1.1. The CNRS Bronze Medal.

The Centre national de la recherche scientifique awards its Bronze Medal rewards researchers in the early stages of their career whose work has established them as specialists in their field. This award is a form of encouragement from the CNRS to continue with ongoing research that has already proved successful.

1.2. The 1988 CNRS Bronze Medal.

In 1988 the Centre national de la recherche scientifique awarded a Bronze Medal to Rémi Brague, François Déroche, Dominique Finon, Eric Gaffet, Bernard Henrissat, Valérie de Lapparent, Pascal Massart, Jean-Paul Montagner, Thierry Poinsot and Claire Voisin.

2.1. The EMS Prize.

The European Mathematical Society prizes were established in 1992. At each European Congress of Mathematics up to ten EMS prizes are awarded to early career researchers not older than 35 years at the time of nomination, of European nationality or working in Europe, in recognition of excellent contributions in mathematics. In the event of maternity, the upper limit is normally extended by 18 months for each child. For other parental, adoption, compassionate or extended sick leave the upper limit is extended to match the documented time spent on leave. The maximal extension is up to the age of 38 years.

Mathematicians are defined to be European if they are of European nationality or their normal place of work is within Europe. Europe is defined to be the union of any country or part of a country which is geographically within Europe or that has a corporate member of the EMS based in that country.

2.2. Claire Voisin wins the 1992 EMS Prize.

The European Mathematical Society prizes were established in 1992 and ten were awarded in the first year. The prizes were presented at a reception held at the Hotel de Ville, Paris, where Jacques Chirac, Mayor of Paris and former Prime Minister of France, presented prizes sponsored by the city of Paris on the occasion of the European Congress of Mathematics. The prizes were awarded to ten mathematicians under the age of thirty-two: Richard Borcherds, Jens Franke, Alexander Goncharov, Maxim Kontsevich, François Labourie, Tomasz Luczak, Stefan Muller, Vladimir Sverak, Gabor Tardos, and Claire Voisin.

3.1. The Cours Peccot.

The Cours Peccot is a semester-long mathematics course given at the Collège de France. Each course is given by a mathematician under 30 years of age who has distinguished themselves by their promising work. The course consists of a series of lectures on the current research of the Peccot lecturer.

3.2. Claire Voisin, 1991-92 Peccot lecturer.

Claire Voisin was chosen as Peccot lecturer for 1991-91 and gave the lecture course Variations de structure de Hodge et cycles algébriques des hypersurfaces. She based the lectures on her paper 'Degenerations de Lefschetz et variations de structures de Hodge', Journal of Differential Geometry (1990).

4.1. The Servant prize.

The Servant Prize is named for French mathematician Maurice Servant (1877-1952) who was awarded the Bordin Prize in 1904. The prize is awarded every two years by the Academy of Sciences alternately in the field of mathematical sciences and in the field of physical sciences. Created in 1952, it became a thematic grand prize in 2001.

4.2. Claire Voisin, the Servant Prize 1996.

The Servant Prize 1996 is awarded by the Academy of Sciences to Claire Voisin:-

... for her work in algebraic geometry, in particular on Torelli's theorem, on the Griffiths-Harris conjectures, on the Noether-Lefschetz locus and on algebraic cycles.

5.1. The Sophie Germain Prize.

The Sophie Germain Prize is an annual prize, created in 2003, awarded on the recommendation of the Academy of Sciences. It is intended to recognise a researcher who has carried out fundamental research work in mathematics.

5.2. Claire Voisin, the Sophie Germain Prize 2003.

The first to be awarded the Sophie Germain Prize by the French Academy of Sciences was Claire Voisin in 2003.

6.1. The CNRS silver medal.

The silver medal of the Centre national de la recherche scientifique honours researchers for the originality, quality and importance of a body of work that is recognised nationally and internationally.

6.2. Claire Voisin, Mathematics considered as one of the fine arts.

In mathematics, the aesthetic aspect is very important. In good papers, we always find a fresh point of view, surprising and attractive ideas that open up new fields. There is nothing worse than laborious work that deploys enormous intellectual resources for unoriginal results.

When Claire Voisin, 44, researcher at the Jussieu Institute of Mathematics, talks about her discipline, we sometimes have the impression of listening to an artist speaking about her creative approach.

At the origin of a problem or a conjecture, there is a kind of reverie which can arise at any moment, when peeling carrots for example. This is the most touching part of mathematics. The important thing for a mathematician is freedom of mind and this inner movement, this underground and unconscious work which suddenly crystallizes.

Claire Voisin discovered mathematics at a very young age thanks to her father with whom she had fun solving simple geometry problems. While she was in college, her older brother lent her his final year mathematics lessons, which she enjoyed studying. A few years passed and, in 1981, she entered the École Normale Supérieure. At this time, she was mainly interested in the philosophy of science.

I had formed a feeling against mathematics because of the teaching in Higher school preparatory classes. It was a very academic course where we only learned dead mathematics.

Her teachers distracted her from epistemology and finally, during her Diplôme d'études approfondies, she developed a taste for the most abstract mathematics.

In 1986, under the direction of Arnaud Beauville, she defended a thesis entitled Le théorème de Torelli pour les cubiques de P5. She thus entered the vast universe of algebraic geometry, a branch of mathematics that can be defined as the study of sets of solutions to systems of algebraic equations. This field has just experienced extraordinary growth under the leadership of visionary researchers like Alexandre Grothendieck.

Claire Voisin chose complex algebraic geometry, dominated by the school of Phillip Griffiths.

We did not want to remain locked in an algebraic geometry separated from other branches of mathematics. We sought to open it up to other methods such as, for example, analytical geometry or differential geometry.

The researcher worked intensively on Hodge theory, one of the aspects of the study of differential topology. This theory represents a framework of thought and an extraordinary tool for addressing central problems in complex algebraic geometry. Her most important articles explored this path and opened new fields, particularly in the study of the topology of Kähler varieties. Her book, Hodge Theory and Complex Algebraic Geometry, on the foundations of Hodge theory, published in 2003, became a reference in the field.

She often allows herself escapades outside her chosen field which prove extremely fruitful. Let us cite, among others, her work on the syzygies of the canonical ring of curves or her research on mirror symmetry, a mathematical phenomenon highlighted by physicists working on string theory.

Claire Voisin finds in the numerous conferences to which she is invited a sort of counterpoint to the retrenchment essential to mathematical research. She is also the editor of several scholarly publications.

It allows me to follow the evolution of my discipline. When we work too much alone, we tend to get locked into mini-subjects.

Claire Voisin also has an intense family life: she is the mother of a family of five children, aged 9 to 19. With them, she likes to share her taste for literature and music.

7.1. The Satter Prize.

The Satter Prize is awarded every two years to recognise an outstanding contribution to mathematics research by a woman in the previous five years. Established in 1990 with funds donated by Joan S Birman, the prize honours the memory of Birman's sister, Ruth Lyttle Satter. Satter earned a bachelor's degree in mathematics and then joined the research staff at AT&T Bell Laboratories during World War II. After raising a family she received a Ph.D. in botany at the age of forty-three from the University of Connecticut at Storrs, where she later became a faculty member. Her research on the biological clocks in plants earned her recognition in the U.S. and abroad. Birman requested that the prize be established to honour her sister's commitment to research and to encouraging women in science. The prize carries a cash award of US$5,000.

7.2. The 2007 Satter Prize.

The 2007 Ruth Lyttle Satter Prize in Mathematics was awarded at the 113th Annual Meeting of the American Mathematical Society in New Orleans in January 2007. The Satter Prize is awarded by the American Mathematical Society Council acting on the recommendation of a selection committee. For the 2007 prize, the members of the selection committee were: Benedict H Gross, Karen E Smith, and Chuu-Lian Terng (chair). Ninth award of the Prize was made to Claire Voisin in 2007:-

... for her deep contributions to algebraic geometry, and in particular for her recent solutions to two long-standing open problems: the Kodaira problem (On the homotopy types of compact Kähler and complex projective manifolds, Inventiones Mathematicae 157 (2) (2004), 329-343) and Green's Conjecture (Green's canonical syzygy conjecture for generic curves of odd genus, Compositio Mathematica 141 (5) (2005), 1163-1190; and Green's generic syzygy conjecture for curves of even genus lying on a K3 surface, Journal of the European Mathematical Society 4 (4) (2002), 363-404).

7.3. Citation for the 2007 Satter Prize.

The Ruth Lyttle Satter Prize is awarded to Claire Voisin of the Institut de Mathématiques de Jussieu for her deep contributions to algebraic geometry, and in particular for her recent solutions to two long-standing open problems. Voisin solved the Kodaira problem in her paper "On the homotopy types of compact Kähler and complex projective manifolds", Invent. Math. 157 (2) (2004), 329-343. There she shows that in every dimension greater than three, there exist compact Kähler manifolds not homotopy equivalent to any smooth projective variety. This problem has been open since the 1950s when Kodaira proved that every compact Kähler surface is diffeomorphic to (and hence homotopy equivalent to) some projective algebraic variety. Her idea is to start with the fact that certain endomorphisms can prevent a complex torus from being realised as a projective variety, and then to construct Kähler manifolds whose Albanese tori must carry such endomorphisms for homological reasons. In a completely different direction, Voisin also solves Green's Conjecture in her papers "Green's canonical syzygy conjecture for generic curves of odd genus", Compos. Math. 141 (5) (2005), 1163-1190, and "Green's generic syzygy conjecture for curves of even genus lying on a K3 surface", J. Eur. Math. Soc. 4 (4) (2002), 363-404.

A century ago, Hilbert saw that syzygies (relations among relations) were important invariants of varieties in projective space, and in the early 1980s, Mark Green conjectured that the syzygies of a general curve canonically embedded in projective space should be as simple as possible. This conjecture attracted a huge amount of effort by algebraic geometers over twenty years before finally being settled by Voisin. Her idea is to work with curves on a suitable K3 surface, where she executes deep calculations with vector bundles (at least in even genus) that lead to the required vanishing theorems.

7.4. Biographical Sketch of Claire Voisin.

Claire Voisin defended her thesis in 1986 under the supervision of Arnaud Beauville. She began employment in the Centre National de la Recherche Scientifique as chargée de recherche in 1986 and since then pursued her career in this institution. She occasionally taught graduate courses but mainly does research and advises students. Her honours include the European Mathematical Society Prize (1992), the Servant Prize (1996) and the Sophie Germain Prize (2003) of the Académie des Sciences de Paris, and the silver medal of the CNRS (2006). She was an invited speaker at the International Congress of Mathematicians in 1994 in Zurich.

Response by Claire Voisin.

I am deeply honoured to have been chosen to receive the 2007 Ruth Lyttle Satter Prize. I feel of course very encouraged by this recognition of my work. I would like to thank the members of the prize committee for selecting me. I am also very grateful to my institution, the CNRS, which made it possible for me to do research in the best conditions.

8.1. The Clay Research Award.

The Clay Mathematics Institute presents the Clay Research Award annually to recognise major breakthroughs in mathematical research. Awardees receive the bronze sculpture "Figureight Knot Complement VII/CMI" by sculptor Helaman Ferguson. The Award was first made in 1999.

8.2. The 2008 Clay Research Award to Clair Voisin.

The 2008 Clay Research Award was made to Claire Voisin:-

... for her disproof of the Kodaira conjecture.

The Kodaira conjecture was formulated in 1960, when Kunihiko Kodaira showed that any compact complex Kähler surface can be deformed to a projective algebraic surface. For the proof, Kodaira used his classification theorem for complex surfaces. The conjecture asks whether Kähler manifolds of higher dimension can be deformed to a projective algebraic manifold. Voisin constructs counterexamples: in each dimension four or greater, there is a compact Kähler manifold which is not homotopy equivalent to a projective one. For dimension at least six, she gives examples which are also simply connected. A later result gives a substantial strengthening: in any even dimension ten or greater, there exist compact Kähler manifolds, no bimeromorphic model of which is homotopy equivalent to a projective algebraic variety. Distinguishing the homotopy type of projective and non-projective Kähler manifolds is achieved through novel Hodge-theoretic arguments that place subtle restrictions on the topological intersection ring of a projective manifold.

8.3. The Kodaira Conjecture.

Geometric structures on a topological manifold often impose restrictions on what kind of manifolds can arise. For example, a symplectic manifold must have nonzero second Betti number, since the symplectic form ω is non-trivial in cohomology. Indeed, if the manifold has dimension $2n$, then ωn has nonzero integral. Yet more restrictive is the notion of a Kähler manifold - a symplectic manifold for which the form $\omega$ has type (1, 1) in a compatible complex structure. In that case many topological conditions are satisfied: the odd Betti numbers are even, the cohomology ring is formal, and there are numerous restrictions on the fundamental group. Kähler manifolds abound: any projective algebraic manifold, that is, any submanifold of complex projective space defined by homogeneous polynomial equations, is a Kähler manifold. In complex dimension one, the converse is true: any Kähler manifold (a Riemann surface) is complex projective. In complex dimension two, the converse is false, but just barely: every complex Kähler manifold is the deformation of a projective algebraic manifold. This fact was proven by Kadaira, using his classification theorem for complex surfaces.

The question then arises: is every compact Kähler manifold deformable to projective algebraic one? Although never explicitly stated by Kodaira, this question has become known as the Kodaira Conjecture. Alas, the proof in dimension two gives no clue about what happens in higher dimension. The crux of the problem, however, is to show that on the given complex manifold $M$, one can deform the complex structure so as to obtain a positive (1, 1) class in the rational cohomology. That is, one must show that the Hodge structure is polarisable. The fundamental theorem here is due to Kodaira: from a closed, rational positive, (1, 1) form, one may construct an imbedding of the underlying manifold into a projective space.

There have been various attempts to prove or disprove the conjecture. Since and deformation of $M$ has the same diffeomorphism as $M$, a disproof requires a topological invariant defined for Kähler manifolds that distinguishes the projective algebraic cones from those that are not.

The starting point for Voisin's counterexample is the construction of a complex torus $T$ which is not projective algebraic because of the existence of a "wild" endomorphism $\phi$. This is an endomorphishm whose eigenvalues are non-reel and distinct, and such that the Galois group of the field generated by the eigenvalues is as large as possible. An example is given by the companion matrix of the polynomial $x^{4} - x + 1$. The second exterior product of a weight one Hodge structure with a wild endomorphism carries no nonzero rational (1, 1) classes, so long as the space of elements of type (1, 0) has dimension strictly greater than one. Therefore the complex manifold $T$ is not projective algebraic, though it can, of course, be deformed to an algebraic torus. The actual counterexample is a suitable blowup of T × T. Consider the subvarieties $T \times \{0\}$,$\{0\} \times T$, the graph of the diagonal, and the graph of $\phi$. Blow up the points of intersection of the diagonals of the identity and of and also the intersection of $T \times \{0\}$ with the graph of $\phi$. Then blow up the proper transforms of the subvarieties to obtain a Kähler manifold $V$ with $H^{2}(V ) \cong \lambda^{2} H^{1}(T)$. Any deformation of a blowup of a complex torus is obtained first by deforming the torus and then deforming the blowup. From this one sees that the wild endomorphism is preserved. Therefore the Hodge structure $H^{2}(V)$ contains no rational (1, 1) classes, and so $V$, and indeed any Kähler manifold with the same cohomology ring as $V$, is not projective algebraic. The same kind of construction yields a disproof of the Kodaira conjecture in dimension four or greater. Voisin also gives simply connected counterexamples in dimension six and greater.

9.1. The Heinz Hopf Prize.

The Heinz Hopf Prize at ETH Zurich was established thanks to a donation received from Dorothee and Alfred Aeppli. The CHF 30,000 award honours outstanding scientific achievements in the field of pure mathematics. It is named after the German mathematician Heinz Hopf, Professor of Mathematics at ETH from 1931 to 1965. The prize is awarded every two years on the occasion of the Heinz Hopf Lectures, which have a long-​​​standing tradition at ETH Zurich. The lectures are given by the laureates. The prize was awarded for the first time in October 2009.

9.2. Heinz Hopf Prize for Claire Voisin.

In the field of research into algebraic geometry, French mathematician Claire Voisin is something of a legend: in recognition of her scientific breakthroughs, the Department of Mathematics at ETH Zurich awarded her the Heinz Hopf Prize on 2 November 2015.

In mathematics, proof is the silver bullet for assured findings. However, in a branch of mathematics such as algebraic geometry, whose fundamental concepts and research topics have long moved away from everyday concepts and forms, conjectures also serves as an essential driver of knowledge advancement. When formulated, conjectures are neither proven nor disproven, but they are well-​substantiated. In this respect, they describe a mathematical problem whose solution is yet to be found.

Claire Voisin is almost legendary in this regard: after all, the 53-​year-old French mathematician disproved the famous 'Kodaira conjecture' in the field of algebraic geometry and was able to produce pioneering interim results for other conjectures.

"At present, Claire Voisin is one of the world's leading researchers in the field of complex algebraic geometry," says Rahul Pandharipande, Professor of Mathematics at ETH Zurich. "Over the last 30 years, she has played a key role in the further development of our field."

For her exceptional services to mathematics, Claire Voisin was awarded the Heinz Hopf Prize 2015 for pure mathematics on 2 November 2015 at ETH Zurich. "Claire Voisin is an excellent choice. The decision to award her the prize was unanimous," says Urs Lang, Professor of Mathematics and Chairman of the selection committee for the Heinz Hopf Prize. Carrying prize money of 30,000 Swiss francs, the prize has been awarded by the Department of Mathematics at ETH Zurich every two years since 2009.

Geometry with algebraic equations

Algebraic geometry is characterised by the analysis of objects such as curves or surfaces using algebraic methods and equations. Historically, this methodology traces its roots back to René Descartes and the introduction of the coordinate system. For example, in this system, the circle is described by the equation $x^{2} + y^{2} = 1$. Today, of course, algebraic geometry has long since moved on from studying one-​ and two-​dimensional objects such as simple curves; now, it also analyses objects with three or more dimensions. At the same time, it addresses not only straightforward characteristics but also invariants, i.e. invariable structural properties of all kinds of shapes, surfaces or bodies.

Similarly, in terms of methodology, it no longer works merely with individual equations but rather with the solutions to arbitrary systems of equations that contain polynomial equations with multiple variables, as well as with algebraic quantities (varieties or rings). For example, this research is of relevance to physics and to the question of how the universe is structured and ordered.

Abstractions to higher dimensions

Claire Voisin's most significant findings include the solution to a 50-​year-old conjecture by Japanese mathematician Kunihiko Kodaira, for which she won the Clay Research Award in 2008. This conjecture relates to so-​called Kähler manifolds, which also play a role in physical string theory. This theory is based on the assumption that fundamental spatial objects are strings instead of elementary particles. The concept of the manifold was introduced in 1854 by German mathematician Bernhard Riemann as a generalisation of surfaces to higher dimensions. Today's customary definition was published in 1913 by his fellow countryman Hermann Weyl, Heinz Hopf's predecessor as professor at ETH Zurich.

Kodaira conjectured that it would be possible not only to transform two-​dimensional Kähler manifolds into an algebraic variety but also to do so in higher dimensions. Claire Voisin solved the problem by presenting counterexamples to the Kodaira conjecture that are not transformations of this kind. "One characteristic feature of Claire Voisin's working method is that she builds her arguments on well-​founded examples," says Rahul Pandharipande.

Author of a reference work

Voisin has also used projective algebraic varieties in a partial proof and an approach to solving the conjectures of the two American mathematicians Mark Green and Spencer Bloch. With regard to the famous British mathematician William Hodge, Voisin's book Hodge Theory and Complex Algebraic Geometry is today considered a reference work. At a symposium on 3 November 2015, talks were given on this and other related problems by three renowned algebraic geometers, among them Arnaud Beauville, Voisin's doctoral supervisor. Eight emerging researchers were also invited.

Claire Voisin works as a Director of Research at the Institut de Mathématiques de Jussieu in Paris. She is a member of various European academies. In 2015, she was appointed Professor of Algebraic Geometry at the Collège de France. Claire Voisin has been invited to return to ETH Zurich in 2017 to conduct research for one year as a fellow at the Institute for Theoretical Studies (ETH-​ITS).

9.3. Heinz Hopf Lectures and prize-​giving ceremony.

Claire Voisin was awarded the Heinz Hopf Prize on 2 November 2015 at 6:15 pm in the Semper Aula at ETH Zurich. The two Heinz Hopf Lectures on the topic of "Diagonals in algebraic geometry" were held on Monday 2 November 2015 in HG G 60 and Tuesday 3 November 2015 in HG G 3, in both cases at 5:15 pm. A symposium featuring three talks by three renowned algebraic geometers on aspects of Claire Voisin's research were held on 3 November 2015 in room HG E 3 at ETH Zurich. The lectures and symposium were given in English.

10.1. The CNRS Gold medal.

Created by the Centre National de la Recherche Scientifique in 1954, the purpose of the gold medal is to recognise the whole body of work of a scientist who has made an exceptional contribution to extending the influence of French research. This award is one of the highest honours in science.

10.2. Claire Voisin, 2016 CNRS Gold medal.

The mathematician, a specialist in the field of algebraic geometry, is awarded the 2016 CNRS Gold Medal, France's highest scientific distinction.

On September 21, 2016, the mathematician Claire Voisin was awarded the CNRS Gold Medal. She follows 2015 laureate Eric Karsenti and joins the exclusive list of winners of France's highest scientific distinction.

A renowned specialist in algebraic geometry, Claire Voisin is recognised by her peers, especially for her research on "the topology of projective varieties of compact Kähler manifolds" and Hodge theory, an important branch of this discipline. The latter was reinvigorated in the 1950s, especially by the French school, notably the work of Jean-Pierre Serre and Alexander Grothendieck, and has seen many recent developments.

For Claire Voisin, 2016 will also have been marked by her inaugural lecture at the Collège de France on June 2. She was appointed new chair of algebraic geometry after having spent 30 years as a senior researcher at the CNRS, and became the first female mathematician to enter the Collège de France.

Born in 1962, Claire Voisin has received many awards during her career, such as the CNRS Silver Medal in 2006 or the Clay Research award in 2008. Following this accolade, CNRS International Magazine published a profile, which we have republished below. Claire Voisin will officially receive the Gold Medal on 14 December during a ceremony at the Sorbonne in Paris.

10.3. Claire Voisin, artist of the abstract.

Very quickly, words no longer suffice. Claire Voisin goes to the board, eraser in one hand, chalk in the other, and draws geometrical figures side by side with complicated calculations. Voisin, a senior researcher at the Institut de Mathématiques de Jussieu in Paris is a specialist in algebraic geometry. More specifically, she works on the study of the "topology of complex algebraic varieties."

To introduce her field, she sketches a sphere that she cuts up in three-dimensional triangles with curved edges, as if they had been shaped by the rounded surface. The result is that you can cover a sphere with triangles, which are themselves the "faces" of a pyramid, for example. "Topologically speaking," Voisin explains, "a sphere and the surface of a pyramid are therefore identical - though saying something like that is an absurdity from the point of view of algebraic geometry," she immediately points out. According to her, "this is also possible with an inner tube that has one or more holes." If "triangulated," the result is a skeleton made up of triangles stuck together along their sides. A metric induced by the ambient space then gives rise to a complex structure, hence to a Riemann surface, which turns out to be a purely algebraic object, a projective curve. And in higher dimensions, the problem becomes even more complex. To get from one figure to the other therefore involves a mathematical trick, the precise details of which are very difficult to grasp for a non-specialist, involving such words as homeomorphism, simplex, Riemann surface, transcendental functions, etc. But the general idea is clear: moving between the "topological," the "algebraic," and "complex geometry," the result is a "multiplicity of perspectives of one and the same object" using different mathematical approaches. "What's exciting about my work is this constant moving back and forth several geometries and several types of tools to prove results in one field or another," Voisin continues.

She resembles the typical mathematician as we often imagine them, with a particular ability for abstract thinking. In fact, though mathematics came easy to Voisin both at school where she was already boning up on final year courses, then at the École Normale Supérieure and while doing her PhD, she knows that for all intents and purposes, she speaks a language that is foreign to most ordinary people. It's not easy to follow what she says. That's true even for the students studying for their Masters in mathematics, to whom she teaches a few courses a year, attempting to "explain these superb ideas." Yet they often drop out, discouraged by the complexity of the field. "It's very frustrating not to be able to get across all the things that mean so much to me in my work and research," says Voisin regretfully. She remembers the six months during which she was an assistant professor, before she got a CNRS position, as being "hellish." "Joining CNRS saved my life!" she jokes.

Becoming a full time researcher at the age of 24, she could at last devote herself entirely to algebraic geometry, the study of the properties of sets defined by algebraic equation systems, which is at the heart of the most abstract mathematics. "There is creative drive in mathematics, it's all about movement trying to express itself," Voisin confides. Nothing to do with the "boring, dead, and dry" mathematics taught in secondary school, where the courses go through an endless series of "definitions, properties, and theorems" using a method that is "always under control, as if on tracks," and which is applied to "simple exercises in logic."

After her doctoral thesis, she became fascinated by a tool that is well known to topology specialists, Hodge theory, which can also be used to tackle complex algebraic geometry. Published in 2003, her book on the subject has rapidly become a reference. She won a number of prizes and awards, such as the CNRS bronze (1988) and silver (2006) medals, and the Clay Research Award (2008) from the Clay Mathematics Institute, for her work on Kodaira's conjecture, another problem in complex algebraic geometry. As an editor of several mathematical journals, she always keeps an eye on the development of her discipline. In her private life, she is also a mother of five, and her eldest daughter started studying mathematics. "But her field is far removed from mine and that of my husband - also a mathematician - so as to avoid any family 'pressure,'" she explains. "In any case," she adds, "we never talk maths at home!"

11.1. The Shaw Prize.

The Shaw Prize, established by the Shaw Prize Foundation in Hong Kong in 2002, is awarded yearly in the three categories Astronomy, Life Sciences and Mathematical Sciences to "individuals who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence."

11.2. Announcement of the Shaw Prize 2017.

The Shaw Prize in Mathematical Sciences 2017 is awarded in equal shares to János Kollár, Professor of Mathematics, Princeton University, USA and Claire Voisin, Professor and Chair in Algebraic Geometry, Collège de France:_

... for their remarkable results in many central areas of algebraic geometry, which have transformed the field and led to the solution of long-standing problems that had appeared out of reach.

11.3. Contribution of Claire Voisin.

The Shaw Prize in Mathematical Sciences 2017 is awarded in equal shares to János Kollár, Professor of Mathematics, Princeton University, USA and Claire Voisin, Professor and Chair in Algebraic Geometry, Collège de France, for their remarkable results in many central areas of algebraic geometry, which have transformed the field and led to the solution of long-standing problems that had appeared out of reach.

Since ancient times, a central theme in mathematics has been the study of polynomials and their solutions. Algebraic geometry is the study of the properties of sets of solutions to polynomial equations in several variables. A simple example of such an equation is $x^{2} + y^{2} + z^{2} = 1$, the solution set of which is the surface of a sphere of radius 1.

As this example demonstrates, solution sets of polynomial equations, which are known as varieties, are geometric objects. Examining the interplay between the algebra and the geometry has turned out to be remarkably fruitful, and algebraic geometry is a major branch of mathematics, the study of which has profound consequences not just for algebra and geometry, but also for several other areas ranging from number theory to mathematical physics.

Two varieties are said to be birationally equivalent if they become the same after excluding a suitably small subset. Rational varieties are those that are birationally equivalent to ordinary n-dimensional space, for some $n$. Some of the most exciting advances in algebraic geometry over the past few decades have been in better understanding the birational classification of higher-dimensional varieties. For example, Shigefumi Mori won the Fields Medal for his Minimal Model Program. Some of the most important breakthroughs have been in characterising rational varieties. Both János Kollár and Claire Voisin have made central contributions to this development.

Among Voisin's major achievements is the solution of the Kodaira problem, which starts with the observation that every deformation of a complex projective manifold is a Kähler manifold (which roughly speaking means a geometric set that locally has a structure compatible with the complex numbers) and asks whether the converse is true. She found counterexamples: that is, Kähler manifolds that not only fail to be deformations of projective manifolds but are not even topologically equivalent to projective manifolds. Another of Voisin's pioneering accomplishments is the establishment a new technique for showing that a variety is not rational, a breakthrough that has led to results that would previously have been unthinkable. A third remarkable result is a counterexample to an extension of the Hodge conjecture, one of the hardest problems in mathematics (being one of the Clay Mathematical Institute's seven Millennium Problems); the counterexample rules out several approaches to the conjecture.

11.4. An Essay on the Prize.

For each type of mathematical structure, there is normally a notion of when two examples are essentially the same. For varieties, the notion is called birational equivalence: two varieties are said to be birationally equivalent if, after removing lower-dimensional subvarieties if necessary, there is a rational map from one to the other with a rational inverse. (A rational map is a ratio of two polynomials.) Rational varieties are those that are birationally equivalent to ordinary $n$-dimensional space, for some $n$. To give an example, for every real number $t$ one can check that $\Large\frac{2t}{(1+t^2)^2}\normalsize + \Large\frac{1-t^2}{(1+t^2)^2}\normalsize = 1$, from which it follows that the map that takes a real number $t$ to the point$(\Large\frac{2t}{(1+t^2)^2}\normalsize, \Large\frac{1-t^2}{(1+t^2)^2}\normalsize)$ sends an infinite line to the unit circle. One can check further that sending a point $(x, y)$ in the unit circle to the real number $\Large\frac{1-y}{x}\normalsize$ inverts this map. There is a small problem here in that no value of $t$ maps to the point $(0, -1)$, but this point on its own is a zero-dimensional subset, which we are allowed to ignore. Thus, a circle is birationally equivalent to ordinary 1-dimensional space, which means that it is a rational variety.

Some of the most exciting advances in algebraic geometry over the past few decades have been in better understanding the birational classification of higher-dimensional varieties. For example, Shigefumi Mori won the Fields Medal for his "Minimal Model Program", which attempts to find in each birational equivalence class a unique simplest variety. There have also been very important breakthroughs in characterizing rational varieties: that is, in finding ways of telling whether a variety is rational.

Along with his work on birational classification, Kollár's most recent work stands out in a direction that will greatly affect algebraic geometry in the decades to come, as an important complement of the Minimal Model Program: the definition and study of moduli of higher-dimensional varieties, which can be thought of as sophisticated geometrical structures whose points represent equivalence classes of these varieties. The importance of this area can be seen in the immense wealth of papers on the moduli problem in dimension 1, which today occupies topologists, combinatorialists and, perhaps most of all, physicists. Already for surfaces the treatment of moduli is an extremely subtle and difficult problem. Kollár's ideas have almost defined the field of higher-dimensional moduli.

Among Voisin's major achievements is the solution of the Kodaira problem, which starts with the observation that every deformation of a complex projective manifold is a Kähler manifold (which roughly speaking means a geometric set that locally has a structure compatible with the complex numbers) and asks whether the converse is true. She found counterexamples: that is, Kähler manifolds that do not just fail to be deformations of projective manifolds but are not even topologically equivalent to projective manifolds. Another of Voisin's pioneering accomplishments is the establishment a new technique for showing that a variety is not rational, a breakthrough that has led to results that would previously have been unthinkable. A third remarkable result is a counterexample to an extension of the Hodge conjecture, one of the hardest problems in mathematics (it is one of the Clay Mathematical Institute's seven Millennium Problems), which rules out several approaches to the conjecture.

Algebraic geometry is a central area of mathematics that has seen many remarkable developments in its history, and these developments continue to the present day. In their different ways, János Kollár and Claire Voisin have made profound contributions to algebraic geometry that will deeply influence the future of the subject.

11.5. Autobiography of Claire Voisin.

I was born in 1962 in a small town in the Northern suburbs of Paris. My parents married in 1945 and had nine children before me (they ended-up with twelve children), so I grew up with many sisters and brothers, most of them older than me, in a curious atmosphere influenced by the 60's spirit but also very much turned towards intellectual life. My father was an engineer who liked science and taught me a lot of classical geometry (circle and triangle) and my mother, who had stopped studying during the Second World War, was very fond of art. She was extremely enthusiastic about Impressionist painting and the subsequent development of Modern Art. She began studying history of art at École du Louvres when she was in her 50's. I left the family home when I was seventeen having gained a Fellowship from the government, and studied first for two years in Classes préparatoires at Lycée Louis-le-Grand. I then entered École normale supérieure of Sèvres: at that time the female branch of ENS was still separated from Ulm, which accepted only male students. There and at Jussieu I studied most sorts of mathematics until I started my PhD thesis at Orsay under the direction of Arnaud Beauville. My thesis, proving the Torelli theorem for cubic fourfolds, already involved the theory of Hodge structures developed by Griffiths, which is still an ingredient of a large part of my research, being a rich tool to study many different sorts of questions concerning algebraic varieties and their moduli.

I defended my PhD thesis in 1986, the same year I got a permanent research position at CNRS, which I kept until 2016. I am now Professor at Collège de France, which is a different institution. In 1984, I married Mathematician Jean-Michel Coron and we have five children, born between 1987 and 1997. In 1987, I met Kollár who at that time was Professor at the University of Utah in Salt Lake City where I was supposed to stay as a postdoc with Herb Clemens but having left my baby in Paris, I found the separation too hard and after one month finally decided to return. Starting from that period, for many years I spent most of my time at home, except of course for attending seminars, and found it very convenient doing research there. Now that our children are grown up, I travel much more, but sometimes I regret the period where my life was mostly centred at home and divided between doing research and raising my children. Of course, I owe much to the French system of childcare and also to my husband who always shared all the family duties with me. In the 2000's, at the week-ends we used to work in a quiet place close to our house: I went there to work in the morning when I felt more fresh, and Jean-Michel worked there in the afternoon.

For a long period, between 1989 and 2012, we lived in Bourg-la-Reine in the South of Paris, in a big family house that was perfect for a large family and had a lot of character, but proved hard to sell when we decided to start a new life in Paris as our children had started to leave. We now live in the 5th arrondissement, a very quiet part of Paris where many institutes are traditionally located, like Sorbonne, Institut Henri Poincaré, Collège de France and École Normale Supérieure. I like this area of Paris, sometimes called Montagne Sainte-Geneviève or Quartier latin, where I had previously lived when I was a student in Lycée Louis-le-Grand, and also later on when I attended courses at Jussieu.

The period 2002-2005 has been quite good for my research. I obtained results on syzygies of curves (the Green conjecture for generic canonical curves), a part of my research which does not involve Hodge theory, then I solved negatively the Kodaira problem, constructing compact Kähler manifolds not homeomorphic to projective ones. This work, to the contrary, was entirely based on the formal study of Hodge structures. The next work I am particularly proud of is my recent contribution to the Lüroth problem, giving a method to detect irrational varieties. This work has known many spectacular further developments. There are many other aspects of algebraic geometry I am interested in, like Chow groups and the Bloch conjectures, hyper-Kähler manifolds (construction and moduli), positivity problems for cycles, variations of Hodge structures etc. In fact, as I get older, I find there are more and more open problems I would like to attack.

What I like in algebraic geometry is a good balance between algebra and geometry and also a good balance between the theory (due to the major foundational work of the 1950-60's) and the objects: the geometry provides a rich sample of classes of objects and discovering the adequate tools to understand these various classes and distinguish them is actually interesting and makes us fully appreciate the general theoretical machinery at our disposal (e.g. Hodge theory, or K-theory and Chow groups, and of course, more general algebraic geometry, schemes, cohomology theory, moduli, Hilbert schemes…).

11.6. Claire Voisin, Forms and Formulas.

A specialist in algebraic geometry, a CNRS senior researcher at the Institut de Mathématiques de Jussieu until last June and now a professor at the Collège de France, 54-year-old Claire Voisin loves to philosophise about her profession. In fact, philosophy was her favourite subject at secondary school. At the time, she also felt drawn to poetry. Mathematics did not seem to hold much interest for her at first, but that would soon change ...

Encouraged by her father, an alumnus of France's prestigious École Polytechnique, Voisin began exploring the myriad implications of the basic theorems of plane geometry (Thales, Pythagoras ...) and, from the age of 15, devoured all the specialised books that were "lying around the house." When her father suddenly lost his job, she was granted a state scholarship to pursue her studies. After two years of preparatory classes, during which she developed a real taste for mathematics, she was admitted to the École Normale Supérieure in 1981, at the age of 19. "I was paid a salary - it was great!" she recalls. "To me, it meant: 'we're happy that you're studying, make the most of it!'"

Reconciling algebra and geometry

After earning a postgraduate teaching diploma in mathematics in 1983, she completed her dissertation three years later under the supervision of Arnaud Beauville. Immediately upon graduation, she was recruited by the CNRS and began what would prove to be a brilliant career. She won numerous prestigious awards, including a European Mathematical Society prize in 1992, and was elected to the Académie des Sciences in 2010. In 2016 she received the 2016 CNRS Gold Medal, France's highest scientific distinction, a few months after becoming the first female mathematician to join the Collège de France, where she holds the chair in algebraic geometry.

Confronted with our difficulty to understand what this high-flying discipline actually entails, she walks to the blackboard and draws a large torus framed by x- and y-axes. "At first sight, all we see here is a thick ring, like a lifebuoy, with one or several holes in the middle," she explains. But in mathematics, we seek to find out more: how do the curves plotted out on this torus intersect? What families of curves can cover the torus? Can the torus be unfolded and superimposed on a sphere? These are topological questions that can be dealt with using the Morse theory or Hodge theory, for example. These objects are also defined by complex algebraic equations, which offers different ways of approaching them."

"What's interesting in algebraic geometry is that you can do geometry to understand algebraic equations and algebra to elucidate geometric figures," adds François Charles, a professor at the Université Paris-Sud in Orsay (Paris) and one of Claire Voisin's former PhD students. There is a constant back and forth between the two approaches. But it also raises a problem: even though algebra and geometry are useful, each in its own way, to understand an object, we cannot always reconcile the two and find the link between them." We are somewhat in the same situation as physicists, who are unable to unite all four fundamental interactions (forces of the Universe) in a single theory. Each one characterises it in a different way, whether it be gravity, or electromagnetism, weak nuclear force and strong nuclear force.

A creative approach

Unlocking the mysteries of the Universe is an appealing prospect for nearly everyone. But what is the point of dissecting a torus or a sphere in mathematical terms? To this deliberately provocative question, Voisin responds with a heartfelt eulogy that reveals her passion for her chosen field: "It was mathematicians who, in ancient times, set out to calculate the radius of the Earth! Without mathematics, none of the progress in physics would have been possible. We need mathematics to study the structure of the Universe! All of the powerful theoretical tools we develop will find applications sooner or later."

One of Voisin's most outstanding findings concerns the Kodaira theorem on surfaces; she has proved that in higher dimensions some compact Kähler manifolds cannot be deformed to complex projective manifolds. She has also proved the impossibility of extending the Hodge conjecture to the Kählerian framework. In projective geometry, she has established Green's conjecture for generic curves, based on a novel construction and complex calculations. Her current research on birational invariants has attracted a great deal of attention and was discussed on November 5, 2015 at the Nicolas Bourbaki seminar, a not-to-be-missed contemporary mathematics event.

"I don't consider myself an artist, but it is true that you must be creative to achieve good results in mathematics. Actually, I painted a lot until the age of 25," she divulges, pointing to the canvases hanging in her living room. "I stopped eventually - my profession calls upon most of one's creative abilities." When she was younger, she would immediately try and solve any intriguing mathematical problem that she came across. Yet this is not always possible today: Voisin has multiple responsibilities in relation to her colleagues and students, as well as... five children and a granddaughter. Her days can be hectic, so she now jots down her ideas in a small notebook - which she extricates from a pile of documents sitting on the floor - for further thought whenever she has time to herself.

11.7. ETH-ITS Senior Fellow Claire Voisin receives the Shaw prize 2017.

Claire Voisin, of the Collège de France, currently Senior Fellow at the ETH Institute for Theoretical Studies, receives the 2017 Shaw Prize in Mathematical Sciences. She shares the prize with János Kollár, of Princeton University.

Professor Claire Voisin, of the Collège de France, currently Senior Fellow at the ETH Institute for Theoretical Studies, receives the 2017 Shaw Prize in Mathematical Sciences. She shares the prize with János Kollár, of Princeton University. According to the citation, Kollár and Voisin are awarded the prize "for their remarkable results in many central areas of algebraic geometry, which have transformed the field and led to the solution of long-​standing problems that had appeared out of reach."

Claire Voisin obtained her Ph.D. in 1986 from the Université Paris-​Sud, under the supervision of Arnaud Beauville. A member of the French Academy of Science, she is the first female mathematician to be elected professor at the Collège de France, where she holds the new chair of algebraic geometry since 2016. Claire Voisin works in algebraic and complex geometry addressing fundamental questions of mathematics involving the subtle interrelations between algebra, geometry and topology.

Her deep results in major questions of algebraic and complex geometry, such as the Hodge conjecture (a millenium problem) on algebraic cycles, the Kodaira conjecture on deformations of Kähler manifolds and the Green conjecture on curves, were recognised by several prizes, most recently the Heinz Hopf prize in 2015 and the Gold Medal of the CNRS, the highest French recognition in Science, in 2016.

Claire Voisin came to the ETH Institute for Theoretical Studies as a Senior Fellow in January 2017 and will stay until the end of the year. She will give a course on hyper-​Kähler manifolds in the Autumn Semester 2017 as part of the Nachdiplom lecture series in mathematics at ETH Zurich.

12.1. The L'Oréal-UNESCO Award.

The L'Oréal-UNESCO For Women in Science International Awards are presented every year to five outstanding women scientists - one per each of the following regions: Africa and the Arab States, Asia and the Pacific, Europe, Latin America and the Caribbean, North America - in recognition of their scientific accomplishments. Each scientist has had a unique career path combining exceptional talent, a deep commitment to her profession and remarkable courage in a field still largely dominated by men.

The scientific fields considered for the awards alternate every other year between Life Sciences (even years) and Physical Sciences, Mathematics and Computer Science (odd years).

12.2. The L'Oréal-UNESCO Award 2019.

On 11 February 2019 Professor Claire Voisin has received the L'Oréal-UNESCO International Award For Women in Science:-

... for her outstanding work in algebraic geometry. Her pioneering discoveries have allowed to resolve fundamental questions on the topology and Hodge structures of complex algebraic varieties.

Claire Voisin is a French mathematician and world leader in algebraic geometry; in her career she has given several outstanding contributions to different areas in the field, such as Noether-Lefschetz theory, Chow groups, syzygies of projective curves, stable rationality and the Lüroth problem; some of her best known results are in the Hodge theory of projective and Kähler manifolds. With her ideas and techniques, ranging between algebra, topology and geometry, she has been able to solve some longstanding open problems, and to give rise to new lines of research.

Let us enter into the subject and briefly describe one of her results: the solution of the Kodaira problem, at the heart of the topology of complex manifolds.

A compact complex manifold is called projective if it is biholomorphic to a projective variety, that is, a subvariety of the complex projective space defined by polynomial equations. It is a very natural question to explore the difference between general complex manifolds and projective ones, for instance at the level of topology. In (complex) dimension one, every compact complex curve (that is, a Riemann surface) is projective, but starting from dimension two, there are non projective examples.

There is a classical intermediate notion, that of Kähler manifold, which is a compact complex manifold admitting a special metric, called a Kähler metric; every projective manifold admits such a metric. By Hodge theory, the cohomology algebra of a Kähler manifold satisfies some special properties. Since the cohomology algebra is a topological (in fact, homotopy) invariant, this gives topological obstructions for a compact complex manifold to admit a Kähler metric.

In dimension two, there are Kähler surfaces which are not projective, but Kunihiko Kodaira proved in 1960 that every compact Kähler surface is homeomorphic to some projective surface, so that the obstructions to projectivity are not of topological type. The Kodaira problem asked whether the same holds in higher dimension.

In a paper published in Inventiones Mathematicae in 2004 Voisin gave a negative answer to this question, in every dimension at least 4. She showed that the cohomology algebra of a projective manifold satisfies some additional properties, and gave examples of Kähler manifolds (in any dimension at least 4) that do not satisfy these properties. This shows that such manifolds cannot be homeomorphic or even homotopy equivalent to any projective manifold.

Born in 1962 near Paris, Voisin studied at the École Normale Supérieure in Paris. In 1986 she got her PhD at Orsay, under the direction of Arnaud Beauville, and entered the Centre National de la Recherche Scientifique (CNRS). She became Directeur de Recherche in 1995, at the Institute of Mathematics of Jussieu. In 2016 she joined the Collège de France, where she holds the chair of algebraic geometry. Voisin won several prestigious awards, including a prize from the European Mathematical Society in 1992, and in 2016 the CNRS Gold Medal, France's highest scientific distinction: she has been the 6th mathematician to receive it (after Borel, Hadamard, Cartan, Serre and Connes). She gave plenary lectures at the European Congress of Mathematics in Stockholm in 2006, and at the International Congress of Mathematics in Hyderabad in 2010. These are only a few of the many recognitions of her outstanding research activity. Voisin is also a mother of 5 children, and has 3 grandchildren.

Voisin is very generous in sharing her ideas and knowledge, and has become an icon and a role model for a whole generation of algebraic geometers, young women and men moving their first steps in science. When it comes to gender issues in mathematics, she believes that "in science there is a cliché which still prevails, that women are not very gifted for abstraction." At the awards ceremony, held on 14 March at UNESCO's Headquarters in Paris, she thanked the L'Oréal Foundation for its action in favour of women's access to the world of science. She dedicated the award to her two granddaughters, "with the wish that they know a world where we do not speak any more of female scientists, but just of scientists, without gender considerations".

The program "For Women in Science", now at its 21st edition, has been created in 1998 by the L'Oréal Foundation in partnership with UNESCO. Besides many fellowships for doctoral and postdoctoral female researchers, every year five International Awards are conferred to outstanding female scientists, working in the five different regions: Africa and the Arab States, Asia-Pacific, Europe, Latin America & North America. Each Laureate receives €100.000.

The research themes for the Awards are, in alternating years, Life Sciences or Physical Sciences. This year, for the first time, Mathematics and Computer Science have joined Physical Sciences. These Awards can be very effective in promoting outside the academic world both the role of mathematics in our society and the role of women in top-level mathematical research. One example: if you have been at Charles de Gaulle airport this spring, you may have seen some big posters on 2019 L'Oréal-Unesco Awards, in particular one with the picture of Claire Voisin, Laureate for Europe, and the program motto: "The world need science. Science needs women."

12.3. Claire Voisin's pioneering discoveries.

Claire Voisin's pioneering discoveries have answered fundamental questions on the geometry, the topology and Hodge structures of complex algebraic varieties.

In today's instantaneous, media-driven world, scientific and mathematical challenges requiring deep thought and intellectual engagement with concepts and abstract theories rarely make the headlines. With the exception of a few more 'celebrated' branches of science, the contribution of scientists to society typically remains hidden, diverting government funding away from science, according to L'Oréal-UNESCO For Women in Science Mathematics Laureate 2019, Prof Claire Voisin.

And with mathematics often presented to school pupils as a 'fixed', unimaginative discipline consisting of 'ready-made' definitions, rule and equations, Prof Claire Voisin believes children are deterred from fully exploring the value of mathematics. "I'm in favour of a more open style of teaching, which would encourage students to push themselves and ask themselves more questions", she recalls.

With a naturally inquisitive mind, Prof Claire Voisin began exploring mathematics from a young age, encouraged by her father. She was delighted, aged 12, to discover the beauty of congruences, the poetry of algebra and the joy of combining conceptual and abstract reasoning to prove a concrete result. "It was wonderful to understand so fully how things work", she says.

In her intellectual elasticity and bold, uncompromising and rigorous approach, Claire Voisin embodies the classic idea of a mathematician. Free from the distraction of a mobile phone, she regularly explores new ideas and conundrums as she walks through the streets of Paris. Her love of finding theories, hidden logic and precise structures extends beyond mathematics to art and poetry too, rendering her unique: an artist of mathematics.

The desire to explore complex subjects and push the boundaries has shaped Claire Voisin's studies and career, leading her to focus on abstract mathematics. She specialises in algebraic geometry, a discipline that studies intrinsic properties of geometric figures starting from their equations. Throughout history, geometry has played an important role in helping us to understand the structure of the world, and more recently, the universe. The first geometers calculated the radius of the Earth, and physicists have built on theories from algebraic geometry to make their own discoveries.

A large part of Prof Claire Voisin's own pioneering research and achievements in complex algebraic geometry relies on the theory of Hodge structures (algebraic structures at the level of linear algebra) and uses it to address fundamental questions on the topology of complex varieties. The most important question in this field is the Hodge conjecture, a major unsolved problem in algebraic geometry.

Speaking about the value of mathematical research, Prof Claire Voisin says that it is hard to predict what will remain important in the long term. The value of research in pure mathematics is not measured by immediate applications, but it is important particularly in terms of developing new ways of thinking. "It might be just one construction, one argument, or the main result that are important for the future", she says.

With a string of awards and accolades to her name, including membership of the Académie des Sciences, Prof Claire Voisin is also the first woman mathematician to enter the Collège de France, the country's most prestigious research institution. She is also one of five women to have been awarded the gold medal from the CNRS (the French National Centre for Scientific Research).

While Prof Claire Voisin has not personally experienced gender discrimination during her career, she recognises that most women scientists do not achieve the recognition they deserve, and their work is often undervalued, or not considered as equal to men's work. "I think women have to make more effort than men, in particular at the beginning of their career, in order to be considered as serious researchers", she says.

Indeed, with too few women becoming mathematicians and scientists, particularly in her field, Prof Claire Voisin believes that women in science are too often treated as a minority, rather than "scientists among scientists". The lack of women in research roles represents "a significant loss for science". She recommends that all young women scientists should be ambitious and overlook external perspectives, and hopes that her recognition as a L'Oréal-UNESCO For Women in Science laureate will help encourage more women to pursue mathematics, including pure mathematics.

13.1. The BBVA Foundation Frontiers of Knowledge Awards.

The BBVA Foundation Frontiers of Knowledge Awards, funded with 400,000 euros in each of their eight categories, recognise 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.

13.2. The 2023 BBVA Foundation Frontiers of Knowledge Award.

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.

13.3. Claire Voisin's contribution.

The awardee researchers have made "outstanding contributions" to algebraic and symplectic geometry, which explore "spaces in high dimensions, which are difficult to visualise and necessitate new mathematical techniques to understand and study," said the committee in its citation.

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 realised 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 realised 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.

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 civilisation," 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."

13.4. Claire Voisin's biography.

Claire Voisin (Saint-Leu-la-Forêt, Valle del Oise, France, 1962) received her PhD in 1986 from Université Paris-Sud, where she studied under Arnaud Beauville. That same year, she joined France's National Centre for Scientific Research (CNRS). She is currently a research professor at the Mathematics Institute of Jussieu-Paris Rive Gauche, a research facility whose parent organizations are the CNRS, Sorbonne University and the University of Paris. Voisin was previously a member of the Centre de Mathématiques Laurent-Schwartz of the École Polytechnique, and was the first female mathematician to enter the Collège de France, where she held the Chair of Algebraic Geometry from 2016 to 2020. An invited speaker at leading research centers and specialist conferences worldwide, she holds or has held editorial positions in publications such as Mathematische Zeitschrift, Journal of Algebraic Geometry, Duke Mathematical Journal or Moduli.

13.5. Breaking down barriers.

The BBVA Foundation Frontiers of Knowledge Award in Basic Sciences has gone to Claire Voisin (National Centre for Scientific Research, CNRS, France) and Yakov Eliashberg (Stanford University, United States) for driving forward mathematical thought by bridging the space between two key areas of geometry with special relevance to quantum physics.

The award committee's citation notes that both researchers have made "outstanding contributions" to algebraic and symplectic geometry. These two areas of research have acquired special importance in recent years because of their potential to provide mathematical foundations for quantum field theory.

This branch of quantum physics, which is used to great success in the study of particle physics, is not entirely well defined mathematically. Therefore, a leading line of research is currently seeking to reconstruct quantum field theory from symplectic or algebraic geometry in order to explore whether the physical consequences deduced from these formulations agree with reality.

"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, and an award committee member.

Mathematics as a source of wisdom and mental discipline

Voisin admits that her research "has no direct application" in solving the practical problems of today. 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.

Whatever the case, Voisin sees mathematics as primarily "a fact of civilisation," 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."

In today's society, inundated with screens and saturated with instant messages that reach us via multiple channels, 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 realise how important it is to know how to concentrate."

14.1. The Crafoord Prize.

The Crafoord Prize is awarded by the Royal Swedish Academy of Sciences and the Crafoord Foundation in Lund, Sweden, with the Academy selecting the winners. A winner from a different discipline is chosen every year among the fields of mathematics and astronomy, geosciences, biosciences and polyarthritis. The Crafoord Prize in Mathematics and Astronomy was first awarded in 1982. The prize is worth six million Swedish crowns and will be awarded this year during the Crafoord days to be held in Lund and Stockholm from 13 May to 16 May 2024.

14.2. The Crafoord Prize in Mathematics 2024.

The Crafoord Prize in Mathematics 2024 is awarded to Claire Voisin, Institut de Mathématiques de Jussieu, France:-

... for outstanding contributions to complex and algebraic geometry, including Hodge theory, algebraic cycles, and hyperkähler geometry.

Claire Voisin is the first woman to receive the Crafoord Prize in Mathematics.

14.3. Claire Voisin's contributions.

Algebraic geometry deals with geometric shapes and structures that can be described as solutions to algebraic equations. Unlike the elementary geometry studied at school, these shapes are often impossible to visualise, and algebraic geometry has developed into one of modern mathematics most theoretically demanding areas.

Claire Voisin has provided important and highly acclaimed contributions in this field, through both counterexamples and strongly positive results for some of the most famous unsolved problems. One such example is the Kodaira problem, about which geometric shapes of higher dimensions that can be described by equations.

Throughout her career, she has also been the leading researcher on the Hodge conjecture, one of the seven Millennium Problems. Recently, she has developed a spectacular method for determining whether geometric shapes are rational, which means they are among the simplest ones. She is also leading in the field of hyperkähler geometry.

Claire Voisin was very surprised when she received the news she was to be awarded such a prestigious prize as the Crafoord Prize.

"If you look at the persons who have got it before me in my field, and in mathematics in general, being part of this group of people is extraordinary!", she says.