theHigherGeometer

Hilbert, by Constance Reid.

A great book, showing David Hilbert's passage from a bold young and ambitious mathematician to an old man surrounded by the ruin of the mathematics department in Göttingen in the 1930s. This helped me place a lot of names of contemporaries, and I can appreciate Minkowski's truncated career much better, I had no idea how big a deal he was in this whole circle, nor that he died early. The author treats the mathematics very well even though she's not trained in it, and from a modern standpoint it helps me connects back from post-1930s work to the previous generation's revolutionary developments.

Electronic version: https://doi.org/10.1007/978-1-4612-0739-9

#Read2026

Apr 8 at 12:28amWeb
Show threadJohn Carlos BaezApr 8

@highergeometer - I may have read this once; I should read it again!

Minkowski was more than Einstein's teacher - the guy who realized time is imaginary - but I don't know his whole institutional role.

Show threadtheHigherGeometerApr 8
@johncarlosbaez He worked on both the geometry of numbers https://en.wikipedia.org/wiki/Geometry_of_numbers as well as mathematical physics later in his life. When he was just 18 he won (jointly, due to his youth) a big prize from the French Academy of Sciences. Was a close friend of Hilbert and was in the Göttingen mathematics department after Zurich, which was a position taken out of necessity to have a job. Also, Carathéodory was his student, also a later Göttingen member.
Geometry of numbers - Wikipedia

Show threadJohn Carlos BaezApr 8

@highergeometer - I'm somewhat familiar with Minkowski's work on the geometry of numbers, though I can never remember the key results. One of them lets us show that a "randomly chosen" lattice packing of spheres in high dimensions is denser than any lattice packing we currently know explicitly. I think that's cool... and frustrating.

(Yes, I can see the key results on Wikipedia!)

Show threadDan PiponiApr 8
@johncarlosbaez @highergeometer I didn't know that went back to Minkowski. I've thought about this kind of thing a lot but I'm not expert enough to make a useful contribution. I think that there are many problems where if you explore the space of possible solutions it turns out the space is full of good solutions. But, bizarrely, short strings of mathematical language are surprisingly good at finding the bad solutions. The unreasonable ineffectiveness of mathematics!
Show threadJohn Carlos BaezApr 8

@dpiponi @highergeometer - Maybe it's just the unsurprising "sometimes effective, sometime ineffective"ness of mathematics.

From page 14 of the third edition of Conway and Sloane's Sphere Packings, Lattices and Groups:

Show threadPozorvlakApr 8
@highergeometer @johncarlosbaez the story behind the prize is a bit more complicated than that, IIRC - the challenge was set specifically to force an elderly Henry Smith (of Smith Normal Form fame) to write up a result he'd announced decades earlier but never fully dotted the i's on. Writing up the proof took a great strain on Smith, who died before the prize was awarded. The fact that there were any other submissions came as a great surprise to the Academy, and there were accusations that Minkowski must have cheated somehow, but I think the final conclusion was that he wasn't even aware of Smith's work on the problem.
Show threadTony VladusichApr 8

@johncarlosbaez @highergeometer

I’ve been reading Gravitation by MTW. They seemed very unimpressed with Minkowski’s ict formulation. They even devote a small section to its “banishment”. 🤣

Show threadDavid QuinteroApr 8
@TonyVladusich @johncarlosbaez @highergeometer yet using time as imaginary was a key ingredient for the no boundaries proposal by Hawking and Hartle about the origin of the universe. You know, ideas obey Nietzsche's eternal return.
Show threadtheHigherGeometerApr 8

@davidsuculum I think it's ok as a technical crutch, but actually doing the Wick rotation using imaginary time is nontrivial. Witness @peterwoit slow burn trying to come to grips with how this is supposed to work on a serious level of modern math physics, even without fancy proposals for universe origins.

@TonyVladusich @johncarlosbaez

Show threadJohn Carlos BaezApr 8

@highergeometer - Woit is trying to come to grips with how the Osterwalder-Schrader theorem doesn't cover spinor fields.

@davidsuculum @peterwoit @TonyVladusich

Show threadJohn Carlos BaezApr 8

@TonyVladusich - in MInkowski spacetime the use of imaginary time is fundamental to a lot of work on quantum field theory - by which I mean not just writing the metric using 𝑖𝑐𝑡 as Minkowski did, but treating Minkowski spacetime and Euclidean spacetime as two different 'slices' of a 4-dimensional COMPLEX spacetime, allowing the use of complex analysis. I'm talking about things like the Osterwalder-Schrader theorem:

https://ncatlab.org/nlab/show/Osterwalder-Schrader+theorem

But in general relativity, where we have no favored coordinates, this is much more problematic, ... though Hawking made extensive use of it, using the relation between imaginary time and inverse temperature.

@highergeometer @davidsuculum

Show threadWeekend EditorApr 8

@highergeometer

Wir müssen wissen.
Wir werden wissen.
(We must know.
We will know.)
-- epitaph of David Hilbert