Twenty Female Mathematicians Hollis Williams Acknowledgements The author would like to thank Alba Carballo González for support and encouragement. 1 Table of Contents Sofia Kovalevskaya ................................................................................................................................. 4 Emmy Noether ..................................................................................................................................... 16 Mary Cartwright ................................................................................................................................... 26 Julia Robinson ....................................................................................................................................... 36 Olga Ladyzhenskaya ............................................................................................................................. 46 Yvonne Choquet-Bruhat ....................................................................................................................... 56 Olga Oleinik .......................................................................................................................................... 67 Charlotte Fischer .................................................................................................................................. 77 Karen Uhlenbeck .................................................................................................................................. 87 Krystyna Kuperberg .............................................................................................................................. 98 Nicole Tomczak-Jaegermann ............................................................................................................. 108 Dusa McDuff ....................................................................................................................................... 118 Karen Vogtmann ................................................................................................................................. 128 Carolyn Gordon .................................................................................................................................. 139 Frances Kirwan ................................................................................................................................... 149 Leila Schneps ...................................................................................................................................... 159 Claire Voisin ........................................................................................................................................ 169 Olga Holtz ........................................................................................................................................... 179 Maryam Mirzakhani ........................................................................................................................... 189 Maryna Viazovska .............................................................................................................................. 199 2 Preface In this piece of writing, I make an attempt to briefly discuss the work of twenty female mathematicians. I am aware that in some sections I have gone off on tangents a bit more or not said as much about the actual work which the mathematician does or the exact details of their achievements. In most cases, it would have been difficult to give more than a few details and it is completely impossible to give a good survey of someone’s work within the confines of 10 pages per person along with added context: what I have tried to do is really give a flavour of the kind of things that the person works on and hopefully inspire you to read on further. The difficulty was particularly obvious in algebraic geometry. By the time one has given all the definitions which are needed, one has run out of space and not had the opportunity to actually apply them to some interesting problems, so I had to try my best to give concise, roughly correct definitions whilst also trying to give an idea of the type of problem which can be tackled with the theory, even if there was not space for more specific examples. In some cases I have skipped some of the definitions, trusting that the reader will look them up in the references if unsure. The work does not necessarily represent the people who I think are the twenty greatest female mathematicians of all time and reflects my own interests in mathematics and the things which I felt I might be to talk about semi-competently for a few pages. Given the range of topics and problems covered, there are likely to be errors and conceptual mistakes. Please email me to discuss mistakes and I will be happy to upload a corrected version. Email: [email protected] 3 Sofia Kovalevskaya Kovalevskaya was a nineteenth-century Russian mathematician, known mainly for her achievements in mechanics and the theory of PDEs. She gained qualifications and held a number of posts which were traditionally reserved only for men and was amongst the first women to work as an editor of a scientific journal1. Probably her most substantial mathematical contribution was the Cauchy-Kovalevskaya theorem, which belongs to PDE theory. PDEs (partial differential equations) are equations which model the rate of change of a quantity with respect to two more other quantities which are also changing, whereas ODEs (ordinary differential equations) model the rate of change with respect to one quantity only (usually a time variable). All of science, engineering and applied mathematics depends on differential equations. PDEs were originally brought into being as tools to describe physical phenomena such as wave motion and heat flow (a role which they still play), but over time PDE theory has also developed in another direction into a rich branch of mathematical theory in its own right (without completely losing sight of the physical origin of the equations). A famous and well-studied example is the one-dimensional wave equation: 휕2푓 휕2푓 = 푐2 . 휕푡2 휕푥2 One sees that there are no mixed derivatives in this expression (ie. derivatives with respect to more than one variable), but it should also be pointed out that mixed derivatives can in fact occur in equations which model physical processes: there are versions of the diffusion equation which contain mixed derivative terms, for example. A very interesting development in the twentieth century was the notion that one might be able to ‘weaken’ the notion of a solution to obtain weak solutions which exist where our traditional idea of a solution does not exist: this is rather surprising from the viewpoint of classical PDE theory, since physical intuition would seem to tell us that the notion of a solution has to be a concrete one which cannot be weakened in this way. Essentially, we take a PDE and rewrite it so that it is in ‘weak’ form: a solution to this equation is known as a weak solution to the original PDE. A strong solution is automatically a weak solution, but it is unfortunately not always obvious that a weak solution should also be strong. I should point out that the act of rewriting an equation is not necessarily by itself a profound one. The notion of having an equation or solution with particular types of operator in it and rewriting it so those operators do not appear is fairly common and trivial. 1 Ann Koblitz, A Convergence of Lives: Sofia Kovalevskaya: Scientist, Writer, Revolutionary (New Jersey, USA: Rutgers University Press, 1993). 4 As an example, one can obtain a fundamental solution to a particular system of equations modelling dilute gas flow, where the solution represents the physical pressure due to a gas: 8훾2Kn푔 푝 = −퐟. ∇휙 + 퐆(∇∇휙) − 2 휇 + 퐆(∇∇휇 ), 15 훾2 훾2 where 퐆 is some arbitrary trace-free symmetric 2-tensor, 휙 is the fundamental solution to the PDE known as Laplace’s equation (derived essentially by considering the symmetries which the solution must have), 휇훾2 is the fundamental solution to the Helmholtz equation (similar to the Laplace equation) and Kn is a dimensionless parameter similar to the Reynolds number2. One notes the repeated appearance of the ∇ operator, known as ‘nabla’ or ‘the gradient’. Using the definition of ∇ as a differential operator, it is a trivial operation to rewrite the pressure term so that it no longer includes nabla: 40 퐫 3 6Kn 푔 1 푝 = 퐟. (− ) + 퐺 푟 푟 ( ) − 푒−훾2푟 4휋|퐫|3 푖푗 푖 푗 4휋|퐫|5 15 4휋|퐫| 푒−훾2푟 (5|퐫| + 3Kn(6Kn + √30|퐫|)) + 퐺 푟 푟 . 푖푗 푖 푗 24Kn2휋|퐫|5 This looks a bit messier, but the differential operator no longer appears and so it might be faster to process the solution with a computer. However, in this case, there is something more powerful going on, since the reformulation we propose allows us to keep a notion of a solution to the equation, even if it cannot be differentiated a sufficient number of times to be counted as a classical solution! At the bottom of things, a variational technique is being employed to find existence of solutions. As the simplest possible example in what is called the ‘elliptic’ theory, one takes a standard second-order elliptic PDE such as the Poisson equation defined for a boundary value problem on a three-dimensional ball 퐵 and then rewrites it as: ∫ −∆푢 ∙ 휙 d푥 = ∫ 푓 ∙ 휙 d푥, 퐵 퐵 where 휙 belongs to 퐶∞(퐵) and 푢 solves the original PDE. The fact that this reformulation is equivalent to the PDE we had before is due to what is known as the fundamental lemma of calculus of variations. The