Sofia Kovalevskaya, Mathematician Of The Year

Pere escaped the office to attend a half-day conference at his alma-mater
Pere Daniel Prieto
Pere Daniel Prieto
March 21, 2019

Earlier this month, The School of Mathematics and Statistics at the Polytechnic University of Catalonia hosted a half-day conference devoted to their ‘Mathematician Of The Year’ - Sofia Kovalevskaya.

FME is my alma mater and Sofia Kovalevskaya is one of my personal heroes so I took the day off to attend the conference and share my experience.

‘Mathematician Of The Year’

Since 2003, which was coincidentally my first year at the university, the FME has chosen a recipient of the ‘Mathematician Of The Year’ award.

During the academic year, the school organises activities aimed at delving deeper into the personal histories of the winner. These activities culminate in a half-day conference devoted to the recipient.

Here’s a list of every ‘Mathematician Of The Year’ recipient since the award was founded:

  • 2003 – 2004: Henri Poincaré, French mathematician, theoretical physicist, engineer and philisopher of science.
  • 2004 – 2005: Albert Einstein, German-born theoretical physicist.
  • 2005 – 2006: Carl Friedrich Gauss, German mathematician and physicist.
  • 2006 – 2007: Leonhard Euler, Swiss mathematician, physicist, astronomer, logician and engineer.
  • 2007 – 2008: Bernhard Riemann, German mathematician.
  • 2008 – 2009: Emmy Noether, German mathematician.
  • 2009 – 2010: John von Neumann, Hungarian-American mathematician, physicist and computer scientist.
  • 2010 – 2011: Paul Erdös, Hungarian mathematician.

Side note: Erdös was so prolific an author that the mathematical community now celebrate the Erdös number. The number pertains to the distance between a mathematician and Erdös, as measured by authorship of mathematical papers. There’s even an xkcd comic about it. FYI, my number is five!

The conference

The day started with a warm welcome from the Dean of the FME. He then gave an introduction which highlighted the importance of celebrating the achievements of women in the field.

The conference then began with a look at Kovalevskaya’s life.

‘Sofia Kovalevskaya’s fascinating life’

View the presentation slides.

The first talk of the day was delivered by Prof. María Molero (Alcalá de Henares University, currently retired) and Prof. Adela Salvador (Polytechnic University of Madrid, also currently retired) and it focused on the biography of Sofia Kovalevskaya.

Kovalevskaya, indeed, lead a fascinating life. For instance, she was the first woman to:

  • earn a PhD, as we now define it, at a European university (Göttingen)
  • be appointed as the editor of a mathematical journal, the Acta Mathematica
  • win the French Prix Bordin de l’Académie des Sciences
  • obtain a tenure position as a professor at a university (Stockholm)
  • be made a Corresponding Member of the Russian Academy of Sciences (no professorship came with this appointment)

Her life is very well documented as she wrote a memoir about her early years, titled ‘A Russian Childhood’ (1889). Her close friend, Anne Charlotte Edgren-Leffler, the Duchess of Cajanello, also wrote a biography all about her.

As fascinating as I found the content of this talk, the organisation and general structure left a lot to be desired. There were too many slides (98 to be exact) and the speakers spent too much time on Kovalevskaya’s childhood.

The focus on her childhood, whilst interesting, meant that there was limited time to discuss the later years of her life - e.g. the time period in which she obtained both her PHD and the Bordin Prize.  

If you’re interested in finding out more about Kovalevskaya, I would highly recommend that you read both of the aforementioned books. You’ll feel as if you were reading a novel as opposed to non-fiction!

A quick look at the Cauchy-Kovalevskaya Theorem’

View the presentation slides.

After a short coffee break, it was time for the second talk of the day. This talk was given by Prof. Alberto Enciso (researcher at the Institute of Mathematical Sciences) whom I’ve known for several years since my stay at the ICMAT in 2013.

The talk focused on what is nowadays referred to as the ‘Cauchy-Kovalevskaya Theorem’. This theorem was first explored by Kovalevskaya in one of the three works she submitted to earn her PhD at Göttingen.

It is usually compared to Picard’s Theorem for ordinary differential equations and is considered its counterpart for partial differential equations, as it looked at the latter and the unique solutions they afforded a neighbourhood at any given point.

The theorem and its uses

Nevertheless, due to the specifics of partial differential equations, this theorem, whilst it’s considered to be oustanding and rare, isn’t particularly useful.

Why is it so oustanding and rare? Partial differential equations are complex and they are usually studied as such - e.g a single equation or system of equations is considered at the same time. In fact, almost every course on partial differential equations that you encounter at universities follows this structure:

‘What is a partial differential equation?’ is a question that’s pondered during the first fortnight and then the rest of the course is devoted to studying particular examples such as heat, sound, diffusion, electrostatics or fluid dynamics. This theorem is the only real general result on partial differential equations which, thus, makes it singular.

So, why is it not useful? Unfortunately, the theorem only applies to the ‘boring’ class of partial differential equations whose coefficients are analytic functions e.g functions that are locally given by a convergent power series. As any first year Mathematics student knows, this class of functions is extremely well-behaved and thus the analyticity is a strong and restrictive condition.

sofia kovalevskaya
The Cauchy-Kovalevskaya theorem.


During the talk, the speaker relayed the result to the audience before exploring guidelines on the original proof made by Kovalevskaya and highlighting the importance of her ideas within the context of the time period. In order to do so, he used a toy model to illustrate every step of the proof which made it much more engaging.

Prof. Enciso also compared the result to Picard’s Theorem. However, in my opinion, the most interesting part of the talk was its conclusion when the speaker explained how many times he had used the result in his own work.

Prof. Enciso is an expert on partial differential equations, with around 70 published works under his belt, and he (in collaboration with his colleague Prof. Daniel Peralta-Saltas) has solved two conjectures (one thanks to V. Arnold in 1965 and the second thanks to Lord Kelvin in 1875). It was really interesting to hear about how he had used the Cauchy-Kovalevskaya Theorem during his career despite its supposed lack of use.

It turns out that he had used the result, by either directly applying it or by being indirectly inspired by it, a total of five times. In its own words, ‘it’s not a really useful result, but it’s kind of remarkable how it does inspire other works’.

In my opinion, this was the best talk of the day.

‘An excursion through integrable systems, around Sofia Kovalevskaya’

View the presentation slide.

After a second break, albeit marked without coffee this time, we convened for the third and last talk of the day. This talk was given by Prof. Mariano Santander from Valladolid University.

Prof. Santander is another of my old acquaintances from my time as a PhD student. I was lucky enough to communicate with him on several occasions due to the commonalities of both our research interests. Prof. Santander’s talk focused on Sofia Kovalevskaya’s 1888 work, ‘Mémoire sur un cas particulier du problème de la rotation d'un corps pesant autour d'un point fixe, où l'intégration s'effectue à l'aide des fonctions ultraelliptiques du temps’.

This work earned Kovalevskaya the Bordin Prize of the French Académie des Sciences, and was later published in a mathematical journal under the following title: ‘Sur le problème de la rotation d'un corps solide autour d'un point fixe’ (Acta Mathematica, 12 (1): 177–232).

Euler and Lagrange

The problems studied by Kovalevskaya in this particular piece of work originate from classical mechanics and consist of studying the motion of a heavy rigid body as it rotates around a fixed point in space.

This problem was previously studied by Leonhard Euler and Joseph-Louis Lagrange and they had both found a particular case in which the problem was integrable. That is that the problem could be solved and the equations that give the motion of the rigid body could be obtained using constants of motion.

These particular cases are now referred to as Euler’s top and Lagrange’s top, respectively. Back in the early 1880s, these were the only cases known to be integrable. In 1888, however, Kovalevskaya famously discovered a new case, nowadays referred to as Kovalevskaya’s top, in which the problem was integrable and she too obtained the solution using constants of motion.

These three tops are now considered to be the only cases in which this problem is integrable (when adding the hypothesis of considering holonomic systems).

Prof. Santander expanded on the historical context of the problem and explained Kovalevskaya’s role in discovering it. He finished by playing several videos in which both a real and 3D model of Kovalevskaya’s top were set in motion.

In watching these videos, it became clear why no-one had previously unearthed this particular solution: the model was extremely non-trivial and the physical representation was very difficult to be created accurately.

This adds an extra layer of value to an already remarkable body of work.

Final thoughts

I really enjoyed attending the conference and I’m really pleased with the choice of Sofia Kovalevskaya as the ‘Mathematician Of The Year’.

Women have played an important role in science across the centuries and denying the impact and importance of their work is foolish. Moreover, their contributions have usually been even more outstanding than their male peers and conferences like this one are a healthy reminder of that reality.

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References:

The video of the conference in full.

The photo gallery from the event.