200 years Bernhard Riemann

Bernhard Riemann was one of the most important mathematicians in history. Born on the 17th of September 1826, he made several huge contributions to geometry, analysis and number theory, before dying on the 2th of July 1866, at only 39 years old.

Bernhard Riemann

As you can see, that means that, this 17th of September, he was born 200 years ago. That's what this post will be about: a celebration of three of Riemann's important contributions.

Riemannian Geometry

The first big discovery of Riemann was his extension of differential geometry to arbitrary dimensions. His advisor, Carl Gauss (yes, that Gauss), had thoroughly studies smooth surfaces in 3D-space. Culminating in his theorema egregium. This was the first proper introduction of higher dimensions in mathematics (though hints of the concept existed before Riemann).

Riemann was, however, definitely the first person to consider curvature in higher dimensions (at least in dimensions >3, since 3-dimensional hyperbolic geometry already existed). This may sound like useless mathematical nonsense, but it is in fact the basis of generale relativity. GR was invented by Einstein using Riemann's theory over 60 years after it was published. Just goes to show how you can never predict if any theory will have practical uses...

When defining curvature of surfaces à la Gauss, we make use of the tangent space at a point. That is, intuitvely, a plane that just barely touches the surface S at the point p.

Tangent vectors to the surface live in this plane. Luckilly for us, surfaces live in R³ (well, Gauss' surfaces did) and so we can let the dot product descend from this ambient space into the tangent space. Since dot products can be used to measure the lengths of and angles between vectors, we can now measure lengths and angles on the surface S. 

For more general manifolds, the dot product cannot be inherited from some ambient space. So, instead, we take some dot product on each tangent space and require that these "varry smoothly" as we move across the manifold. So now, we can measure angles and lengths on these special Riemannian (hey) manifolds.

Riemann Surfaces

Suppose we take a complex number z = r exp(iθ) and we wish to compute its square root. Luckily, this is pretty easy, just √z = √r exp(iθ/2). There, solved it. But, hold on, if z = r exp(iθ), then we also have z = r exp(iθ + 2πi), so √z = √r exp(iθ/2 + π). Damn, we've got two square roots...

Positive real numbers also have two square roots and we can work with those just fine. Just pick one to be the principal root. However, over C, the situation is much worse. Suppose that z moves around the origin in a circle. Now √z will also move around 0 but at half the speed (since arg(√z) = arg(z)/2). In other words, when z gets back where it started, √z will only have gone halfway around. So it is impossible to define a continuous square-root function over C.

Riemann's solution? Instead of considering the square root function, consider{(z, w) ∈ C² | w² = z} together with the maps {(z, w) ∈ C² | w² = z, z ∉ R₀⁺} → C: (z, w) ↦ z and {(z, w) ∈ C² | w² = z, z ∉ R₀⁻} → C: (z, w) ↦ z. These are coordinate charts! We've just turned this set (almost) into a manifold.

Effectively, we have two copies of the complex minus (minus 0) floating on top of each other, defined in such a way that if you move across the negative real axis, you jump from one plane to the other.

The one point still missing is (0,0), which is where the two planes "intersect". So, let's take U to be all those points (z, w) on sheet one with |z| < 1 together with all (z, w) on the second sheet with |z| < 1. We now define the map (z, w) ↦ w. Now we can study the square root over C by studying this manifold (or, more often, its compactification).

Of course, this is not limited to the square root. Any algebraic equation P(z, w) = 0 can be used to define a Riemann surface, as we just did for the equation w² = z.

Riemann Hypothesis

And then there's the big one, probably the most famous thing Riemann ever did, at least for math-enthousiasts. Riemann, famously, wrote one (1) paper on number theory. If we denote the amount of primes less than or equal to x by π(x), we have that π(x)/Li(x) → 1 as x → ∞. Here,

Riemann, however, figured out an even better result, known as Riemann's explicit formula. First of all, we take π'(x) to be π(x) for non-prime x and π'(x) = π(x) - 1/2 for prime x. In essence, we just take the average of π(y) for y near x. Next up, define the functions

Then, Riemann proved that

Where ρ runs over all non-trivial roots of the (in)famous Riemann zeta function. Since |x^ρ| = x^Re(ρ) for x > 0 real, understanding the real part of the non-trivial zeroes of ζ puts very strong bounds on the error term for π(x) = li(x) + E(x).

The Riemann hypothesis tells us that Re(ρ) = 1/2 for every single ρ in that sum. It is currently one of the Millenium prize problems.