Currently having fun at @johncarlosbaez' Edinburgh seminar on This Week's Finds and Representation Theory.

https://golem.ph.utexas.edu/category/2022/09/seminar_on_this_weeks_finds.html

Today's topic seems to be Young diagrams.

I have recognized some names I have seen on Twitter before.

If you've missed this talk, don't worry, there will be a recording. Or see if you can make it next week...

Grothendieck dreamt of turning algebraic varieties into objects called 'motives' that behave more like vector spaces or abelian groups. This would let you chop algebraic varieties - or really their motives - into more fundamental building blocks.

(2/n)https://en.wikipedia.org/wiki/Motive_(algebraic_geometry)

Or maybe you're thinking "This is the kind of math I like!" Then take a look at her paper.

She defines derivations of operads, and shows entropy is a derivation of the operad whose space of n-ary operations is the (n-1)-simplex!

(4/n)

@RefurioAnachro @johncarlosbaez I do not know much about the classification of 3D Lie groups but if you look at Solv, it is easy to see that you can also get a different geometry by changing a parameter (like in https://arxiv.org/abs/2005.06430 ). For some values of alpha you get Thurston geometries (H2xR, H3 and Solv) and for others you get geometries that are not Thurston because there are no closed manifolds with these geometries.

@RefurioAnachro @johncarlosbaez ... But only if the if the proportionality constant is 1. Otherwise you get the less symmetric Berger sphere. Anything with Berger sphere geometry can be also given S3 geometry, so it is useless in the context of Thurston geometrization. (In case of twisted H2xR you can also change the proportionality constant.) I have some visualizations here: https://twitter.com/ZenoRogue/status/1258035231996682244 (we did not know the name Berger sphere then)

@johncarlosbaez @RefurioAnachro Also "there are 8 homogenoeous geometries in 3d" is misleading IMO -- there are 8 geometries useful in the context of geometrization theorem (classifying closed manifolds), but there are also homogeneous complete simple-connected manifolds which are not useful there, because they are less symmetric variants of Thurston ones (Berger sphere) or because there are no closed manifolds with a given geometry.

#introduction

Hi all! I'm Cody, a refugee from Twitter math.

Mostly interested in proof theory and logic, but interested in a lot of mathy things.

Mostly, my variable width ascii art came out okay'ish, but here it failed badly. I suppose it's still kind of readable.

Anyways, here's a photo of that bit from my blog's LaTeX rendering. I know, mathstodon.xyz can render LaTeX as well, but I didn't like the look of matrices on other clients, so...

Maybe I should simply take pictures right away next time.

Show thread

Weyl algebra - https://en.wikipedia.org/wiki/Weyl_algebra

Group center - https://en.wikipedia.org/wiki/Center_%28group_theory%29

Quantizaton - https://en.wikipedia.org/wiki/Quantization_%28physics%29

Schrödinger picture - https://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture

Heisenberg picture - https://en.wikipedia.org/wiki/Heisenberg_picture

Wigner-Weyl transform - https://en.wikipedia.org/wiki/Wigner%e2%80%93Weyl_transform

18/END

Links:

Here's a seminar with two speakers talking about the Heisenberg group: History for Physics - "The Weyl-Heisenberg group: from quantum mechanics to quantum information" - https://www.youtube.com/watch?v=atgyvRvS1IM

The first part delves into a specific problem set, the second half gives a more bird's eye account on the history of Heisenberg group.

Edward Teller "Understanding Group theory with Heisenberg" - https://www.youtube.com/watch?v=2OiRL6DfFw4

17/

I secretly wanted to make this post about Weyl algebra, also called symplectic Clifford algebra, another name I find irresistable. Let me say just this much: Herrmann Weyl wanted to understand why the Schrödinger picture of quantum mechanics is equivalent to Heisenberg's, and that's why he came up with this stuff.

This post is already too long, so I will stop here.

Thanks to ZenoRogue for showing me the relation between Nil-geometry and the Heisenberg group, which got me started!

16/

If that doesn't mean much to you, at least notice that the center is the part of a group which is commuative. Look what happens if we substitute the identity for z and right-multiply first by y and then by x:

Id = xyx^-1y^-1

yx = xy

15/

We can fix generators for the small Heisenberg group like this:

/ 1 1 0 \

x = | 0 1 0 |

\ 0 0 1 /

/ 1 0 0 \

y = | 0 1 1 |

\ 0 0 1 /

That leads to the center generated by z = xyx^-1y^-1:

/ 1 0 1 \

z = | 0 1 0 |

\ 0 0 1 /

14/

The Heisenberg group turns up in quantum mechanics.

> [In] Weyl quantization, proposed by Hermann Weyl, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. - https://en.wikipedia.org/wiki/Quantization_%28physics%29

13/

In any case, there's a representation of the Nilmanifold which, again, looks almost like the things we did earlier! Of course these ideas make sense in dimensions other than three, see here for a more complete account:

Nil manifold on Wikipedia - https://en.wikipedia.org/wiki/Nil_manifold

Nilgeometry explained - https://www.youtube.com/watch?v=FNX1rZotjjI

12/

The notion of Nilgeometry was introduced by Anatoly Mal'cev in 1951 as the quotient N/H of a nilpotent Lie group N modulo a closed subgroup H. Just so you know where the funny name comes from. For solvable Lie groups there is an analog, and there's a corresponding Thurston geometry beautifully named Sol!

11/

Thurston's space sort of looks like a plane, but if you move in a circle, you also move perpendicular to the plane! It is similar to an example for an effect called holonomy: when you roll a ball in a small circle on a plane, the ball ends up twisted along an axis pointing away from the plane. While moving in two dimensions on a plane, we have also moved using a third degree of freedom. The difference is that in nilgeometry that third direction does not repeat every full turn.

10/

I like to learn and popularize higher maths. I'm not a pro, but I can stomach corrections!

Joined Apr 2017