Abel Prize Laureate 2019: Karen Uhlenbeck

Karen Uhlenbeck was selected by a committee of five mathematicians nominated by the European Mathematical Society and the International Mathematics Union. Her work involves the study of partial differential equations, calculus of variations, gauge theory, topological quantum field theory, and integrable systems. The full citation from the announcement can be found here and a short biography by Jim Al-Khalili here.

“Karen Uhlenbeck receives the Abel Prize 2019 for her fundamental work in geometric analysis and gauge theory, which has dramatically changed the mathematical landscape. Her theories have revolutionised our understanding of minimal surfaces, such as those formed by soap bubbles, and more general minimisation problems in higher dimensions.” – Hans Munthe-Kaas, Chair of the Abel Committee.

Karen’s work covers minimisation problems, such as solving for the shape of a soap bubble acting to minimise its energy under gravity. Here’s a fantastic slow-motion experiment from Ray Goldstein at the University of Cambridge demonstrating the change in the shape of a soap bubble as the two supporting wires are pulled apart.

Karen also works in topological quantum field theory which has very important consequences for physicists, not least in relation to the Yang-Mills Mass Gap Hypothesis – one of the 7 million-dollar Millennium Problems. You can read more about the problem here.

“If I really understand something, I’m bored.” Karen Uhlenbeck

Throughout her career Karen has been very active in the area of mentorship and furthering the cause of women in mathematics. She is the founder of the Institute of Advanced Study Women’s Program, now entering its 25th year, and the Park City Mathematics Institute Summer Session, which places a huge emphasis on interdisciplinary research and collaboration between mathematicians from all areas.

The Abel Prize was established on 1 January 2002 – 200 years after the birth of Niels Henrik Abel. The purpose is to award the Abel Prize for outstanding scientific work in the field of mathematics. The prize amount is 6 million NOK (about 750,000 Euro) and was awarded for the first time on 3 June 2003.

You can read the official announcement from the Norwegian Academy of Science and Letters here.

What is the blast radius of an atomic bomb?

Picture the scene: you’re a scientist working for the US military in the early 1940’s and you’ve just been tasked with calculating the blast radius of this incredibly powerful new weapon called an ‘atomic bomb’. Apparently, the plan is to use it to attack the enemies of the United States, but you want to make sure that when it goes off any friendly soldiers are a safe distance away. How do you work out the size of the fireball?

One solution might be to do a series of experiments. Set off several bombs of different sizes, weights, strengths and measure the size of the blast to see how each property affects the distance the fireball travels. This is exactly what the US military did (see images below for examples of the data collected).

Picture1 Picture2

These experiments led the scientists to conclude that were three major variables that have an effect on the radius of the explosion. Number 1 – time. The longer the time after the explosion, the further the fireball will have travelled. Number 2 – energy. Perhaps as expected, increasing the energy of the explosion leads to an increased fireball radius. The third and final variable was a little less obvious – air density. For a higher air density the resultant fireball is smaller. If you think of density as how ‘thick’ the air feels, then a higher air density will slow down the fireball faster and therefore cause it to stop at a shorter distance.

Now, the exact relationship between these three variables, time t, energy E, density p, and the radius r of the fireball, was a closely guarded military secret. To be able to accurately predict how a 5% increase in the energy of a bomb will affect the radius of the explosion you need a lot of data. Which ultimately means carrying out a lot of experiments. That is, unless you happen to be a British mathematician named G. I. Taylor…

Taylor worked in the field of fluid mechanics – the study of the motion of liquids, gases and some solids such as ice, which behave like a fluid. On hearing of the destructive and dangerous experiments being conducted in the US, Taylor set out to solve the problem instead using maths. His ingenious approach was to use the method of scaling analysis. For the three variables identified as having an important effect on the blast radius, we have the following units:

Time = [T],       Energy = [M L2 T-2],      Density = [M L-3],

where T represents time in seconds, M represents mass in kilograms and L represents distance in metres. The quantity that we want to work out – the radius of the explosion – also has units of length L in metres. Taylor’s idea was to simply multiply the units of the three variables together in such a way that he obtained an answer with units of length L. Since there is only one way to do this using the three given variables, the answer must tell you exactly how the fireball radius depends on these parameters! It may sound like magic, but let’s give it a go and see how we get on.

To eliminate M, we must divide energy by density (this is the only way to do this):

eqn1.png

Now to eliminate T we must multiply by time squared (again this is the only option without changing the two variables we have already used):

eqn2.png

And finally, taking the whole equation to the power of 1/5 we get an answer with units equal to length L:

eqn3

This gives the final result that can be used to calculate the radius r of the fireball created by an exploding atomic bomb:

eqn4

And that’s it! At the time this equation was deemed top secret by the US military and the fact that Taylor was able to work it out by simply considering the units caused great embarrassment for our friends across the pond.

I love this story because it demonstrates the immense power of the technique of scaling analysis in mathematical modelling and in science in general. Units can often be seen as an afterthought or as a secondary part of a problem but as we’ve seen here they actually contain a lot of very important information that can be used to deduce the solution to an equation without the need to conduct any experiments or perform any in-depth calculations. This is a particularly important skill in higher level study of maths and science at university, as for many problems the equations will be too difficult for you to solve explicitly and you have to rely on techniques such as this to be able to gain some insight into the solution.

If you’re yet to be convinced just how amazing scaling analysis is, check out an article here explaining the use of scaling analysis in my PhD thesis on river outflows into the ocean.

And if that doesn’t do it, then I wish you the best of luck with those atomic bomb experiments…

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