Why algebraic geometry is interesting

The first few items concern classical algebraic geometry, which studies polynomials over the reals and complex numbers.

  1. Classical algebraic geometry is the ‘next step’ after linear algebra. Linear algebra allows only addition and multiplication by scalars. If you allow multiplication of coordinates, you get polynomials in multiple variables, and that’s exactly what classical algebraic geometry is. There, one studies ‘varieties’ which are like linear subspaces. Instead of being defined by a linear equation (like x+y+z=0), they are defined by algebraic equations (like $latex y+x3-z=0$). Despite some hiccups, dimension is well-behaved, and each extra equation usually cuts down the dimension by 1.
  2. A lot of very interesting theorems came out of algebraic geometry, especially when projective space is thrown in. Projective space adds points at infinity to Euclidean space. So, for instance, two parallel lines in the plane intersect at infinity. This gives you theorems like Bezout’s theorem, which says that two varieties (i.e. zero sets of algebraic functions) that aren’t degenerate (like sharing factors) will intersect in the projective complex numbers a number of times equal to the product of their degrees (so two quadric surfaces in projective C3 intersect 4 times).
  3. Classical algebraic equations come up all the time. Matrix multiplication is algebraic, the determinant equation is an algebraic equation, etc.
  4. Now we get to crazier things. When abstract algebra was being developed, mathematicians began to see connections between polynomial rings and algebraic equations. In particular, every variety corresponded to a ‘prime ideal’ in the complex numbers and points corresponded to maximal ideals. What does this mean? Look at the complex numbers C. Take any polynomial in C. By the fundamental theorem of algebra, it factors into linear polynomials (x-a)(x-b)\cdots(x-c). So the only ‘prime’ polynomials are the linear ones (x-a). But these are in 1-1 correspondence with points. It gets more complicated in higher dimensions.
  5. So now mathematicians knew that you could make prime ideals in a ring correspond to points in a space. So they took it and ran with it: what if you took ‘any’ ring and made a space whose points corresponded to prime ideals? This is the idea of a ‘spectrum’. It turns out that commutative rings of ‘finite dimension’ have the most tractable spectra. These spaces have weird properties. For instance, the element 0 usually forms its own ideal, but corresponds to no classical point. It’s an extra point that is somehow close to all other points, because every other ideal contains it. The integers have a countable number of points, one for 0 and one for each prime. Things get really, really weird if you look at the prime ideals of Z[X], the ring of integer polynomials.
  6. Mathematicians quickly discovered that almost everything you did with polynomial rings had a geometric analogue. Quotienting by a prime ideal corresponded to restricting to a subspace. Inverting a prime ideal corresponded to taking the complement. Quotienting by only the linear terms of an ideal gave the tangent space. Tensor products gave ‘fiber products’. The possibilities were endless!
  7. It even extended to number theory, giving number theory a geometric setting. In many ways, number theory could be done by working with countable spaces like the integers in the same way that you work with uncountable spaces like the complex numbers. This explains many of the connections between finite fields and complex numbers, such as the Riemann hypothesis’s connection to certain finite fields.
  8. Now algebraic geometry is even being applied to physics in the form of string theory. String theory is deeply involved in algebraic geometry.

These are just a few reasons that algebraic geometry is interesting to me.

This post was originally posted on reddit.

Image by Jbourjai.

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