Eva Miranda: "A man advised me to read Miranda's work: I had to tell him that Miranda was me."

In the office ofEva Miranda (Reus, 1973), you're welcomed by a sea of ​​rubber duckies of different sizes and outfits; a very curious orange plush toy; and a suspiciously Newtonian plastic apple. In the background, blackboards are covered in mathematical formulas, indecipherable to those outside the field. A professor at the Polytechnic University of Catalonia and a member of the Center for Mathematical Research, she has been distinguished as an ICREA Academia member and has received the most prestigious international awards. Her research focuses on the areas of geometry and topology, the study of shapes, and their interactions with mathematical physics.

She says that she has been fascinated by the stars since she was little and dreamed of working at NASA. Perhaps that's why part of her research is linked to the movement of satellites and the famous three-body problem. Recently, she participated in a conversation with J. Doyne Farmer, one of the fathers of chaos theory, organized by the Museum of Natural Sciences as part of the activities associated withexposure The Invention of Time.

At six years old, children associate being very bright with being a boy.

— And this is tantamount to thinking that men are the only ones good at science. If you repeatedly tell a group that they're not good at something, it's clear they'll believe it and won't pursue those fields of knowledge. We continue to talk a lot about women's bodies, about their roles, but not about their abilities. So, here we have a very serious social problem that, moreover, despite efforts to improve it in recent years, is getting worse. Let me tell you an anecdote.

Forward.

— I have been invited to become the Karen Uhlenbeck Professor [the first woman to win the Abel Prize, considered the Nobel Prize of mathematics], which is a very prestigious award, at Princeton University in 2027. This distinction is part of a program focused on promoting female talent called WAM (women and mathematics, Women and Mathematics). But now the acronym no longer means this. Trump. In fact, much support has been withdrawn from initiatives promoting women scientists and diversity in science. The situation is becoming increasingly complicated. Generally, it's a bad time for research, and this is noticeable worldwide. Being a woman is, frankly, very difficult in the world we live in. I still encounter surprising situations.

But you are a full professor at the UPC, a position held by just over one in four women. Furthermore, you are an ICREA Academy member and have received the most prestigious international awards in mathematics.

I am in a privileged position, yes, however, despite the awards I have received, I still feel very questioned. When I was younger, I didn't realize it. I remember, for example, in a talk where I was presenting my research, a man raised his hand to ask me if I had read Miranda's work. And I had to answer that I was Miranda. Women have it hard; we always have to justify ourselves. And even now, when I participate in scientific articles with male colleagues, the new ideas in the articles are, in a way, attributed to the men. But I'm optimistic that we can improve.

Given this scenario, how can we foster an interest in mathematics among girls?

— The teacher is the most important part; they are key. Because they have to inspire all the students, they must motivate them. They must be their role model. The problem is that there aren't many mathematicians teaching in secondary school because, honestly, the salaries aren't very attractive compared to other career paths available to those with these degrees. This means that only those with a very strong vocation for teaching dedicate themselves to it, and mathematics... They end up being taught by people who come from other backgrounds, They may have varying degrees of passion for the subject, and they can provide real-life examples of how the mathematics they teach students can be applied. In secondary school, I was crazy about math, and poetry too; I even wrote some, and one day I dared to show it to my language teacher, who snapped at me: "Too predictable!" That was the final straw. And it made me choose math. I don't regret that choice, mind you! But I think it's a demonstration of the influence of teachers.

As a child, she also dreamed of working at NASA.

— I imagined myself doing calculations; it was one of my obsessions. And I would make up stories to tell my mother about the jobs I would do there. And, interestingly enough, what I'm doing now is related: I'm trying to solve a rocket fuel-saving problem using symplectic geometry, which is the area of ​​mathematics I work in. Geometry is the study of shapes and the sizes of shapes. In symplectic geometry, instead of measuring with a tape measure, we measure area.

What does this have to do with rockets?

— Imagine a rocket traveling along a certain path; think about the trail it leaves behind. If the rocket moves a certain distance—measured from its launch point—the resulting geometric shape is a triangle. Interestingly, as the trajectory progresses, the area remains constant because it follows Hamilton's equations of physics. These equations measure a field, which, to greatly simplify, is like the velocity of a particle, and they are the equations that govern the world, and they are symplectic! In fact, all trajectories studied since the beginning of time using Hamilton's equations actually follow symplectic geometry.

An example of its application is the Japanese Hiten probe and what the mathematician Edward Belbruno did.

— He is a fascinating scientist, mathematician, and artist who works in the field of symplectic geometry with applications in space mission design. In this particular case, the Japanese probe Hiten had run out of fuel to enter lunar orbit using a conventional trajectory. Belbruno applied chaos theory to spaceflight and managed to make the spacecraft jump out of orbit. He saved it! And, for that reason, that trajectory has borne his name ever since. Symplectic geometry is, in a way, the language of physics and has applications in many fields, particularly in the design of space trajectories.

Elon Musk must have mastered these equations, because he keeps launching satellites into space...

— And it's a huge problem because it creates an overload of space debris; we have a lot of objects lost in space that could fall to Earth. In fact, we've already had a few close calls... Space debris is monitored using symplectic geometry. Both NASA and the ESA apply a formula to calculate the probable trajectory of objects when they re-enter Earth's atmosphere. And I ended up working on all this somewhat by chance, because I've always been involved in more abstract research.

Such as the work he did together with fellow mathematicians Daniel Peralta, Robert Cardona and Francisco Presas, with whom they proved that it is impossible to know where a glass bottle with a message that we threw into the Mediterranean today would end up.

— There's a funny story behind this. In February 2020, shortly before the pandemic, I was returning by train from Madrid when I saw a question on the X network posed by the Australian mathematician Terence Tao, which was related to one of the Millennium challenges. These challenges are extremely complex problems published in 2000 by the Clay Mathematics Institute, a US foundation. Tao announced he would award one million euros to anyone who solved any of them. One of these problems involves the Navier-Stokes equations, which describe the motion of liquids and gases. Even now, two centuries after they were first posed, we don't know if they have a long-term solution.

What does it mean?

— If I told you right now that a tsunami was coming, you wouldn't believe it; that only happens in movies. In reality, we have systems capable of detecting this type of event minutes or even hours in advance, allowing us to send alerts to the population. Well, mathematicians don't know how to prove whether the equations that model the movement of water and fluids—the ones engineers use in a simplified version—have a solution, or whether, on the contrary, a tsunami can occur suddenly, a "singularity," and therefore these equations aren't good enough to describe it. This is what Tao proposed in an article, and what he set out to prove. He wanted to abstractly force the liquid to accumulate energy until a tsunami occurred, which he could then capture using the Navier-Stokes equations. And when I read it, a lightbulb went off in my head. We could approach the question with the tools we have, which are geometric.

As?

— We associate fluid motion with a computer, a kind of abstract water machine that takes a point in space as input and outputs the point to which the fluid has moved. As the rubber duck we threw into the sea moves, you calculate its position with this computer. To do this, we use contact geometry and rely on a Turing machine. And we have shown that there could be undecidable trajectories.

What does it mean?

— If we associate the rubber ducks with our water computer, trajectories appear for which no computer can determine whether or not they will reach a specific area in a finite time. This is a physical manifestation of the halting problem, which Alan Turing proved to be undecidable; there are questions that no algorithm can answer. In this case, no algorithm allows us to say whether the toy will pass through a specific point at a given time. Fluids are so complex that not even a computer can decide! We know that there are limits in mathematics, and we know this because of the theory of logical chaos, which is related to what I'm explaining. There are places where we could say that "calculations lose coverage," like cell phones. Paradoxically, understanding these limits is what gives us humans the most strength; knowing that there are things that computers cannot do, like calculating where the space debris orbiting Earth will end up (how we want to demonstrate this!) or where a rubber duck will eventually land, shows that people and our creativity still have much to contribute.

Isn't it frustrating not being able to predict where the duckling will go?

— Ultimately, unpredictability simply means that things change. If I had gotten up fifteen minutes earlier, I would have caught an earlier bus and wouldn't have been right on time for this interview. In other words, I can alter things in the past that will affect the future, and a small change can cause a major long-term change, because if you miss the three o'clock bus, you might have to wait an hour and miss the interview. Chaos theory studies this. And the issue of undecidability complicates everything even further because it goes beyond chaos; it tells you that you can't measure it, that there's no way to compute or calculate it. We published this conclusion in the journal Proceedings of the National Academy of Sciences (PNASand it had a great impact.

Did they solve the challenge posed by Tao?

— We tried, but our solution didn't prove what Tao was trying to prove. And we didn't win the million dollars either.

And yet, from here also arises the idea of ​​making a water computer.

A water-based computer is not actually possible to build because the necessary conditions are not stable. However, we have a new computing model that we call a hybrid machine or Topological Kleene Field Theory (TKFT) model. This is a much more robust version that uses the concept of movement, of trajectory velocities, which must exist. We have already created an abstract design, together with Ángel González-Prieto and Daniel Peralta-Salas. Our hybrid computer can be visualized as a system of pipes through which water flows. And the shape of these pipes can be extraordinarily complex. We are convinced that our new computing model will surpass all the types of computers we have now, including quantum computers. Proving this is very difficult, but we are working on it and are at a very exciting point.