To be invited to speak at the International Congress of Mathematicians is a great honour, and to be forty years of age or younger is doubly impressive, for it means that you might be very well in with a chance to win the Fields Medal. I thought I might scan through the list of speakers at the ICM 2010 site and see who the young guns are. In particular, I thought I'd focus on the women.
Now, at the absolute top rank of mathematics, there have been historically very few women. Think about this: between the first ICM and the first address by a woman, almost 30 years passed. Then another 60-odd years went by until Karen Uhlenbeck spoke at ICM 1990.
Fortunately, though, this is all changing. And so it is heartening indeed to see women's names pop up in this list. Even more wondrously, these young scientists are not restricted to the traditional powerhouses of mathematics - Russia, France, USA. You'll find Iranians and Spaniards and Taiwanese as well.
So here goes.
Maryam Mirzakhani: this mathematician from Iran, now based in the USA, is doubly honoured - she addresses a session in Topology as well as Dynamical Systems. Like several previous winners of the Fields Medal, she was very successful in her youth at the International Mathematics Olympiad. More recently, she was awarded the Blumenthal Award (2009), which is awarded quadrennially, and is for the best PhD thesis published in preceding four years. Her work - among others - is in the geometric structures and their deformations in all sorts of spaces, and she brings in an interdisciplinary approach to solving problems in the field, by using insights from combinatorics and mathematical physics.
Irit Dinur: is a theoretical computer scientist from Israel, where she has been working on problems in proof theory - how to establish formally that a proof is correct? In particular, by making random inspections of a formally written-out proof, is it possible to verify it? This is a deep problem in theoretical computer science, with a fundamental result (that it is, indeed, possible) established in 1992. She was able to establish a much simpler and radically new proof of this theorem in 2005, which has resulted in fresh pastures for investigation. As we know, it's not enough to solve a problem - what's better is to do so in such a way that a whole new domain of research opens up, with exciting new possibilities. Dinur has done this with aplomb. And so there's some gossip that she might win the Rolf Nevanlinna Prize (which is also awarded quadrennially at an ICM) for the applications of mathematics in the information sciences.
Sophie Morel: is from France, and her PhD thesis was an important development in the Langlands Program, solving a problem that had remained open for over twenty years. (You may recall that Laurent Lafforgue won the Fields Medal in 2002 for his contributions in this area.) This is cutting-edge work at the intersection of number theory and algebraic geometry. She was made a full professor of mathematics at Harvard last year, a notably rare and distinguished achievement made especially so when you realise that she's the first woman to be tenured in mathematics at that university! To boot, she is a skilled polyglot, conversant in French, English, Russian, German, Spanish, and now learning Korean.
Chiu-Chu Liu: is a mathematical physicist from Taiwan. She, again, is a multidisciplinarian, combining techniques from topology, differential geometry, and algebraic geometry to answer open problems in theoretical physics. In particular, her work in establishing the Marino-Vafa conjecture has been well-recognised. This has deep ramifications in string theory.
Anna Erschler: is a Russian mathematician based in France. Along with Mirzakhani, she too has two addresses at the ICM (Probability and Geometry). Her work is at the conjunction of probability and group theory.
Isabel Fernández: is a Spanish professor of mathematics at the University of Seville, and has received much attention in her native country for being the first ever Spanish woman to be invited to an ICM. Her work has been termed, loosely, soap-bubble geometry, because she investigates the geometric properties of curved objects. It is at once a classical field in mathematics, but equally cutting-edge, combining results from differential equations, complex analysis and variational calculus. Interestingly, her work has found immediate practical application in architecture, notably at the Olympic stadium in Munich, where surfaces she studied have been found to be light-weight, use little by way of materials, and are notably resilient. And, having solved one of the open major problems in the field - minimal surfaces in homogeneous spaces - the invitation to the ICM celebrates her achievement (to be sure, with her colleague Paul Mira).
Catharina Stroppel: is a German mathematician, winner of the 2007 Whitehead prize for her work in representation theory and its applications to low-dimensional topology.
Marianna Csörnyei: is Hungarian, another Whitehead prizewinner (2002), and works in geometric measure theory. "Central to her work is the analysis of viable definitions of ‘negligible’ in the context of infinite-dimensional situations, with a view to applications in non-linear geometric functional analysis. Technically difficult, the judges described her work as characterised by the ‘startling nature of many of her results’. A particularly ‘spectacular achievement’ highlighted was her proof that the three main notions of ‘negligibility’ coincide, and her revelation of delicate phenomena in the theory of Lipschitz quotients even in the finite dimensional case." (from here)
Nalini Anantharaman: is French; her work is in mathematical physics, and she attempts to answer questions about the phenomenon of dispersion: "A wave propagates in a closed cavity. It will bounce off the walls. I'm trying to understand how it will dissolve: will it remain compartmentalized, contained in a portion of the cavity or will it be dispersed throughout the cavity?" (from here) One of her major contributions is in quantum chaos, where she established some results supporting the Quantum Unique Ergodicity Conjecture.
Katrin Wendland: is German; her work is in mathematical physics, particularly in the nature of particles. Notably, she "constructed a large class of examples of mirror symmetry using orbifold methods and Kummer K3 surfaces" (from here) Her research also unfolds deep connections between non-commutative geometry and algebraic geometry, and she has many contributions in topological quantum field theories.
Dorit Aharonov: is an Israeli computer scientist. In 2005, she was profiled in Nature. One of her major achievements was to show that even in the presence of interfering noise, a quantum computer could still achieve reliable results. Because quantum computers require considerable isolation from their surroundings (the 'quantum processors' should not interact with their surroundings, or the resultant errors will rapidly degrade the computation), it was thought that these would remain theoretical curiosities. Her work in quantum error correction went a long way in establishing the domain as technologically viable. In addition, she has worked in the quantum scale problem - why do quantum effects manifest only at subatomic levels but appear to vanish at human scales? "Aharonov showed that for many noisy quantum systems, there is a level of noise above which a transition to classical behaviour is inevitable. Such transitions are much sharper than expected from other theories that predict a gradual shift away from quantum behaviour."
Hopefully, at least one of them wins the medal.