If you have read Cixin Liu’s The Three Body Problem, the name Henri Poincaré might ring a bell. Poincaré was an early twentieth century mathematical master. One of his feats was an analysis of how three masses in mutual orbit behave. This analysis provides the foundation for chaos theory. In Liu’s science fiction book, the main character becomes obsessed with an online game that hinges on a world orbiting three suns — a cosmic version of Poincaré’s problem.
Poincaré is also the central figure in modern topology. He made an educated guess — a conjecture — that any 3 dimensional manifold ( I promise that you’ll get the feels for what a manifold is if you follow the link) that is finite in size and without boundary can be described by a 3 dimensional sphere. Basically if you have a shape that has no holes and can be fit into a box, then you can mush the shape into a sphere. Ok, just watch this video or try these games.
It took about a hundred years to prove Poincaré’s guess — Grigori Perelman did so in 2002 and was awarded (but refused) a Fields Medal and won (but refused) the $1 million Millenium Challenge. Poincaré accomplished intellectual gold in his short lifetime, giving us the formalization for gravitational waves and chaos theory among many other thins. He championed human rights and the Poincaré Institut actively continues his work.
On a whim I visited Montparnasse Cemetery where he rests with other family. The groundskeeper was so excited to point me to the location. I took some photos during a brief lull in the rain as the day’s national strike unfolded.
Last weekend we spent a beautiful afternoon with cousins at the Sri Aurobindo Ashram in Delhi.
There was a lecture going on about the divine inspiration of Calculus — a meditation on how both Newton and Leibniz came to discoveries of infinite series and limits that led to the starting point for advanced maths.
As I pondered the spirit of The Mother, my mind went back to the visit we’d taken to the Jantar Mantar in Jaipur — an astronomical observatory built around the time of Newton’s discovery. Surely, a society that had the capacity to develop highly accurate astronomical predictions had the sophistication to develop the machinery for dealing with infinitesimal rates of change.
Newton — or more likely Leibniz — was indeed late to the game by at least 200 years!
Keralan mathematician Nilakantha Somayaji in the 1400s seems to have worked out machinery for dealing with infinitesimal velocity and converging series.
What are those ideas? Langlands has spent his life looking for connections between number theory and real analysis. The featured image is a rendering of an automorphic form, one of the kinds of functions that Langlands has been interested in. As far as I could understand, Beilinson and Drinfeld found ways of connecting this work to modern physics. Maybe a deeper understanding is my goal for 2018. This Quartz article is a good quick read as is this short piece on the fundamental lemma.
Or, you can let the distinguished Dr Langlands explain it himself.
Whether or not you have a liking for numbers, seeing an 81 year old still in the thick of things is infectiously inspiring. Perhaps you’ll allow him to re-acquaint you with Pythagorus?
Yesterday I came across a photo of two gentlemen sitting outside of my old grad-school student lounge. They are Sasha Beilinson and Vladimir Drinfeld, two mathematicians from my alma mater who were awarded this year’s Wolf Prize in Mathematics.
The CS department at the University of Chicago shared space with Mathematics and Statistics in my day, so it was not unusual to encounter mathematicians while having lunch (or a nap) in the lounge. There have been many usefulcollaborations and intersections between these departments.
I have no idea what Sasha or Vladimir do. I tried to understand. I glanced at their ground breaking work, a book called Chiral algebras. They state in the introduction “Chiral algebras have their origin in mathematical physics;” and “Chiral algebras are “quantum” objects.” Ok.
Drinfeld and Beilinson still run the Geometric Langlands Seminar that of course captures the essence of what they care about most. As best I can figure, Langlands,himself a 1996 Wolf prize recipient, is a mathematician who envisioned building links between algebra and modern physics. Drinfeld and Beilinson have extended that work. Maybe the best explanation of this undertaking is provided by Edward Frenkel.
It is really heartening that discrete geometry and other branches of advanced mathematics can be use to preserve democracy — much in the spirit of the 1964 voting rights act (being signed in the featured image).
Tufts University mathematician Moon Duchin has done a lot of work in this area, leading the effort to train mathematicians to be expert witnesses in gerrymandering cases. Duchin’s Metric Geometry and Gerrymandering Group page has a lot of useful resources.
Consider registering for one of the gerrymandering trainings if you’re a mathematician, statistician, or data scientist based in the Bay Area!