/sci/ - Science & Math

previous thread >>11568171

Why take something classical like differential geometry when you can take something more contemporary, like discrete geometry?

See https://en.wikipedia.org/wiki/Outline_of_geometry

>>8350028
A pack of losers that cannot deal with dynamic divergences let alone pratical.

I recently fell in love with algebraic geometry.

Now I'm wondering, what are its applications? Other than number theory where rational curves are apparently used frequently? I know that differential geometry is heavily applied in physics. Is the same true about algebraic geometry? Is it useful for computing stuff too?

>>6136540
The essence of mathematics is generality. Take rings as an example: Many mathematical questions naturally involve objects with some notion of "addition" and "multiplication". For example:
(1) Integers, rational numbers, real numbers, complex numbers, etc.
(2) Polynomials in one or more variables
(3) Linear transformations of vector spaces, or equivalently, matrices
So, it's useful to have a unified framework for studying all of these, since often, theory developed for one can be applied to the others.

Consider polynomials. The set of solutions to a system of polynomial equations is a classical question, and since these solution sets look like curves, surfaces (see image), 3-folds, etc. — called "algebraic varieties" in general — this is also very geometrically motivated.

So, how do we study this question algebraically? It turns out that rings of polynomials over a field have an analogue of prime numbers — called irreducible polynomials — and a unique factorization theorem analogous to the fundamental theorem of algebra. Furthermore, this picture transfers over to the study of algebraic varieties, where the variety described as the zero set of a polynomial decomposes into irreducible pieces based on the polynomial's factorization. In other words, irreducible polynomials can be thought of as geometric objects.

The algebraic concept of unique factorization in rings, originally seen in the integers with prime numbers, naturally lends itself to the study of polynomials, and in turn to the study of geometric questions. This is just one example of the power of mathematical abstraction.

Similarly, with groups, one can view permutations, symmetries of objects, rotations of space, rigid motions of the plane, and changes of coordinates as all being special cases of the same concept. This perspective then makes the common structure more apparent and reveals things that would otherwise be hard to see.

>>6070694
To elaborate on this, here's a starting point for algebraic geometry: by Hilbert's Nullstellensatz, the maximal prime ideals of the polynomial ring <span class="math">\mathbf{C}[x_1, x_2, \ldots, x_n][/spoiler] correspond in a natural way to points in complex n-dimensional space <span class="math">\mathbf{C}^n[/spoiler].

The non-maximal prime ideals correspond to algebraic curves, surfaces, 3-folds, etc., embedded in <span class="math">\mathbf{C}^n[/spoiler].

Thus, the notion of prime ideals of a ring allows much of algebra, geometry, and number theory to be viewed in a common framework.

>>6059879
If you've just studied linear algebra, you haven't really gotten to the core of algebra, usually called "abstract" or "modern" algebra. This is the theory of algebraic structures like groups, rings, fields, and modules. (Vector spaces, the main object of study in linear algebra, are the "nicest case" of modules, which is why they're taught first.)

Algebra is an incredibly vast field of study, so it's impossible to summarize it all in a single post. But, I can briefly describe the motivation behind the main objects:
- Groups generalize the idea of symmetries; they can be thought of as collections of reversible operations that can be combined by applying them in order. (Technically, this describes a group action, but that's the historical motivation.) Examples: symmetries of a regular polygon, permutations of a set, invertible linear transformations of a vector space.
- Rings generalize number systems; they're sets where you have addition and multiplication operations that behave somewhat like the integers. Examples: integers, rational numbers, real numbers, complex numbers, quaternions, matrices with entries in any of the rings just mentioned, polynomials in several variables.
- Commutative rings (rings where multiplication is commutative) are of particular interest due to their deep connection with algebraic geometry — there is a powerful and far-reaching equivalence between algebra and geometry behind the study of systems of polynomial equations.
- Fields are commutative rings that behave like the rational numbers: division by anything but zero is possible. These are the basis (pun maybe slightly intended) of linear algebra. Fields themselves have something called a Galois theory, which allows the structure of roots of polynomials to be interpreted via group theory.
- Modules generalize vector spaces; they're like "vector spaces over a ring", but their structure can be far more complicated due to the more complicated structure of arbitrary rings.

>>5943496
It's not any specific topic that's important so much as having both a solid grounding in the fundamentals of the major areas and some knowledge at a more advanced level. So, make sure you have a good understanding of algebra, analysis, and topology at least at an undergraduate level. Then, pick some topics that interest you and take more courses in it.

If you can't decide on a course, take algebraic geometry — it's awesome.

Also, this is now an algebraic geometry thread.

https://en.wikipedia.org/wiki/Conical_surface
More generally, n-dimensional projective varieties correspond to (n + 1)-dimensional affine cones (i.e., affine varieties that consist of a union of lines through the origin).

So, for example, elliptic curves in the projective plane correspond to degree-3 conical affine surfaces in 3-dimensional affine space.

Pic tangentially related.

Algebraic number theory - A
Algebraic geometry - grades aren't up yet
Everything else - not math, doesn't matter