In the Basic Sciences category

# The Frontiers of Knowledge Award goes to Claire Voisin and Yakov Eliashberg for work that has driven forward mathematical thought by building bridges between two key areas of geometry

The BBVA Foundation Frontiers of Knowledge Award in Basic Sciences has gone in this sixteenth edition to Claire Voisin (National Centre for Scientific Research, CNRS, France) and Yakov Eliashberg (Stanford University, United States) for work that has driven forward mathematical thought by breaking down barriers and bridging the space between two key areas of geometry. The awardee researchers have made “outstanding contributions” to algebraic and and symplectic geometry, which explore “spaces in high dimensions, which are difficult to visualize and necessitate new mathematical techniques to understand and study,” said the committee in its citation.

14 February, 2024

These two fields have gained growing importance in recent years through their links with theories of quantum physics, which explores the fundamental properties of matter and energy on the subatomic scale. Working independently, the awardee mathematicians “have played essential roles in developing these different aspects of geometry, in particular by adapting and relating concepts from either side, crossing the boundary between the two disciplines,” the citation continues. Their work, it adds, “has inspired a high level of activity in international research in both areas of mathematics.”

“For researchers in our discipline, there’s no greater stimulus than when you break down the barriers between two areas, because by doing that you can adopt a new language, possibly a new framework, a new way of looking at things from the other side, which enables you to make further progress. If you can frame something you have a problem about in another format, then sometimes you can see the way forward. This has been among the main contributions of Voisin and Eliashberg, who have furthered the progress of mathematics by dismantling the barriers between areas of geometry,” explains committee member Nigel Hitchin, Emeritus Savilian Professor of Geometry in the Mathematical Institute at the University of Oxford (United Kingdom).

Algebraic geometry is a classical mathematical discipline that starts from a class of simple equations, those defined by polynomials, and studies their solutions from the standpoint of geometry. “It’s a discipline that has a certain rigidity,” Hitchin explains, because even the slightest modification to the geometric objects under study can change their properties out of all recognition. Symplectic geometry, which numbers Eliashberg among its founders, arises from the geometric objects that describe motion in physics. It is, in theory, a “more flexible” discipline, says Hitchin, whose roots lie in the study of how position and velocity vary with time.

Voisin and Eliashberg have drawn parallels between algebraic and symplectic geometry, bringing to light the former’s more flexible side and the more rigid aspects of the latter, while applying tools from each discipline to study problems routinely assigned to the other.

**The discovery of “mirror symmetry” between two areas of geometry**

Looking back, Voisin is clear that her feeling for mathematics was not “love at first sight” but a passion forged slowly over the course of many years. In fact, although she was “very good” in the subject at school, she tended to dismiss it as “not really deep.” Her attitude would change on being lent an algebra textbook by an older brother, and it was then, during her teenage years, that she began “learning maths for pleasure.” Even so, she had no notion of making it her profession, but was simply “following my interest, I didn’t think to relate it to the future.” It was at university, particularly when starting work on her doctoral thesis, that she discovered “the wonderful ideas” of algebraic topology, a field that uses algebraic tools to study certain properties of geometric objects, and began to be “deeply interested” in mathematical thought. “I just followed my path and suddenly, gradually, I realized that it was all becoming extremely interesting.”

Thus began a prolific research career that would yield notable new insights in the field of algebraic geometry. The researcher soon realized that the phenomenon known as mirror symmetry, already described by other authors, could be a way to build bridges between algebraic and symplectic geometry. Given that both geometries are important in some areas of physics, there was already a suspicion that some relationship must exist between the mathematical objects of one and the other. “I was really shocked when I came across the mirror symmetry conjecture,” Voisin recalls today, and it was this element of surprise that inspired her to study it in detail. She set out her conclusions in the book *Mirror Symmetry*, published in 1996, helping bring about what she describes as “the dynamics of exchange between symplectic geometry and algebraic geometry.”

Today, both symplectic and algebraic geometry have acquired renewed importance, because of their potential to provide mathematical foundations for quantum field theory – a branch of quantum physics that is being used with great success in particle physics, but which lacks a firm mathematical footing. To address this shortfall, a highly active line of research is seeking to reconstruct quantum field theory using the formulations of symplectic or algebraic geometry, to explore whether the deducible physical consequences match with reality.

Of her later work, the mathematician says she has taken most pleasure from the articles in which she has obtained “an important result, but one that was easy to state and with a method that was elegant, simply because I found a new way of thinking about the problem.”

For instance, in a 2004 paper, published in *Inventiones Mathematicae*, she concluded that there were objects within algebraic geometry, known as Kähler manifolds, that could not be obtained by deforming other, apparently related manifolds. To prove this impossibility, she used tools from topology, a branch more closely related to symplectic than to algebraic geometry.

**From “internal exile” in the USSR to transforming mathematics at Stanford**

Born in Leningrad (now St. Petersburg) in 1946, Eliashberg’s first passion was not numbers and equations, but music. “When I was a small child, I thought I would devote my life to playing the violin,” he recalls. But at around 13, after doing well at school Olympiad, he was invited to join a young people’s mathematical circle. It was there that an “excellent teacher” woke in him a fascination for the subject, which he would go on to choose for his degree course at Leningrad University.

Despite being a brilliant student and performing outstandingly with his doctoral thesis, he relates how “different circumstances” in the Soviet Union of the time forced him into a kind of internal exile. In 1972, specifically, he was sent to work at a university in the city of Syktyvkar, almost 1,300 kilometers northwest of Moscow, where winter temperatures could drop to many degrees below zero. He made his first application for a foreign travel visa in 1979, but was not only rejected, but sentenced in response to “a kind of eight-year limbo,” during which he was expelled from the university and cut off from mathematical life.

It was not until 1987, “thanks to the more open climate of Perestroika,” that he was granted a travel visa for the United States, where he would reignite his brilliant research career, firstly at Berkeley and, since 1989, at Stanford University, where he was appointed to the professorship he holds today.

Eliashberg’s research “has fundamentally transformed several areas of geometry, created some new ones, and revealed unexpected connections between previously unrelated fields,” in the words of his nominator Kai Cieliebak, Professor of Analysis and Geometry at the University of Augsburg (Germany). The awardee helped found symplectic geometry and a related field, symplectic topology, likewise concerned with the objects that describe motion but with a focus on those of their properties that do not change when the objects deform. Although the underlying ideas of both fields emerged in the 19th and early 20th centuries, it was not until the 1980s that Eliashberg’s insights cemented the existence of symplectic topology, ushering in the prolific research field we know today.

However, due to the discrimination he suffered under the Soviet regime, he was unable to publish his work until 1987, when it appeared in *Functional Analysis and its Applications*. A year earlier, he had received an invitation to speak at the International Congress of Mathematicians (the most important global gathering of the mathematical community), but “of course I was not allowed to go, so my talk was presented by someone else.”

Eliashberg was also responsible, with Mikhail Gromov, for establishing the homotopy principle, which allows to solve differential equations and differential relations, and has applications in differential geometry, including symplectic topology and isometric immersion problems, as well as in partial differential equations and fluid dynamics.

With Helmut Hofer and Alexander Givental, he shaped the new branch of contact geometry and, within symplectic geometry and topology, pioneered an entire subfield closer to algebraic geometry known as symplectic field theory. “There are some questions you can approach from the symplectic side and others from the algebraic,” Eliashberg remarks. “They are kind of complementary.” And what his work has shown is that combining the two is often the best route to a solution.

**The utility of a “source of knowledge” with “a very precise notion of what is true”**

Voisin admits that her research “has no direct application” in solving practical problems. But then again, she adds, in mathematics “you never know what will be useful,” whether it be to inspire new advances in our basic knowledge of nature or to hasten technological development.

In effect, she sees mathematics as primarily “a fact of civilization,” with a cultural value comparable to that of music: “Doing mathematics,” she affirms, “is a source of knowledge, a way of attaining knowledge that is at the root of something fundamental in human activity.” On the one hand, she says, is the fact that mathematicians “have a very precise notion of what is true, of what is known and what is not known. For us, a key point is to prove. And when something is not proved, we cannot call it a statement.”

Then of course we have today’s society, inundated with screens and saturated with instant messages that reach us via multiple channels. Against this dizzying reality, Voisin stakes a claim for mathematics as an essential mental discipline: “For me, it’s a form of concentration and I think many people today do not realize how important it is to know how to concentrate.”

Eliashberg, meantime, talks of “the beauty of the new worlds created by mathematics,” frequently the product of a creative dialogue between people from different fields: “I believe that both in mathematics and science in general, the most wonderful results come from discovering the connections between seemingly different things. I am excited by different areas of mathematics working together and the fruits that may come from discovering unsuspected links.”

Like Voisin, he points up the fact that, as the history of science has repeatedly shown, “if you have a great mathematical idea, then it’s almost sure at some point to bring some kind of application that is useful to society.” Indeed his own research is being put to use in the design of future space missions: “Some of my colleagues are working with NASA to try to apply tools derived from my work in the optimization of spaceship or satellite trajectories, consuming as little fuel as possible and taking advantage of the gravitational pull of planets.”

But his most cherished hope is that his work as a professor will serve to inspire new generations of young mathematicians who “can benefit society in many ways, perhaps in ways that I am not even capable of imagining.”

**Nominators**

A total of 83 nominations were received in this edition. The awardee researchers were nominated by Daniel Álvarez-Gavela, Assistant Professor of Mathematics at the Massachusetts Institute of Technology (United States) and 2019 Vicent Caselles Mathematical Research Prize laureate; Kai Cieliebak, Professor in the Institute of Mathematics at the University of Augsburg (Germany); Radu Laza, Professor in the Department of Mathematics at Stoney Brook University (United States); and Rafe Mazzeo, Professor in the Department of Mathematics at Stanford University (United States).

**Basic Sciences committee and evaluation support panel**

The committee in this category was chaired by **Theodor Hänsch**, Director of the Division of Laser Spectroscopy at the Max Planck Institute of Quantum Optics (Germany) and the 2005 Nobel Laureate in Physics, with **Aitziber López Cortajarena,** Ikerbasque Research Professor, Scientific Director and Biomolecular Nanotechnology Group Leader at CIC biomaGUNE, Center for Cooperative Research in Biomaterials (Spain)**, **acting as secretary. Remaining members were **Emmanuel Candès**, Barnum-Simons Professor of Mathematics and Statistics at Stanford University (United States),** María José García Borge**, Research Professor at the Institute for the Structure of Matter (IEM), CSIC (Spain), **Nigel Hitchin**, Emeritus Savilian Professor of Geometry in the Mathematical Institute at the University of Oxford (United Kingdom), **Martin Quack**, Professor and Head of the Molecular Kinetics and Spectroscopy Group at ETH Zurich (Switzerland), and **Sandip Tiwari**, Charles N. Mellowes Professor in Engineering, Emeritus at Cornell University (United States) and Distinguished Visiting Professor at the Indian Institute of Technology, Kanpur (India).

The **evaluation support panel** was organized into three groups. The Physics Group was coordinated by **Marisol Martín González**, coordinator of the MATERIA Global Area and Research Professor at the Institute of Micro and Nanotechnology (INM-CNM, CSIC) and formed by **Alberto Casas González**, Research Professor at the Institute for Theoretical Physics (IFT, CSIC-UAM); **Alfonso Cebollada Navarro**, Research Professor at the Institute of Micro and Nanotechnology (IMN-CNM, CSIC); **Lourdes Fábrega Sánchez**, Tenured Scientist at the Institute of Materials Science of Barcelona (ICMAB, CSIC); and **Alejandro Luque Estepa**, Tenured Scientist at the Institute of Astrophysics of Andalusia (IAA, CSIC). The Chemistry Group was coordinated by **José M. Mato**, General Director of CIC bioGUNE and CIC biomaGUNE, and formed by **Miguel Ángel Bañares González**, Research Professor at the Institute of Catalysis and Petrochemistry (ICP, CSIC); **Ethel Eljarrat Essebag**,** Director and **Scientific Researcher at the Institute of Environmental Assessment and Water Research (IDAEA, CSIC); **Francisco García Labiano**, Deputy Coordinator of the MATERIA Global Area and Scientific Researcher at the Institute of Carbochemistry (ICB, CSIC); **Jesús Jiménez-Barbero**, Scientific Director of CIC bioGUNE and Ikerbasque Research Professor in the Chemical Glycobiology Lab; **Gonzalo Jiménez-Osés**, Principal Investigator in the Computational Chemistry Lab at CIC bioGUNE; **Luis Liz-Marzán**, Principal Investigator in the Bionanoplasmonics Lab at CIC biomaGUNE; **Aitziber López Cortajarena**, Ikerbasque Research Professor, Scientific Director and Principal Investigator in the Biomolecular Nanotechnology Lab at CIC biomaGUNE; and **María Luz Sanz Murias**, Scientific Researcher at the Institute of General Organic Chemistry (IQOG, CSIC). The Mathematics Group was coordinated by **José María Martell Berrocal**, CSIC Vice-President for Scientific and Technical Research, and formed by **María Jesús Carro Rosell**, Professor of Mathematical Analysis at the Universidad Complutense de Madrid; **Alberto Enciso Carrasco**, Research Professor at the Institute of Mathematical Sciences (ICMAT, CSIC); **Francisco Martín Serrano**, Professor of Differential Geometry at the University of Granada; and **Rosa María Miró Roig**, Professor in the Department of Algebra and Geometry at the University of Barcelona.

**About the BBVA Foundation Frontiers of Knowledge Awards**

The BBVA Foundation centers its activity on the promotion of world-class scientific research and cultural creation, and the recognition of talent.

The BBVA Foundation Frontiers of Knowledge Awards, funded with 400,000 euros in each of their eight categories, recognize and reward contributions of singular impact in physics and chemistry, mathematics, biology and biomedicine, technology, environmental sciences (climate change, ecology and conservation biology), economics, social sciences, the humanities and music, privileging those that significantly enlarge the stock of knowledge in a discipline, open up new fields, or build bridges between disciplinary areas. The goal of the awards, established in 2008, is to celebrate and promote the value of knowledge as a public good without frontiers, the best instrument to take on the great global challenges of our time and expand the worldviews of each individual. Their eight categories address the knowledge map of the 21st century, from basic knowledge to fields devoted to understanding and interrelating the natural environment by way of closely connected domains such as biology and medicine or economics, information technologies, social sciences and the humanities, and the universal art of music.

The BBVA Foundation has been aided in the evaluation of nominees for the Frontiers Award in Climate Change by the Spanish National Research Council (CSIC), the country’s premier public research organization. CSIC appoints evaluation support panels made up of leading experts in the corresponding knowledge area, who are charged with undertaking an initial assessment of the candidates proposed by numerous institutions across the world, and drawing up a reasoned shortlist for the consideration of the award committees. CSIC is also responsible for designating each committee’s chair across the eight prize categories and participates in the selection of remaining members, helping to ensure objectivity in the recognition of innovation and scientific excellence.