How I Learned to Love Algebraic Geometry
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How I Learned to Love Algebraic Geometry
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Re: How I Learned to Love Algebraic Geometry
#2I liked the part about Alexander
This one:
One of these geniuses was Hartshorne’s thesis advisor, Alexander Grothendieck. From about 1960 to 1970, Grothendieck revolutionized algebraic geometry as part of an epic quest to prove some conjectures about number theory, the Weil Conjectures. He had the idea that these could be translated into questions about geometry and settled that way. But making this idea precise required a huge amount of work. To carry it out, he started a seminar. He gave talks almost every day, and enlisted the help of some of the best mathematicians in Paris.
Re: How I Learned to Love Algebraic Geometry
#3I liked the part about Alexander This one: One of these geniuses was Hartshorne’s thesis advisor, Alexander Grothendieck. From about 1960 to 1970, Grothendieck revolutionized algebraic geometry as part of an epic quest to prove some conjectures about number theory, the Weil Conjectures. He had the idea that these could be translated into questions about geometry and settled that way. But making this idea precise requ…
Indeed. Grothendieck is one of the few names I associate with the word "genius" without qualification. Up there with people like von Neumann and Witten. There are a lot of really smart people in the world, but they stand out even among the best.
Re: How I Learned to Love Algebraic Geometry
#4“Mathematicians study curves described by all sorts of equations – but sines, cosines and other fancy functions are only a distraction from the fundamental mysteries of the relation between geometry and algebra.”
Is this statement backed up by theorems or is it being made because there are just more proven theorems in algebraic geometry and there is not much work on solution sets to more general equations?
Re: How I Learned to Love Algebraic Geometry
#5Algebraic Geometry is a powerful tool of number theory because much of it works over any field. It allows one to translate geometric intuition (algebraic geometry over the complex numbers) into a more algebraic environment (finite, p-adic, or number fields).
Going back further, algebraic geometry over the complex numbers was shown in the early 20th century to be in many ways equivalent to more classical analytic geometry: https://en.wikipedia.org/wiki/Algebraic_geometry_and_analyti...
Re: How I Learned to Love Algebraic Geometry
#6“Mathematicians study curves described by all sorts of equations – but sines, cosines and other fancy functions are only a distraction from the fundamental mysteries of the relation between geometry and algebra.” Is this statement backed up by theorems or is it being made because there are just more proven theorems in algebraic geometry and there is not much work on solution sets to more general equations?
I wasn't crazy about that line, but basically, sines and cosines make much less sense over say, the rational numbers, whereas polynomials work just fine. They're the most general "nice" function one can define over an arbitrary commutative ring.
Going back to the 1800's, in the theory of Riemann surfaces (one-dimensional complex manifolds), the only meromorphic (complex differentiable) functions are ratios of polynomials! And all closed projective complex manifolds are algebraic varieties. See https://en.wikipedia.org/wiki/Algebraic_geometry_and_analyti...
Re: How I Learned to Love Algebraic Geometry
#7It would be nice if they went in to a bit more detail about what they see as the connection between these two subjects.
As they say, a variety is the zero set of some polynomial equations. In particular a Riemann surface can be represented as a polynomial equation P(x,y)=0. This can quantised as you might expect x->x, y-> h d/dx, and you can study the equation P(x,h d/dx) \psi(x) = 0. The solution \psi can be written as a formal power series. For a nice choice of P(x,y), these power series can be a generating function for "something", and can contain a lot of interesting algebraic information.
Re: How I Learned to Love Algebraic Geometry
#8I like that you can use latex in the comments section..
Re: How I Learned to Love Algebraic Geometry
#9I really liked his first paragraphs! I also liked physics and was a math grad student to learn the math for physics!
But I tend to agree with his first statements about polynomials: They look too restrictive to be highly promising for physics.
It appears that he drifted off a central focus on physics and got interested in some math that may, long shot, have something to do with some detailed aspects of string theory. It looks like he is more interested in the math than the physics.
Okay, but to me it's a loss for physics.
Re: How I Learned to Love Algebraic Geometry
#10“Mathematicians study curves described by all sorts of equations – but sines, cosines and other fancy functions are only a distraction from the fundamental mysteries of the relation between geometry and algebra.” Is this statement backed up by theorems or is it being made because there are just more proven theorems in algebraic geometry and there is not much work on solution sets to more general equations?
Solution sets to more general equations are so difficult and pathological that a lot of modern mathematics just entirely rules them out as objects of study. For example, any time you hear the word “manifold”, it refers to a space which has none of these pathologies and is entirely smooth. So the entire theory of differentiable manifolds will never encourage anything which “pinches”, or drops down a dimension, etc.
On the other hand, by restricting ourselves only to polynomials and their solution sets, it turns out that the singularities which arise are not too bad, and we can study them in detail. In other words, restricting to polynomials is the _only_ restriction we have to make, pretty much every solution set can be studied from there on.