary background topics for NCG?

  • Thread starter ehrenfest
  • Start date
  • Tags
    Geometry
In summary, the prerequisites to study NCG are: Real Analysis at the level of Rudin's "Principles of Mathematical Analysis", Complex Analysis at the level of Marsden's "Basic Complex Analysis," Topology at the level of Munkres, Algebra at the level of Fraleigh's "An Introduction to Abstract Algebra", and knowledge in algebraic geometry, homology, cohomology, and possibly k-theory. For studying Lie groups, one needs knowledge in both algebra and differential geometry.
  • #1
ehrenfest
2,020
1
What are the prerequisites to study NCG?

I know Real Analysis at the level of Rudin's "Principles of Mathematical Analysis", Complex Analysis at the level of Marsden's "Basic Complex Analysis," Topology at the level of Munkres, Algebra at the level of Fraleigh's "An Introduction to Abstract Algebra".
 
Physics news on Phys.org
  • #2
Probably algebraic geometry, homology and cohomology as pre-pre-requisites.

You can also see some remarks about k-theory in Connes text. So I guess that's another prerequisite.
 
Last edited by a moderator:
  • #3
OK. Guess I should wait a bit.
 
  • #4
What are the prerequisites to study Lie groups?
 
  • #5
ehrenfest said:
What are the prerequisites to study Lie groups?

Lie group = group + differentiable manifold, hence knowledge in both algebra and differential geometry is needed.
 

1. What is non-commutative geometry?

Non-commutative geometry is a mathematical framework that studies geometric objects using algebraic methods, where the order of operations matters. It is a generalization of classical geometry that allows for the study of non-commutative spaces, such as quantum spaces.

2. How is non-commutative geometry different from classical geometry?

The main difference between non-commutative geometry and classical geometry is that in classical geometry, the order of operations does not affect the final result, whereas in non-commutative geometry, it does. This means that the algebraic structure of non-commutative spaces is more complex and requires different mathematical tools for analysis.

3. What are some applications of non-commutative geometry?

Non-commutative geometry has applications in various fields, including theoretical physics, where it is used to describe quantum systems, and topology, where it is used to study non-commutative spaces. It also has applications in cryptography, signal processing, and data analysis.

4. Who developed non-commutative geometry?

Non-commutative geometry was developed by the French mathematician Alain Connes in the late 20th century. He was inspired by the work of mathematicians such as Alexander Grothendieck and Georges De Rham, and his ideas have since been further developed by other mathematicians and physicists.

5. What are some current research topics in non-commutative geometry?

Some current research topics in non-commutative geometry include the development of non-commutative versions of classical geometric objects, such as non-commutative manifolds, and the study of non-commutative spaces in relation to quantum field theory and string theory. There is also ongoing research on the applications of non-commutative geometry to other fields, such as computer science and economics.

Similar threads

Recommendations for study materials to prepare for undergrad Physics
    • STEM Academic Advising
Replies
7
Views
864
Masters in physics with a bachelors in liberal arts
    • STEM Academic Advising
Replies
7
Views
1K
Studying Specific guidance suggestions?
    • STEM Academic Advising
Replies
14
Views
694
Courses Mathematics Bachelor's Degree: Choices ahead
    • STEM Academic Advising
Replies
1
Views
726
Recommended mathematical courses for a Physics major
    • STEM Academic Advising
Replies
9
Views
1K
Admissions I Need some advice about my physics undergraduate options
    • STEM Academic Advising
Replies
11
Views
667
Help me use my library of textbooks to form a study plan
    • STEM Academic Advising
Replies
3
Views
942
I Highschooler looking for math research/projects
    • General Math
Replies
8
Views
500
Introduction to Modern Astrophysics -- What are the prerequisites?
    • STEM Academic Advising
Replies
2
Views
867
Getting into an engineering career with no degree...
    • STEM Academic Advising
Replies
12
Views
1K

Hot Threads

Recent Insights

Back
Top