E8 and Catastrophe Theory(Bifurcation) - help

[MTd2]

I've been looking to what kind of bifurcation E8 corresponds to, but I didn´t find anythng. The only place I've found anything was on wikipedia, but it doesn't say anything. Can anyone help me?

http://en.wikipedia.org/wiki/Catastrophe_theory

 

[tiny-tim]

I've been looking to what kind of bifurcation E8 corresponds to, but I didn´t find anythng. The only place I've found anything was on wikipedia, but it doesn't say anything. Can anyone help me?

http://en.wikipedia.org/wiki/Catastrophe_theory

Hi MTd2! Are you any relation of R2d2?

As I expect you found, the A,D,E classification of catastrophes by names beginning with A D and E makes the As correspond to the A series of Lie groups, the Ds to the D series, and the Es to the E series … http://en.wikipedia.org/wiki/ADE_classification

In turn, the A series of Lie groups correspond to SU groups, and the B series to SL groups.

There are also C and D series of Lie groups (not connected with catastrophes), and finally the E F and G series, officially known as "exceptional" cases … http://en.wikipedia.org/wiki/Simple_Lie_group#Infinite_series

so I guess the E₈ catastrophe has to be categorised as exceptional, and not corresponding to anything other than itself.

 

[MTd2]

Well, people used to call me R2D2 at college! But that's because I use a wheelchair!

Now, seriously, I am asking somthing a little bit different. According to this list provided in the wiki (http://en.wikipedia.org/wiki/Catastrophe_theory):

Vladimir Arnold gave the catastrophes the ADE classification, due to a deep connection with simple Lie groups.

* A0 - a non-singular point: V = x.
* A1 - a local extrema, either a stable minimum or unstable maximum V = \pm x^2 + a x.
* A2 - the fold
* A3 - the cusp
* A4 - the swallowtail
* A5 - the butterfly
* Ak - an infinite sequence of one variable forms V=x^{k+1}+\cdots
* D4- - the elliptical umbilic
* D4+ - the hyperbolic umbilic
* D5 - the parabolic umbilic
* Dk - an infinite sequence of further umbilic forms
* E6 - the symbolic umbilic V = x3 + y4 + axy2 + bxy + cx + dy + ey2
* E7
* E8
//////


There is just one exceptional explained, but I couldn't find the other 2. I'd like to know what is the stable polynomial of the other 2 Es as well as its sketched behavior.

 

[yyat]

I don't know too much about this myself, but I found http://www.math.purdue.edu/~agabriel/dynkin.pdf" which gives the singularities corresponding to E6, E7, E8 (for analytic functions C^3->C). You probably want to check the source cited there (particularly [1] by Arnold) for more info.

 

[MTd2]

Yes, but I've seen that on several places. Note for example the tetrahedral formula
x^4 + y^3 + z^2, is more complete on wikipedia, because it has the full stable form with 5 control parameters. I said above that this one was explained, well, sorry. It wasn't, except for expliciting the 5 parameters.

But there is nothing for the other groups. And I checked Arnold's book, in the end, and it is just explained in a very sketchy way the icosahedron/dodecahedron group. I'd like something more complete.

 

[yyat]

The codimension 6 singularity

$$x^3+xy^3+z^2$$

has a universal unfolding ("stable form")

$$x^3+xy^3+z^2+ax+bx^2+cy+dy^2+exy+fx^2y$$.

The codimension 7 singularity

$$x^3+y^5+z^2$$

has a universal unfolding

$$x^3+y^5+z^2+ax+by+cy^2+dy^3+exy+fxy^2+gxy^3$$.

Also note that [1] in the link I gave is not Arnold's book but an article in Functional Analysis and Applications.

 

[MTd2]

Yes, but I can't get that article.

 

[yyat]

This is also contained in http://www.mat.univie.ac.at/~michor/catastrophes.pdf" (written by one of my profs in the year I was born). Check out p. 39 and 46.

 

[MTd2]

Hmm, Thank you!