Dynamics on Graphs13
For a general graph X and a vertex v, we can consider starting with the singleton population at v and performing mutations on X to increase the population total. If we consider X=D5 then there are 5 starting positions, and each yields a mutation poset. In this video we also introduce a closely related poset of positive roots obtained from the singleton at v, whose edges are labelled by mutations.
This is really the starting point of a massive possible investigation of combinatorial structures that arise from this dynamics on graph point of view, featuring the Mutation and Numbers Games.
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0:01
Hello everyone I'm Norman Wildberger thanks for joining me on this adventure Dynamics on graphs
0:07
so last time I talked about the D5 mutation post set that we get by starting with a
0:14
Singleton population namely this one here a single single one at the vertex one and then performing
0:24
successive mutations to increase the population up to the maximum population which is this one here
0:30
and we recorded the mutations that we are performing in ultimately this Post-it kind of
0:40
fashion so allowing for the possibilities that we can do these mutations in different orders
0:48
now instead of using S2 as 0 it's S3 etc we can also just record this if the context is clear
0:54
with a simplified Post-it with just the labels of the vertices where the mutations are happening
1:03
okay so this is all well and good but it's really a special case because we are concentrating on
1:11
this particular Singleton what happens if we start with a Singleton at one of the other vertices what
1:15
kind of Post-it if it all do we get in those cases so that's what we want to investigate today so
1:22
let's start with um well actually let's first of all let's make this the following observation that
1:27
we could record you know the initial population also so here's the initial population one so
1:33
we kind of make a an extended post set by adding the sort of one above it okay then what we get is
1:40
um something that actually contains a copy of the original graph here in green so up at the top
1:49
and in some sense this is a sort of a hanging of this this tree because
1:54
we're really taking the vertex one sort of the letting the tree hang from it as if by gravity
2:02
so let's just observe that and uh now let's have a look at what happens if we consider starting at
2:08
Singletons with uh other vertices so if we start with a Singleton at vertex zero so let me remind
2:13
you in this story here that's that vertex there okay so then we get a similar mutation posted so
2:19
if we just run through it so imagine a Singleton here so then we can mutate at two and then we
2:23
can mutated three and then at four and then at one or we actually do the one first and then
2:29
and then the three and the four and after we've done some of that then we can perform the another
2:34
two mutation to bump that up to two and then another three mutation to bump that up to two
2:38
Okay so that's all encoded by this particular uh mutation poset which which is very similar
2:47
and that's not a surprise because there's a symmetry in the graph between the vertex of
2:52
what we're calling one this one here and the vertex zero so we expect these two to be uh
2:57
so what happens if we say start with the vertex four
3:01
so imagine starting with a Singleton at four we could then perform mutation at three and then add
3:06
two and then at either one or zero and after we've done both of those then we can perform
3:13
another mutation at two and then one more at three so we get this kind of mutation posted
3:20
or extended posted and notice also that both these two cases we get a hanging of the original
3:27
graph from the vertex that we're starting with here starting with zero here starting with four
3:35
what about if we start with vertex three so here I've only gone part of the way so let's suppose we
3:41
start with the vertex three here of the Singleton there okay then we can well we can either mutate
3:48
that two or a four this there's a you know some Choice there so if we do two first uh then we
3:53
could do a one and zero and then once we've done it we can either even pump this thing up to two
4:00
Okay so that would correspond to start looking at this part of the mutation post set okay and
4:07
then at that point there we could then mutate at four but in principle in fact we could mutate it
4:14
for sort of any time so the actual post head looks more like this with a my line from three
4:19
to four and then um and finally after we've done all of that then we can mutate once more
4:24
at three to bump up to the maximum population so again we see uh a copy of the original
4:34
graph if we include this sort of initial Vertex or apart from the other ones we see it sort of
4:40
hanging from the vertex three cool and perhaps the most interesting what happens if we start
4:47
with vertex two the sort of central one here all right well then let's see what happens we
4:54
get this thing here so just have a look at two okay we're starting there then we can mutate
4:58
either at zero at zero or at one or at three okay so that's represented with this diagram
5:04
here we start with the two so that's in in red and then we can go either a one mutation or a
5:09
zero mutation or a three mutation after we've done the three then we can do a four and after
5:16
we've done one zero and three then we have enough to do a mutation at two the name of this one here
5:22
and then we can do the mutation three so we get a someone more interesting uh kind of a post set
5:35
all right great so this uh that opens up a huge area of potential research and something
5:41
that you can now spend a lot of interesting time so it just by investigating the various
5:46
corresponding mutation pose sets that we get Associated to vertices of graphs so you start
5:52
with a graph okay and you pick up vertex and you say all right let's pretend we have a Singleton
5:57
population at that vertex and we just start mutating trying to increase the population
6:04
so what kind of what you know what kind of story emerges and can we encode that that
6:11
story of mutations or potential mutations with the pole set now we know that the Ade graphs are
6:22
really special and one of the reasons is because we can associate in this question
6:27
in this way finite process that there's a maximum population that we get to and
6:33
then we cannot go further so these posts are necessarily finite if we start with an Ade graph
6:38
but in principle you could ask the same question for other graphs but certainly in the context of
6:44
what we're doing if we start with an Ade graph okay you can start with any one of them choose
6:48
a vertex and consider what happens if you just try to increase the population starting with a
6:53
slings and that that that thing and do you get some kind of Post-it structure
6:59
so you know there's an underlying question here why do we actually always get post sets you know
7:03
and what role do these partially ordered sets play clearly it looks like we're going to have
7:09
to learn a little bit more about the theory of partially where it sets because it's making
7:14
direct contact here with this Theory in fact there is actually another rule for Post-its in
7:22
the above story so if we go back to the very first thing that we did where we had this as
7:27
the mutation posing you recall that was when we started with the Singleton at the vertex one
7:31
so what we can do instead of just recording the mutations that are possible we can actually do a
7:37
little bit something a little bit more in depth and record the various positive Roots which are
7:42
obtained from this singleton population starting at one by applying these mutations in some order
7:50
and happily what we get also forms of Post-it but it's a little bit more complicated one
7:55
that's how it work okay so here have a look at this beautiful little thing all right what do
8:03
we see here we see a post set whose vertices are are labeled by essentially root populations okay
8:11
so I'm not drawing the graph I'm just drawing a sequence of values uh in the in this sort
8:18
of D5 shape so here this represents the Singleton population at vertex one and then there's an edge
8:27
because we're increasing the population by doing a mutation at vertex two so that's
8:34
represented by this blue 2 along this Edge okay and that mutation of two results in a new
8:41
root population this one here one one zero zero with a zero also okay and then these
8:46
various possibilities that that were encoded in the previously Now sort of allow us to uh
8:52
to create this bigger thing so here here is the population where you just have one one and one
8:58
at the vertex one two and zero here at vertex one two and three once we've done both of these
9:06
then we can get to one one actually I suppose we should start counting with one up here the
9:11
zero one right one one one one one zero and over here is zero one one one one and then
9:18
you can just check from here and here once you got those and you can get this one one one one
9:22
one one one and from here you can also before you do the mutation four you can do the mutation two
9:29
you have enough to to bump that up get mutation two we'll bump this one up to a two and then a
9:37
subsequent 4 will give us this and then finally limitation three is giving us this thing here
9:43
so this is a very cool post set okay very cool and you see that pleasantly in fact maybe even
9:52
remarkably this thing has some some symmetry that's a number of properties as we'll see but the
9:59
most obvious one I guess is that it's symmetrical he might ask why is that you know is that is that
10:05
a feature that we're going to expect for for more generals not stories okay um and let's give this
10:12
thing a name okay so it's obviously a sort of a subset or a sub post set of the set of all
10:17
positive roots of D5 and we might call it R plus D5 comma one to denote the fact that we're really
10:25
starting with this population the Singleton at one and then performing increases Okay so
10:34
it's like an increasing um an increasing uh family of root populations starting with the Singleton
10:43
at one okay there's a number of questions here um so um one is how many increasing mutation
10:51
sequences will get you from the uh the start to at the end here so so one way of getting from
10:57
um from this initial one to this final one just with a single sequence of mutations so you say
11:02
to to go mutate two and then zero and then three and then four and then two and then three that'll
11:10
that'll get you from there to there but another way is just to you know find some other latter
11:14
down here uh so starting with this one say two and then three and then zero and then two and then
11:21
four and then three okay so how many such mutation sequences um are are present in in such a posit
11:31
and of course a very interesting question what are the corresponding objects or Post-its that we get
11:38
by changing this initial Singleton at one to an initial Singleton at any of the other vertices
11:45
right so we're going to get sort of analogs of what we did above but in this more in this bigger
11:52
fashion where we see not just the mutations but also their root populations so I'm hoping that you
11:59
can start to see that there is a huge amount of possible investigation you can do here just with
12:04
a pen and paper okay and uh you could literally spend weeks I'm not kidding okay you can literally
12:11
spend weeks and weeks investigating this and the more you look at this and the more you think about
12:16
it you you you see more and more structure being revealed and more and more remarkable
12:22
things happening really you know it's it's it's absolutely incredible and that's really why Lie
12:27
theory is is such a a profound and essential uh area of mathematics is because the actual examples
12:34
are so rich and that's exactly exactly what we're seeing here and in fact a lot of the richness of
12:41
Lie Theory ends up being reflected in some fashion or another by things that were we were creating
12:48
and it is emerging from this mutation point of view this this Dynamics on graph point of view.