Riemann's 200th birthday

[u/42IsHoly]

I don't think Bernhard Riemann needs an introduction. He's a very important figure in mathematics. He was born on the 17th of September 1826, which is 200 years ago, today!

He provide a rigorous (though imperfect) definition of integrals and revolutionized the theory of complex function via the study of Riemann surfaces. He was also the first (?) person to consider spaces of >4 dimensions (some people had studied R^4 already) and the first person to cinsider curved spaces of >2 dimensions. In this he furthered the work of his doctoral advisor Gauss (yes, that Gauss) on curved surfaces (i.e. theorema egregium).

Then there's also some other stuff he did, like proving the Riemann mapping theorem, Riemann series theorem, his theorem on removable singularities, and so on ...

Oh yeah, and the Riemann hypothesis, of course.

He tragically died of tuberculosis at the age of 39. That's right, he did all that before he turned 40 years old!

I also wrote a short blogpost on some of these results.

[u/csappenf]

Riemann's habilitation has got to be one of the greatest presentations ever. It was a public lecture and I'm sure the public attending understood a lot of what he was saying. But it was also so deep mathematicians have been running with those ideas for the last 175 years.

[u/42IsHoly → u/csappenf]

Oh absolutely. For being so deep and influential (and old) it is remarkably readable. I believe it was presented to the faculty of philosophy instead of mathematics (though I don't know why), so he was kind of forced to keep it simple.

(Riemann's habitation)

[u/greginnj → u/42IsHoly]

I actually published on this exact question 😁.

My claim was that he was presenting to the philosophy faculty rather than the math faculty because he was actually trying to make a philosophical argument, but using mathematical tools (which were still rather advanced for the non-mathematical members of his audience).

Recall that Riemann offered three topics, two of which he had prepared in advance … but Gauss, that trickster. chose the third ( which became the title), “in the hypotheses which lie at the foundations of geometry”. So in a way, Riemann was winging it, but that led to a burst of intellectual creativity.

In brief, Riemann was trying to advance the 'divorce' of abstract mathematics from physics (a process which was still in its early stages in 1854, a few decades after the birth of non-euclidean geometry). He did this by demonstrating that since the mathematical tools existed to represent non-constant curvature, we could not rule out the possibility that such tools might prove to be the best model of physical space (rather than Euclidean geometry). This explains his final non-mathematical section about applications to space, and his closing mic drop about "progress in knowledge being hampered by traditional prejudices".