Differences between Algebraic Topology and Algebraic Geometry

In summary, the individual is unsure which subject they prefer between algebraic topology and algebraic geometry, but they do enjoy algebra. They have decided to take a course in algebraic topology because it is more like analysis, which they also enjoy. They are wondering if it is possible to study topology without any background in geometry and prefer to not take courses that they deem not useful for their interests. The person also mentions that algebraic topology involves examining loops in a topological space, while algebraic geometry studies zero sets of polynomials. They have made the decision to take courses in topology, algebraic topology, and differential topology, while not taking courses in geometry, algebraic geometry, and differential geometry.
  • #1
annoymage
362
0
i don't know if i can post it here, like this man https://www.physicsforums.com/showthread.php?t=397395, there's a lot of usefull comment for me.

anyway, I'm still don't really know which one i like, either algebraic topology, or algebraic geometry. but i really do like algebra... so I'm planning to take my next year course. for now I've decided go to algebraic "topology"( because I've heard topology is more like analysis, and i love analysis and also, geometry are motivated by low dimensional(more to practical thing i guess which i not preferred) but topology defined based on nature),

but i was wondering, is it ok if i did'nt take ANY geometry courses.. or rather is it possible for me to study rigorously in topology without any fundamental on geometry? it's not that i don't want to learn them, just sometimes it's really redundant for me to learn something which is not really usefull for me to pursue something i like,
(example: like there's course on magic and latin square, which have nothing to do with algebraic topology, sooner or later, i'll forget about that course)

thanks for the time, and sorry if anything wrong with my english.
 
Last edited:
Physics news on Phys.org
  • #2
Neither of these courses are going to look like a classical geometry course, and wouldn't require any such background.

Algebraic topology starts by taking a topological space and examining all the loops contained in it. For example, in the plane every loop can be contracted to a single point. But on a torus, if you have a loop going around it through the middle, this cannot be contracted to a single point. Algebraic topology makes this rigorous by constructing a group consisting of all "distinct" loops (they can't be wiggled to form another one) I don't see how taking an algebraic topology class before taking a normal topology class makes sense to be honest, so you might want to look into how that would work

Algebraic geometry is the study of the zero sets of polynomials. For example, y-x2=0 just gives the parabola, x2+y2-1=0 just gives the unit circle. Of course you can do this in arbitrary dimensions. You can look at the set of polynomials which are zero on such a set - for example on the parabola, the polynomial y4-x2y3 is always zero as well. These polynomials form an ideal in the ring of all polynomials, and properties about this ring correspond to properties about the set of points where the polynomials are zero.

So to summarize, the two subjects have very little to do with each other, besides the fact that they both involve algebra (not even the same type of algebraic object) and that the words topology and geometry are often assumed to be talking about similar things
 
  • #3
i've made my decision, I'm taking topology, algebraic topology and differential topology. without geometry, algebraic geometry and differential geometry. it's ok right?, i hope its ok
 

1. What is the main difference between algebraic topology and algebraic geometry?

The main difference between algebraic topology and algebraic geometry is the type of mathematical objects they study. Algebraic topology focuses on topological spaces, which are objects that can be deformed without tearing or gluing. On the other hand, algebraic geometry studies algebraic varieties, which are sets of points defined by polynomial equations.

2. How do algebraic topology and algebraic geometry use algebraic tools?

Both algebraic topology and algebraic geometry make use of algebraic tools, such as groups, rings, and fields, to study their respective objects. In algebraic topology, algebraic tools are used to classify topological spaces and to study their properties. In algebraic geometry, algebraic tools are used to define and study algebraic varieties, as well as to solve equations and understand their geometric properties.

3. What are some common techniques used in algebraic topology and algebraic geometry?

Some common techniques used in algebraic topology include homotopy theory, spectral sequences, and cohomology. In algebraic geometry, techniques such as sheaf theory, schemes, and intersection theory are commonly used. Both fields also make use of category theory and homological algebra to study their respective objects.

4. How do the goals of algebraic topology and algebraic geometry differ?

The main goal of algebraic topology is to study topological spaces and their properties, such as connectedness, compactness, and homotopy. In contrast, the goal of algebraic geometry is to study algebraic varieties and their properties, such as dimension, singularities, and intersection numbers. However, both fields share the common goal of using algebraic techniques to understand and classify these mathematical objects.

5. Can algebraic topology and algebraic geometry be applied in other areas of science?

Yes, both algebraic topology and algebraic geometry have applications in various areas of science, including physics, engineering, and computer science. In physics, algebraic topology is used to study the topology of space-time and in string theory. In engineering, algebraic geometry is used in robotics and control theory. In computer science, both fields are used in areas such as image processing, machine learning, and data analysis.

Similar threads

Replies
3
Views
2K
  • Differential Geometry
Replies
16
Views
2K
  • Topology and Analysis
Replies
2
Views
359
  • STEM Academic Advising
Replies
3
Views
1K
  • Science and Math Textbooks
Replies
5
Views
1K
  • Sticky
  • Science and Math Textbooks
Replies
10
Views
5K
  • Differential Geometry
Replies
1
Views
2K
  • Differential Geometry
2
Replies
69
Views
10K
  • STEM Academic Advising
Replies
7
Views
2K
  • STEM Academic Advising
Replies
10
Views
2K
Back
Top