Algebraic Geometry is Impossible Without These 6 Things

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Video on Solvability

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[willnewman]

There are several issues with this video.

1. You say Algebraic Geometry is the "cornerstone of modern mathematrics." While it has many connections to other areas, so do most fields of mathematics. Even Algebraic Geometers would not agree with this statement.
2. While your first two examples are somewhat related to algebraic geometry, the second two are not, at least in the way you describe them. "Finding" roots of a single variable polynomial is not a focus of algebraic geometry. Note that these formulas have roots in them, while algebraic geometry deals strictly with rational functions.
3. Your sterographic projection example is related to algebraic geometry, but not birational geometry. The map you define extends an isomorphism between the conic and the line, so nothing birational is happening.
4. Your sentence "Birational geometery lacked the tools to analyze complex structures and local properties of surfaces in higher dimenisons..." is either very wrong, or does not make any sense.
5. You conflate the notion of Riemann surfaces with algebraic sufraces. Riemann surfaces are the same as algebraic curve, meaning they have dimension 1.
6. You say that y/x doesn't make sense when x=0, which is true, but blowing up the origin in A^2 does not fix anything. What you probably want to be doing is considering the (rational) map A^2 to P^1 which takes a point (x, y) to [x:y], which is the slope through (x, y) and (0, 0). This map is not well defined at the origin, but blowing up the origin gives a well defined morphism.
7. You do not say what algebraic geoemtry is. While the first two examples are related to algebraic geoemtry, they do not give a sense of what it is past configurations of points and lines in the plane. And, as mentioned, the next two examples are not really part of algebraic geometry as you explaned them. So only the last two feel like algebraic geoemtery, and my issues stated above show that they are poor explanations of concpets from algebraic geometery. Moreover, in these examples, you use terms which you do not define, which I feel leads to viewers not understanding what is happening.

When you make videos like these, you speak with authority, which you clearly lack. You need to learn more abouut a subject before making a video like this.

[jonathandawson]

I do not understand. The infinite slope problem can be solved if we specify the line as such, ax+by = 0 where (a, b) is the point we want the line to be perpendicular to. Granted this is a shift of thinking, from "if x is this then y is that", to "set of all (x, y) points such as this holds", but it's a small shift and no Riemann surface is needed.

[plranisch]

Beautiful! 🎉Thanks, carry on, please.

[KaliFissure]

Surface(cos(u/2)cos(v), cos(u/2)sin(v), sin(u)), u, 0, 2pi, v, 0, 2pi

Radially symmetric Klein bottle?
Single sided surface?

[ajmergill]

Thanks for this.I have a request to you can you make a video on what is homogenisation of second degree curve actually means? What we do actually when we homogenise equation with another equation like parabola equation with straight line equation. How this is related to conic sections? I should be very thankful to you for this.

[m_c_]

yo, that's the cornerstone, yo. Fir real, that's what's up.

[CategoryTheorist]

Please some more videos on Galois theory!

[mayankthakur]

How does this video not have like a 100k or 1mil views it has great explanations