A short explanation of my PhD results

I here give a short introduction to my PhD thesis, aimed at a broad audience.
You may find the thesis here: Non-affine horocycle orbit closures on strata of translation surfaces: new examples.



Like most of the Mathematics in the last 50 years, my work is at a crosspath between several theories (Geometry and Dynamics) and different levels of objects (the basic objects I study: translation surfaces, and the universe of all such surfaces: moduli space of translation surfaces). Let me start by explaining both of these points.

Geometry and Dynamics
Geometry is roughly the part of Mathematics concerned with looking at spaces with some structure ("structure" means something that can be used to determine what a straight line is), and studying the shapes and interactions of sub-objects.
Example 1: in Euclidean geometry (the geometry we do in High School), you care about points, lines, circles but also and foremost how lines with some property behave with each other. Something like "Given a line D and a point P, there is exactly one line parallel to D and passing throuhg P"
Example 2: in Differential geometry, the definition of a "straight" line is less strict: lines can be curved, but they cannot have corners (they are differentiable). Again, it is very interesting to see what lines can look like and how different lines (or sub-surfaces) interact with eath other
Example 3: if you know what the complex numbers are, you can imagine doing the same as in Example 2 but by only allowing complex-differentiable things. This again gives rise to a rich and interesting field of Mathematics.

Dynamics is the part of Mathematics that studies how points move if you apply a process to them many times in a row.
Example 1: take a point in the plane, and tell it to move randomly by 1 foot every second in either of the 4 cardinal direction (N-S-E-W). What is going to happen if you wait for a very long time? Is the point going to come back to its starting position? Or is it going to go to infinity? If so, in what way and how fast? This process is called a random walk, and you can do something like it in a much broader generality than just the plane.
Example 2: take any object in your favorite geometry (for instance a Euclidean surface like a plane, a sphere or a torus) and push a point in a given direction, i.e. following some straight line. Then what is happening to your point? Is it going away to infinity? Is it coming back to its starting point? Is it at least coming back very closely to its starting point? Does it spend more time in one part of your surface rather than the rest of it?

How Geometry and Dynamics are connected together
Imagine that you are a point on a surface that you know to be either a plane, a sphere or a torus. You are wondering about the universe around you. If you had access to sophisticated tools (such as the ability to measure curvature around you), you could easily tell in which of these spaces you might live. What if you don't have access to such tools? For instance if your geometry is not rigid enough to define curvature, or if your measuring tools are simply not good enough? (this actually happen in real life: if you look outside your window it's hard to see if Earth is a sphere or a plane).

Then you could look at the problem from a dynamical view-point and ask: what would happen if I started moving in a straight line on my planet?
Answer: if you were on a plane, you would go forever further and further away from your starting position. On a sphere you would go around an equator and get back safely to your starting point. On a torus you would always get back very close to your starting position but maybe not exactly to it.
Therefore by simply moving in some direction, you will know which one of the three universes you live in. In other words, you just solved a geometric problem by using Dynamics. The beauty of this technique is that the only two things you need are a way to define lines and infinite time. That's not very much to ask in the world of Mathematics.

The objects I study
I study something called translation surfaces: these are surfaces you can build out of lined paper, by cutting out polygons and regluing all of the sides by respecting one simple rule: you can only glue sides with same length and same direction. You can obtain a torus this way (take a square and glue the opposite sides togother) but you cannot obtain a sphere or a cone.
The study of such objects arose when people started asking dynamical questions about orbits of a ball on billiard tables, or of gas particles inside cubes. There are also some interesting applications of the theory of translation surfaces to something called quasi-crystals.

Now while these surfaces themselves present interesting behaviors to study, the most intriguing phenomena arise when you look at the space of all such surfaces. We call that space the Moduli space of translation surfaces.
In order to understand why it might be a good idea to look at that space, remember your Physics classes from High School or so: when physicists want to visualize the way some system evolves through time, they usually introduce something called a phase space. That is defined to be the space of all possible states of that evolving system.
Example: you want to study a ball launched in a pit with some initial force and some spin. To describe your system as time goes by, you can plot your ball on a dimension-9 space: 3 dimensions for the position of the ball (x, y and z), 3 for its speed in each direction, 3 for its spin (amount the ball rotates in each direction). This gives rise to a dynamical system inside a 9-dimensional space, in which interesting facts can be proven in an easier way.

The same goes for translation surfaces: you will usually obtain more powerful techniques and results by considering the dynamics on the moduli space of translation surfaces rather than on the translation surfaces themselves. And just to be clear: a "point" in the moduli space is actually a translation surface, and "moving along a straight line" in moduli space means "changing the initial translation surface in a linear way". Similarly, a point approaches infinity on the moduli space if the surface it represents degenerates into something which is not a translation surface. In moduli space, this actually happens in finite time, like if you were walking in a straight line inside a forest instead of a plane: whenever you hit a tree, you cannot walk further passed that direction. Something even more peculiar about moduli spaces of translation surfaces is that by running a full circle around a tree you might end up in a radically different spot than you started with.

My contribution
Let me start by being a bit more specific about what problem I work on. Moduli spaces can be split down into layers, much like a ram of paper can be split into many sheets of paper. These layers are called strata and it is on these strata that I do Dynamics. These strata look a lot like some classical objects called homogeneous manifolds. In this setting, Marina Ratner proved in the 90's a series of theorems showing that if you throw a point in a certain way (called unipotent) then this point will fill a nice sub-surface. By comparison, other ways of throwing this point might have led to fractal-like surfaces.

In my thesis, I elaborate tools to study a similar problem on strata. More explicitely, I show that if you start with a Veech surface (a specific type of translation surface) then pushing along unipotent ways can give rise to surfaces that are not "that" nice, but not fractal-like either. You can think of a "nice" surface here as some plain cheese like Comté, while a not-that-nice surface would be like Emmenthal and present many small holes (possibly infinely many of them, but all fitting nicely together). In contrast, a fractal surface would not look like any type of cheese you ever encounter in real life.
It is interesting to note that Alex Eskin, Maryam Mirzakhani and Amir Mohammadi proved in a ground-breaking paper that these types of small holes never appear if you push in 2 dimensions at the same time. Disclaimer: this is a very vague and somewhat erroneous way of stating what they actually showed. I invite the interested reader to look up the article on arXiv.

I am now working as a Consultant at Deloitte Touche Tohmatsu, in the Luxembourg office. I would love to discuss about translation surfaces with anyone interested! Feel free to reach out to me at lpc49 [at] cornell [dot] edu if you would like to chat about this further.

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