Lagrangian of the Standard Model by Matilde... • shapes, figures & forms

Lagrangian of the Standard Model by Matilde Marcolli (via Lieven Le Bruyn)

This is her illustration of why higher mathematics needs to reduce the complication of something we can read but probably not understand.

group theory
quotienting
equivalence-classes
simple observations like “2+2=4 in how many different ways?”

are the kinds of tools that can reduce, reduce, simplify, reduce something like the above into what, with a minor encyclopædic knowledge, can finally be comprehensible.

For higher level examples let’s look at Michi Johanssons’ blog and Jacob Lurie’s undergraduate thesis.

$\begin{diagram} R^4 &\rTo^{\begin{pmatrix}A&0&BAB&0\\0&B&0&ABA\\0&0&A&B\end{pmatrix}} & R^3 &\rTo^{\begin{pmatrix}A&0&BAB\\0&B&ABA\end{pmatrix}} & R^2 &\rTo^{(A \quad B)} & R & \rTo\\ \dTo^{f_3} && \dTo^{f_2} && \dTo^{f_1} & \rdTo^{(a \quad b)\circ\epsilon} \\ R^3 &\rTo^{\begin{pmatrix}A&0&BAB\\0&B&ABA\end{pmatrix}} & R^2 &\rTo^{\begin{pmatrix}A&B\end{pmatrix}} & R & \rTo^\epsilon &\mathbb F_2 & \rTo & 0 \end{diagram}$

$H^*(D_8,\mathbb F_2)=\mathbb F_2[x^1,y^1,z^2]/\langle xy\rangle$

Even if you don’t know what the symbols mean, you can see that their arrangement is simple–not sequential in the way of the Lagrangian of the standard model of particle physics, but you see pieces like ABA and BAB–simple patterns embedded in this vast encyclopædic framework, still ABA BAB is understandable by a human.

the Lie algebra E6 can be represented with restricted bits of {27 complex numbers}

Likewise in Lurie’s example we hear about “lacing” and 27 complex variables at once–but it’s all been reduced, reduced, reduced so that the pieces can fit in a logic that a human (who knows the vocabulary) can understand through its simplicity.

(Source: math.fsu.edu)

isomorphismes

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