Seeing Through Walls: María Ángeles García Ferrero at CRM

From October to November 2025, María Ángeles García Ferrero held the CRM Chair of Excellence, collaborating with Joaquim Ortega-Cerdà on concentration inequalities and teaching a BGSMath course on the topic. Her main research focuses on the Calderón problem, determining a medium’s internal electrical conductivity from surface measurements, which has applications in medical imaging for distinguishing stroke types and monitoring brain function.

In October and November 2025, María Ángeles García Ferrero (ICMAT – Institute of Mathematical Sciences) spent two months at CRM as the Maria de Maeztu Chair of Excellence, working on problems that sound almost like magic: how to see what’s hidden, how to know what you can’t measure directly, how to reconstruct the invisible.

Her main field is inverse problems, the mathematics of gaining information about an unknown object from related data. “In particular, I work on the Calderón problem, which attempts to determine the electrical conductivity of a medium from measurements of voltage and current at the boundary,” she explains. It’s the kind of problem that only makes sense when you think about what it enables. If you can measure electrical signals on the surface of a human skull, can you map what’s happening inside the brain? If you know how electricity flows through different tissues, can you distinguish a stroke caused by bleeding from one caused by blockage?

She came to the field during her first postdoc. “It was connected with what I had done previously, but it was something different,” she says. The shift opened a new door, and what pulled her through was the applicability. “The applications of these problems, which in particular involve medical imaging techniques, were one of the reasons that made me stick with this topic. So, it interests me mathematically, but the real-world motivation gives me additional incentives to dedicate myself to it.”

Working with Joaquim Ortega-Cerdà

However, at CRM, García Ferrero worked on something different: concentration inequalities for holomorphic functions, a topic she developed through her collaboration with Joaquim Ortega-Cerdà (Universitat de Barcelona – CRM). “It’s a topic I got to know thanks to him,” she says. “Honestly, listening to him talk about any subject and not becoming interested is very difficult.”

The work they’re pursuing is both a continuation and an expansion. In a previous paper, they studied how concentrated polynomials can be when restricted to domains of the complex plane. Now they’re generalising that result, though not all the tools that worked before are available in the new setting. It’s the kind of problem that requires building new machinery.

From October 27 to November 5, García Ferrero taught a BGSMath course on precisely this topic: concentration inequalities for holomorphic functions. Five sessions, each two hours long. The course covered uncertainty principles, spaces of holomorphic functions, the Lieb conjecture on generalised Wehrl entropy, and the stability of these inequalities. It’s a field where conjectures from the late 1970s have only recently been proven and where quantitative stability results are still emerging.

The Calderón Problem

On November 19, García Ferrero gave a colloquium at the Maria de Maeztu Closing Conference, a gathering of the CRM research community marking the completion of a four-year strategic research program. The María de Maeztu distinction, awarded by Spain’s Ministry of Science and Innovation, recognises research units that demonstrate exceptional international scientific leadership. It provides substantial funding and connects CRM to the SOMMa alliance of Spain’s top research centres and units of excellence. It’s one of the highest honours in Spanish research.

Her talk returned to her main research: the inverse conductivity problem, also known as the Calderón problem. Named after Argentine mathematician Alberto Calderón, who posed it in 1980, the setup sounds deceptively simple. If you can measure voltage and current on the surface of an object, can you determine its internal electrical conductivity?

Think of it as X-rays, but instead of using radiation, you’re using electrical measurements. You apply voltage at the boundary, measure the resulting current, and try to reconstruct what must be happening inside. In medical imaging, this translates to Electrical Impedance Tomography (EIT). Different tissues conduct electricity differently: blood, bone, healthy tissue, tumorous tissue, bleeding, and blockages. If you can solve the Calderón problem, you can monitor brain function or monitor the lung activity of premature infants without invasive procedures or the risks of repeated X-rays.

“I think initiatives such as the MdM Chair of Excellence are important because sometimes we look for role models in our day-to-day life. This opportunity I’ve had to come here and talk with students, with people visiting CRM and also the universities in Barcelona, I think it helps a lot to give visibility to this role of women in mathematics, but in an approachable way.”

García Ferrero’s talk walked through the main questions: Can you uniquely determine the conductivity? If so, how stable is the reconstruction? Does a small measurement error lead to a catastrophic error in the recovered conductivity? And can you actually compute it? In one dimension, the problem has no solution at all: completely different conductivities can produce identical boundary measurements. In two dimensions, uniqueness was proven in 2006 for bounded conductivities. In three or more dimensions, the breakthrough came in the 1980s from John Sylvester and Gunther Uhlmann, but only for very smooth conductivities.

The talk also touched on what makes the problem mathematically delicate. The stability is logarithmic, meaning that to improve your reconstruction accuracy by a factor of 10, you need to improve your measurement accuracy exponentially. This isn’t just a technical annoyance; it is fundamental to the problem’s structure. Under certain special assumptions (for instance, if the conductivity is piecewise constant with known jump locations), the stability improves dramatically. But in the general case, you’re stuck with logarithmic stability.

The colloquium covered the full arc: the problem’s origins in geophysical prospecting and medical imaging, the mathematical framework of Dirichlet-to-Neumann maps that encode the boundary measurements, the construction of CGO solutions that make reconstruction possible, and the ongoing work to handle ever-more-realistic conductivity profiles. It was technical but accessible, pitched to an audience of mathematicians across specialities. And it showcased exactly what the María de Maeztu program aims to support: frontier research with both mathematical depth and real-world impact.

Visibility from the day-to-day

The Chair of Excellence program doesn’t just fund research visits. It’s designed to bring senior women mathematicians into sustained contact with the CRM community. “I think it’s important because sometimes we look for role models in our day-to-day life,” García Ferrero says. “This opportunity I’ve had to come here and talk with students, with people visiting CRM and also the universities in Barcelona, I think it helps a lot to give visibility to the role of women in mathematics, but in an approachable way.”

Visibility isn’t just about seeing someone at a conference. It’s about proximity. It’s about the conversations that happen in corridors, the questions asked after a course, the presence that becomes ordinary rather than exceptional. For García Ferrero, the Chair also carried personal significance. “At a personal level, it’s an honour because of the centre that grants it, because of the people I know are behind it. It’s a very important recognition. And at an even more personal level, the opportunity to return to the mathematical community in Barcelona, where I already spent time before, it’s an opportunity I couldn’t pass up.”

She’s based at ICMAT in Madrid now, a tenured researcher, elected to the Executive Committee of the European Mathematical Society in 2024. But for two months, her office was at CRM, her collaboration was with Ortega-Cerdà, and her work was on polynomials and concentration inequalities. The inverse problem she’s always working on (how to reconstruct the hidden from the visible) took on a different form. And the visibility she provided, simply by being present, working through problems in real time, is exactly what the program was designed for.

You can watch the full interview with María Ángeles García Ferrero on the CRM YouTube channel: