# Explaining maths without using them

At the time I was finishing my thesis some non mathematician friends asked me to explain them what I was actually doing.  In this post I write the explanation I elaborated for them. In one sentence, the first problem one need to know how to solve is the following.

How do you explain a mathematical result without using maths at all?

The answer is well known to everybody teaching maths,  however, I will elaborate a bit on this. The first step is to understand the historical roots of the problem you solved, together with the main motivations that people had to need to develop maths in such a determined direction that your problems appeared naturally. Once, you have understood completely this part, you need to associate each problem you solved to something that the people you want to explain the problem to can understand well. This means to associate each result to something completely different but that behaves exactly the same. Finally, you need to somehow transport all this information and deliver it to your audience in a comprehensive way. The last part of the process is well performed by an ”artistic drawing” (whatever that means) together with an explanation of each element. The hardest part of the communication is the explanation of the proof of each result, it happened indeed, that this becomes an issue of time.

In what follows I will answer a more specific question.

Problem:

Explain your thesis to your friends with time constrains conditions.

My thesis can be coded by the following drawing.

The drawing has 5 main elements: a black disk at the top left, a black disconnected curve at the top and bottom, a red apple shape at the left, a blue swirl at the bottom and a green tree shape in the centre and right of the drawing. I will explain each of the five element in what follows.

1. Black disk: If one goes back in time from the questions I faced in my thesis, we arrive to the three body problem. This is represented by the three white bodies inside the black disk. I can suggest you some nice reference to read about, like [Wikipedia, Celestial Encounters by F. Diacu and P. Holmes]. In a few words:  The n-body problem consists in determining the position of $n$ planet at any time $t$, given the initial conditions of known position and velocity at time $t=0.$ For $n=2$ the problem can be easily solved, one can prove indeed, that the trajectory of one planet with respect to the other always lies along a conic section. The problem for $n=3$ was addressed by Henri Poincare and literally he found the chaos in it. He was not able to find a solution, but instead, introduces qualitative methods to understand the solution, like, periodic solutions, recurrence theorem, non existence of uniform integrals, asymptotic solutions, dependence of the solutions with respect to a parameter, homoclinic orbits, first return map, invariant curves and homoclinic tangles. These qualitative methods are still our tools to understand dynamic systems that are too complicated to be understand in a deterministic way, i.e. those dynamical systems which equations can be  solved.
2. Black disconnected curve:  This curve represents the continuous trajectory of a particle that moves because of some physical laws acting on it. We suppose that there is a finite measure on the system drew that its invariant under the physical laws, in this case,  by the Poincare recurrence theorem, almost every particle will enter in a finite amount of time into a set $A,$ providing the measure of this set is strictly positive. The set $A$ in the picture corresponds to the black disk, and the black disconnected curve eventually enters $A$ after some (unknown) finite amount of time. The question that we face is how can we estimate the time that takes the particle to enter for first time $A.$ We solved this problem under very strong hypothesis on the physical laws. Estimations for this problem under our hypothesis are known, however, we refined the existing bound.
3. Blue swirl: This represents chaos in the classic sense of the diagram of phase of ODE´s (Ordinary differential equation). Recall that when we have an ODE $x'=Ax$ in $\mathbb{R}^2,$ the diagram of phase corresponds to the plot of the vectors $(y_1,y_2)$ in the $(x_1,x_2)$-plane, where we draw at the point $x= (x_1,x_2)$ the vector $(y_1,y_2)=Ax.$ This represent the velocity field of the solutions of the ODE.  We assume $A$ is not singular, so that the equilibrium is $x=0.$  The diagram of phase will be determined by the eigenvalues and eigenvectors $\lambda_1,\lambda_2$ of the matrix $A.$ There are 7 cases for the equilibrium state. The equilibrium state is:
1. Stable if $\lambda_1<\lambda_2<0.$
2. Unstable if $0<\lambda_1<\lambda_2.$
3. Saddle if $\lambda_1<0<\lambda_2.$
4. Degenerate if $\lambda_1=\lambda_2\in \mathbb{R}.$
5. Center if $\lambda_1=i \beta, \beta\in\mathbb{R}\setminus \{0\}.$
6. Stable spiral if $\lambda_1=\alpha+i \beta, \alpha<0, \beta\in\mathbb{R}\setminus \{0\}.$
7. Unstable spiral if $\lambda_1=\alpha+i \beta, \alpha>0, \beta\in\mathbb{R}\setminus \{0\}.$ In the drawing we represented this case.
4. Red apple shape: This represents a seed that gives birth to a tree after a process that allows to ”build” a complex structure from iterations of a simpler one. This is analogous to the system build by a simple iterated function scheme that consists of two disjoint contractions on the unit interval, these contractions have associated a unique non-empty closed invariant set. This sets comes from a structure similar to the one of a tree, where one branch divided into to smaller, and each smaller one into two smaller and so on.
5. Green tree shape: The green structure represent a tree that we consider analogous to the invariant set of an iterated function system. The problem that we face in the last chapter is: What happens if we perturb a little bit the seed of the tree, how different is going to be our tree? The analogy with this is: perturbing a little bit both maps of our iterative function and study how smoothly that affect the invariant set. A mathematical way to do this is by considering the for example the ”measure” (called Hausdorff dimension) of the invariant set.

# Metro Linea 5

This is the Santiago Metro Line 5:

There are three kind of stations, $R$ red, $G$ green and $M$ red/green. Between 6 and 9 hours and between 18 and 21 hours a train goes either along $R$ and $M$ stations, or along $G$ and $M$ stations. The rest of the time, trains stop at every station. This systems is part a “route express” program.

Is the route express system efficient ?

This turns out to be a highly non-trivial maths problem. First of all, we require to design an accurate model of the Santiago Metro Line 5.

Mathematical model:

1. We need to identify the passenger entrance rate to each station. It is certainly not a Poisson distribution, as there is not constant rate (it highly depends on the time) and moreover, the arrivals are not independent, because usually people enters in group. Actually, I currently do not know which discrete distribution would be the most appropriate. An obvious simplification is to consider it is Poisson with different rates depending on the time (example from 6 to 7, 7 to 8, etc.). In this case we could use some statistical test to adjust the rates at each station and at each hour.
2. We need to identify the passenger exit rate to each station. An ideal supposition is it that the exit rate at each station at the time $n+12 \pmod{24}$ is equal to its entry rate at the time $n.$ This would allows us to consider only entrance rates.
3. Once we have established the entrance and exit rate we can evaluate the efficiency of the system. We consider a random passenger $X,$ its expected travel time is $\mathbb{E}(X)= \sum \mathbb{P}$(going from the station $a$ to the station $b$)$\times$ Time(going from the station $a$ to the station $b$), where the sum is over all possible combinations of $a$ and $b.$ The function Time does depend on the strategy of the model (i.e. which stations are $R, G$ or $M$).

Choosing the optimal transportation plan:

A similar problem was addressed by the french mathematician Gaspard Monge in 1781.

1. Each strategy of colouring stations by $R, G, M$ will give a (optimal) travel time from the station $a$ to the station $b.$ We are no force in principle either to assign the same colouring for routes in opposite directions.
2. In order to solve the problem we would need to check $2 \times 3^{30}$ different possibilities and choose the one that gives the smaller value of $\mathbb{E}(X).$ As this is clearly impossible with modern computers, we need to design an efficient algorithm to solve this problem in a reasonable amount of time.

There is still the harder part of the work to be done in order to answer our original question. I wonder to know how strong are the assumptions by Metro de Santiago in order to determine its route express system.