In his thesis, Eduardo Tablate – adjunct professor at CEU San Pablo University – was able to bring together the two areas of mathematics that most interested him: group theory and functional analysis. After several research initiation grants, he began working in these fields within the group led by Javier Parcet (ICMAT-CSIC), during his master’s degree, thanks to an ICMAT Master’s SO grant. Parcet and José Conde (ICMAT-UAM) were the directors of his thesis. Now, for this work, he receives one of the Vicent Caselles awards granted by the Spanish Royal Mathematical Society and the BBVA Foundation.

Ágata Timón García-Longoria (ICMAT)

What does receiving the Vicent Caselles Award mean to you?

It has already helped me apply for postdoctoral research projects. It will allow me to have more professional opportunities. In addition, it is recognition of all the work done in the thesis, and also the effort both Javier and José made as supervisors. And well, it encourages me to keep moving forward in research; it seems we are on the right track, and we must continue this way.

Another important recognition of your work is that part of the results obtained during your thesis were published in Annals of Mathematics.

Yes, it was the first publication we made during the thesis, in 2023. It was the result of many factors, although I honestly believe the main one was luck. The research group I joined had a clear understanding of the important problems to work on and had a lot of experience and good ideas to tackle them. We addressed the main question of the article and the thesis at a time when the previous theory had developed enough to make the leap to solving it possible. Tackling that problem ten years earlier wouldn’t have been fruitful.

Weren’t you overwhelmed by facing such a big problem right at the start?

My supervisors always told me it was an important problem and that we had to be open to the possibility of not solving it, but they passed on more excitement than stress. A very important part of mathematical work is related to mood and motivation. Being motivated and happy doing what you’re doing are some of the keys to succeeding in what you’re trying to solve.

What is the central problem of your thesis about?

We studied so-called Schur multipliers, which are mathematical objects that transform matrices by altering their entries — for example, by multiplying each entry by a number defined by what we call a symbol. But despite this simple definition, little was known about their properties, especially their boundedness properties. That is, if you have a matrix and apply a Schur multiplier, how large is the resulting matrix compared to the original one? Of course, this depends on the multiplier. The central question of my thesis was to understand this relationship between how entries are transformed and the changes produced in the matrices.

“Schur multipliers appeared in the background of many important problems in various areas of mathematics.”

Why is this problem interesting to the community?

The previous question has many connections: with geometric group theory, operator algebras, perturbation theory… It was a question that appeared in the background of many important problems across various fields of mathematics. Even if not in its full general form, as we addressed it in the thesis, understanding certain families of these multipliers was key to solving open problems in different areas. For example, in solving the Krein conjecture in 2011, the fundamental ingredient was understanding a specific family of these multipliers. Additionally, Uffe Haagerup and Michael Cowling showed, back in the 1980s, that there is a connection between the properties of Schur multipliers and the geometry of the operator spaces where they act, or of the groups associated with those operator spaces.

They are also linked to problems in quantum physics, right?

Yes, they are related to what are known as quantum channels, and also to quantum harmonic analysis, to Fourier multipliers. Quantum harmonic analysis is conducted on operator algebras, and for the development of the theory, Schur multipliers are a cornerstone. In my thesis, we extended key results from classical harmonic analysis to the setting of Schur multipliers and noncommutative harmonic analysis. We proposed a theory parallel to the theory of singular integrals for smooth Schur multipliers. On the other hand, we studied the interaction between curvature and classical harmonic analysis and examined how it translated into the context of Schur multipliers. If we also restrict ourselves to a certain class of multipliers – the so-called idempotents – it is possible to characterize their boundedness properties, and there is an interaction with curvature.

What was your working process like in solving the problem?

Overall, very good, although there are always ups and downs. Often, you think you’ve solved something and then realize it’s wrong. You notice it yourself, you don’t even need to discuss it. You leave the office saying “that’s it,” and you’re on the train, happy, thinking you can solve anything, and suddenly you realize you were wrong. And you go back to the beginning. That didn’t happen in this case, but it has happened in others we worked on. We had a draft ready to send, and I realized half of it was wrong. We had to redo it. For days, nothing worked. On the other hand, there are hard moments related to job uncertainty. At the beginning of a PhD, you’re very comfortable, but when the contract end approaches, around four years in, you need to find another one. And if you don’t, you fall off the track. That was tough for me.

“Mental health is fundamental, and it’s a reflection the research community has come to late.”

It seems the scientific community is now beginning to recognize how these pressures affect its members.

Mental health is essential, and it’s a reflection the research community has arrived at too late. Mental health problems are common. Talking to colleagues helped me. Most of them are sincere. I’ve had lunch with colleagues on days they were depressed, when things weren’t working out. And vice versa. We’ve had many long conversations about these topics. I think it helps, because it gives you perspective. And if necessary, one can seek professional help without any problem. In our group, we’ve never had any prejudice about that. In the moments when I’ve been the saddest or needed the most help, my two supervisors understood and supported me. I would do the same. I believe there’s still a long road ahead, but we’re moving in the right direction.

What other problems are you currently working on?

Together with Javier, in the context of the Gilles Pisier–Mikael de la Salle Laboratory, we are trying to understand intermediate families between smooth Schur multipliers and idempotents. This is closely related to studying the geometry of certain spaces and operator algebras. Just as groups have been classified, classifying operator algebras is an important problem. For instance, algebras associated with simple, semisimple, or semisimple lattice groups appear to be linked to the properties of these objects. It’s a more ambitious goal than what we tackled in the thesis. In fact, we’ve been working hard for months and haven’t made much progress yet. We’ve hit a wall several times, but that’s the natural path forward.

“I spent three summers at ICMAT as an undergraduate student with Intro–Severo Ochoa grants.”

How did you become interested in this field?

During my studies, I spent three summers at ICMAT as an undergraduate with Intro–Severo Ochoa research initiation grants. One year with Yago Antolín, another with Aníbal Rodríguez, and another with Jesús Munarriz. Each year I chose a different area to learn, and it was a great way to connect with the research world. Then, in my undergraduate thesis, I started working on classical harmonic analysis and dispersive equations, specifically the Schrödinger equation, with Ana Vargas, through a collaboration grant at the Autonomous University of Madrid. That’s when I realized I loved classical harmonic analysis, but also group theory and functional analysis. I was fascinated by non-standard, non-commutative structures. I loved seeing how geometric intuition from classical cases translated into these structures. Ana recommended I speak with José [Conde] and Javier [Parcet], who kindly welcomed me. I applied for a Severo Ochoa master’s grant from ICMAT to do my master’s thesis with them and got it, so I began working with them. Then we applied for a PhD grant and… here we are.

“The Intro grants were my first real contact with serious mathematics.”

What would you highlight about the Intro–Severo Ochoa grants?

They were my first real contact with advanced mathematics, with mathematical research, and they guided me a lot. The math taught in university classes is very closed-ended, everything works. There’s no space to ask the professor something and be told, “I don’t know, we’ll have to think about it.” At first, it’s a scary feeling. But little by little, you adapt and become drawn to that context.

When did you realize you wanted to pursue mathematics?

I’ve always liked mathematics a lot, even in high school. But I also really liked philosophy. In fact, I debated whether to study one or the other. What attracted me most about mathematics is that it’s a field where you’re very autonomous. You can read on your own, and things have reasons. I also felt it was where I could best use my abilities and do my best work.

“I debated whether to study philosophy or mathematics.”

What are the next steps in your career?

I have several possibilities on the table. The natural path is to do a postdoctoral stay in another place, another country. Ideally, with a related researcher who doesn’t do exactly the same, to learn new things and continue growing as a mathematician. Later, I’d like to return to Spain and reintegrate into the national research community.

Is there any advice you wish you had received at the beginning of your thesis?

My supervisors gave me excellent advice, so I would pass on the same. First, choose your thesis advisors very carefully. Make sure the group you join works on topics you like, as much as possible. There will be many things you don’t understand, but at least be attracted to the themes, since you’ll be spending a lot of time on them. Also, make sure the group has a strong publication record, since it’s the context where you’ll learn and connect with good mathematicians. Then, try to communicate a lot, ask questions when you don’t understand, without fear of admitting ignorance. At first, you’ll attend meetings and understand almost nothing, but over time, you’ll grasp more and more. And lastly, enjoy it — the PhD is a beautiful stage. You don’t need to impose excessively high demands on yourself or think too far ahead, especially in the beginning — just take it day by day.

What do you enjoy outside of mathematics?

I’m a fan of music, chess, and football.

Any particular music genre?

I like all kinds — classical, rock. I often work with music. I associate certain songs with parts of the thesis work.

Could you recommend one?

Sultans of Swing, by Dire Straits.

Note: This is an automatic translation. Some nuances may differ from the original Spanish version.