You may be wondering, somewhere deep in the back of your mind, what happened to the Network Theory series on this blog. It’s nowhere near done! I plan to revive it, since soon I’ll be teaching a seminar on network theory at U.C. Riverside. It will start in October and go on at least until the end of March.
Here’s a version of the advertisement I sent to the math grad students:
Network Theory Seminar
I’ll be running a seminar on Network Theory on Mondays from 3:10 to 4:30 pm in Surge 268 starting on October 6th.
Network theory uses the tools of modern math—categories, operads and more—to study complex systems made of interacting parts. The idea of this seminar is to start from scratch, explain what my grad students have been doing, and outline new projects that they and other students might want to work on. A lot has happened since I left town in January.
I hope to see you there!
If you want more detail, here is a sketch of what’s been happening.
1) Franciscus Rebro has been working on “cospans”, a very general way to treat a physical system with inputs and outputs and treat it as a morphism in category. This underlies all the other projects.
2) Brendan Fong, a student of mine at Oxford, is working on a category where the morphisms are electrical circuits, and composing morphisms is sticking together circuits:
• Brendan Fong, A compositional approach to control theory.
• John Baez and Brendan Fong, A compositional framework for passive linear networks.
The first paper here is an outline of the project; the second one actually carries out the project, or at least a large chunk of it.
3) Blake Pollard, a student in the physics department, has been studying Markov processes. In a Markov process, things randomly hop from one vertex of a graph to another along edges. Blake has created a category where morphisms are ‘open’ Markov process, in which things flow in and out of certain special vertices called ‘terminals’.
4) Jason Erbele has been working on categories in control theory, the branch of math used to study physical systems that interact with the outside world via inputs and outputs. After finishing this paper:
• John Baez and Brendan Fong, Categories in control.
he’s been taking more concept from control theory and formalizing them using categories.
5) Oh yeah, and what about me? I gave a series of lectures on network theory at Oxford, and you can see videos of them here:
• John Baez, Network Theory.
Jacob Biamonte and I have also more or less finished a book on chemical reaction networks and Petri nets:
• John Baez and Jacob Biamonte, Quantum Techniques for Stochastic Mechanics.
But I won’t be talking about that; I want to talk about the new work my students are doing!
Posted by John Baez 
states, the unitary time evolution of a state 
is generated by a self-adjoint 
matrix 
called the Hamiltonian. In other words, 
and evolving for a time 
is then given by
the expected value of the observable associated to a self-adjoint operator 
is
![[O, H] = 0 \quad \iff \quad \displaystyle{\frac{d}{d t}} \langle O \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t.](https://s0.wp.com/latex.php?latex=%5BO%2C+H%5D+%3D+0+%5Cquad+%5Ciff+%5Cquad+%5Cdisplaystyle%7B%5Cfrac%7Bd%7D%7Bd+t%7D%7D+%5Clangle+O+%5Crangle_%7B%5Cpsi%28t%29%7D+%3D+0+%5Cquad+%5Cforall+%5C%3A+%7C%5Cpsi_0+%5Crangle%2C+%5Cforall+t.+&bg=ffffff&fg=333333&s=0 "[O, H] = 0 \quad \iff \quad \displaystyle{\frac{d}{d t}} \langle O \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t.")
map the set of probability distributions to itself whenever 
and we call this family of operators a continuous-time Markov process, or more precisely a Markov semigroup. 
is the vector built from the diagonal entries of 
More concretely, 
is the 
th component of the vector 
![[O,H] =0](https://s0.wp.com/latex.php?latex=%5BO%2CH%5D+%3D0+&bg=ffffff&fg=333333&s=0 "[O,H] =0")
and all 
obeying the master equation. █![[O,H]=0](https://s0.wp.com/latex.php?latex=%5BO%2CH%5D%3D0&bg=ffffff&fg=333333&s=0 "[O,H]=0")
the expected value of 
and all 
However, John and Brendan saw that—unlike in quantum mechanics—you need more than just the expectation value of the observable ![[O,H]=0.](https://s0.wp.com/latex.php?latex=%5BO%2CH%5D%3D0.&bg=ffffff&fg=333333&s=0 "[O,H]=0.")
You really need both
? Then it’s called a Dirichlet operator… and we found that in this case, we get a stochastic version of Noether’s theorem that more closely resembles the usual quantum one:![[O, H] = 0 \quad \iff \quad \displaystyle{\frac{d}{d t}} \langle O \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t \ge 0](https://s0.wp.com/latex.php?latex=%5BO%2C+H%5D+%3D+0+%5Cquad+%5Ciff+%5Cquad+%5Cdisplaystyle%7B%5Cfrac%7Bd%7D%7Bd+t%7D%7D+%5Clangle+O+%5Crangle_%7B%5Cpsi%28t%29%7D+%3D+0+%5Cquad+%5Cforall+%5C%3A+%7C%5Cpsi_0+%5Crangle%2C+%5Cforall+t+%5Cge+0+&bg=ffffff&fg=333333&s=0 "[O, H] = 0 \quad \iff \quad \displaystyle{\frac{d}{d t}} \langle O \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t \ge 0")
direction is easy to show, and it follows from John and Brendan’s theorem. The point is to show the 
direction. Since 
are an orthonormal basis of eigenvectors, and 
are the corresponding eigenvalues. We then have:
being linearly independent. For any infinitesimal stochastic operator 
connected components iff we can reorder (permute) the states of the system such that 
which implies![[O, H] = 0](https://s0.wp.com/latex.php?latex=%5BO%2C+H%5D+%3D+0&bg=ffffff&fg=333333&s=0 "[O, H] = 0")
![[O, H] = 0.](https://s0.wp.com/latex.php?latex=%5BO%2C+H%5D+%3D+0.&bg=ffffff&fg=333333&s=0 "[O, H] = 0.")
█
so that 
generates a 1-parameter unitary group. Show that the 
is 
sums to 1: 
be a possibly time-dependent observable, and write 
for its expected value with respect to some initial state ![\displaystyle{ \frac{d}{d t}\langle O(t)\rangle_{\psi(t)} = \left\langle [O(t), H] \right\rangle_{\psi(t)} + \left\langle \frac{\partial O(t)}{\partial t}\right\rangle_{\psi(t)} }](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cfrac%7Bd%7D%7Bd+t%7D%5Clangle+O%28t%29%5Crangle_%7B%5Cpsi%28t%29%7D+%3D+%5Cleft%5Clangle+%5BO%28t%29%2C+H%5D+%5Cright%5Crangle_%7B%5Cpsi%28t%29%7D+%2B+%5Cleft%5Clangle+%5Cfrac%7B%5Cpartial+O%28t%29%7D%7B%5Cpartial+t%7D%5Cright%5Crangle_%7B%5Cpsi%28t%29%7D+%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{ \frac{d}{d t}\langle O(t)\rangle_{\psi(t)} = \left\langle [O(t), H] \right\rangle_{\psi(t)} + \left\langle \frac{\partial O(t)}{\partial t}\right\rangle_{\psi(t)} }")
It is easy to see that a function 
solving
that preserves the differential equations and fixes the diagonal 
and thus could account for this invariant subsystem. It turns out that what matters is the structure of the mutual dependencies of the three subsystems making up the big system. That is, the evolution of 
depends only on 
the evolution of 
depends only on 
and the evolution of 
depends only on 
has no symmetries. Nonetheless, the existence of the invariant subsystem living on the diagonal 
can be deduced from certain properties of the graph 
The key is the existence of a surjective map of graphs 
with exactly one node, call it 
and one arrow. It is also crucial that 
has the following lifting property: there is a unique way to lift the one and only arrow of 
is the set of edges, 
is the set of nodes, and the two arrows assign to an edge its source and target, respectively. To each node we attach a phase space (or more formally a manifold, perhaps with boundary or corners). Here ‘attach’ means that we choose a function 
it assigns to each node 
a phase space 
(If we think of the set 
as a category of manifolds and smooth maps, then 
is simply a functor.) 
where 
as the collection of phase spaces of the subsystems constituting the network 
We refer to 
is 
while the phase space of the network 
is the real line 
in a graph 
depends on the states of the subsystem associated to the node 
To make this precise requires the notion of an ‘open system’, or ‘control system’, which we define below. It also requires a way to associate an open system to the set of arrows coming into a node/vertex 
controlled by the the phase space 
is a map 
is the tangent bundle of the space 
so that for all 
is a vector tangent to 
Given phase spaces 
For us, submersions of the form 
are general enough.)
(this is a very short tree, a corolla if you like). The input tree of a node 
In the example the input tree 
of the one node of 
For instance for the input tree in the example we just discussed, 
is the map
then 
is also a network. This allows us to talk about the phase space 
of an input tree. In our running example,
of open systems associated with every node 
is
to each one of them.
by 
and the space of vector fields on 
). In our running example the interconnection map for the network 
where 
is a phase space function and 
is a collection of open systems compatible with the graph and the phase space function. The corresponding vector field on the total space of the network is obtained by interconnecting the open systems.
In this category, by definition, a morphism from a network 
which is compatible with the phase space functions:
defines a map 
of total phase spaces in the opposite direction:
The objects of this category are pairs consisting of a phase space and a vector field on this phase space. A morphism in this category is a smooth map of phase spaces that intertwines the two vector fields. That is:
in 
then a morphism from 
to some dynamical system 
is simply a piece of an integral curve of the vector field 
defined on an interval 
And if 
is the constant vector field on the circle then a morphism from 
Proving that a given dynamical system has a periodic orbit is usually hard.
so that 
and the two interconnection maps 
and 
are compatible:
of open systems on the network 
is the free category with products on certain objects and morphisms coming from a directed acyclic graph 
