I want to study algebraic geometry and differential geometry

In summary: Yes, there are many abstract algebra books you can read. "Algebra" by Artin is a good place to start.
  • #1
pswongaa
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I want to study algebraic geometry and differential geometry, what should I learn beforehand
what is the relation between abstract algebra and homological algebra:confused:
 
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  • #2
As for differential geometry, the minimum prerequisites are linear algebra and a rigorous calc III course. I suggest you go through something like "calculus on manifolds" by Spivak.
Then you can read books like Do Carmo's "Curves and surfaces".

Although not strictly necessary, I would suggest learning topology as well. Thinks are going to make much more sense with some topology knowledge. Either way, if you want to read advanced books on differential geometry, then you will need topology anyway.

As for algebraic geometry, you will need quite a solid knowledge of abstract and commutative algebra. You should be very comfortable with things like ideals, prime ideals, modules, etc.
You don't need to be an expert in commutative algebra however, it is possible to learn the commutative algebra while you are reading algebraic geometry. Just make sure you know the basics.
A basic knowledge of topology is handy as well.
A good book is Harris "Algebraic geometry: a first course".

As for homological algebra, it is a subset of abstract algebra. But you don't need to know homological algebra right away. Besides, it is going to be very weird and unmotivating to you. Of course, one you go deeper into differential geometry, algebraic geometry or algebraic topology, then homological algebra will show up. But by then, it might be more motivated to you.

Another excellent first book in algebraic geometry is Miranda's "algebraic curves and Riemann surfaces". It is totally different than the Harris book, but it is really great in motivating all the concepts. To handle it, you will need to know complex analysis, differential geometry and some abstract algebra.
 
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  • #3
could you suggest some abstract algebra books? I only have artin's algebra, it is enough?
 

FAQ: I want to study algebraic geometry and differential geometry

What is the difference between algebraic geometry and differential geometry?

Algebraic geometry is the study of geometric objects defined by algebraic equations, while differential geometry is concerned with the study of smooth curves and surfaces using the tools of calculus and differential equations.

Why is algebraic geometry useful?

Algebraic geometry has numerous applications in fields such as physics, computer science, and statistics. It is also a powerful tool for understanding and solving problems in number theory and cryptography.

What are some key concepts in differential geometry?

Some key concepts in differential geometry include manifolds, tangent spaces, curvature, and metrics. These concepts are essential for understanding the properties of curved spaces and their applications.

What mathematical background is needed for studying algebraic and differential geometry?

A strong foundation in algebra, calculus, and linear algebra is necessary for studying both algebraic and differential geometry. Some knowledge of abstract algebra and topology is also helpful.

Are there any real-world applications of algebraic and differential geometry?

Yes, there are many real-world applications of these fields, including in engineering, computer graphics, and robotics. For example, differential geometry is used in designing cars and airplanes with optimal aerodynamics, and algebraic geometry is used in computer vision algorithms for object recognition.

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