Category: Mathematics


Remembrance for Poincaré

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.

Fresh flowers — the Poincaré’s are not forgotten.


Happy 1.1919 day!

Today is January 19, 2019, 1/19/19!

The number 1.1919 can be expressed as the fraction 354/297 and the repeated fraction 1.\overline{1919} is 118/99 . Such a rational day!

The number 11919 is itself composite, expressible in terms of the primes 3, 29, and 137.

Let’s dive into 11919’s 19 side!

The featured image is a 19-sided star, a design by my daughter that was inspired by the ceiling of one of the Taj Mahal’s entrances

This tiling from the tomb of Shaikh Salim Chisti comes close also

Maybe 19 sides.

Whether they are 19 sided or not, they are still amazing.

19 is the 8th prime number. It is also a Pierpont prime, a number that can be written as 2^u 3^v + 1 where in this case u = 1 and v = 2 .

19 has a special significance in the Bahá’í world where the are 19 days in each month and 19 months in the year, with about \sqrt{19} (intercalary) days leftover.

The sum of the integers 1 to 19 is 190 and the sum of the primes up to 19 is 100!

Any day is a good day to meditate on the amazing patterns around us, especially a cold and rainy January Saturday!


Calculus and The Mother

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.

This was in service of improving the accuracy of astronomical calculations. I’m not even sure if Somayaji’s work was used in the Jantar Mantar observatories, but there is now speculation that the Kerala school might well have been known to Leibniz.

Chalk another wonder up to globalization, I’ll give props to The Mother for the inspirations.


A good year for Robert Langlands

I just saw that Robert Langlands has won this year’s Abel Prize in mathematics. A month back I had noted that two University of Chicago mathematicians –Sasha Beilinson and Vladimir Drinfeld — had received the Wolf prize for work that builds upon Langlands’ ideas.

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?

I feel such a blessing to have the optimistic spirit of my 80-something mother still present to bring uplift, laughter, and fresh greens from the garden to us — all served with divinely channeled love.  I think of the many 70+ year olds who passionately hold the world accountable,  try to make a difference with their material success, fathom prime numbers like Langlands, weave saxophone melodies, and make the world a beautiful place with their wisdom and selflessness. Spring persists in the garden of the ageless mind. I’ll leave you with some Sonny Rollins


The geometrical beauty of Doha

As we passed through Doha on the way to Gaborone, I was amazed by the architectural beauty of so many Islamic inspired structures. It was truly a feast for the eyes and mind.

Though we did not have time to visit many of the older architectural treasures, I discovered that a lot of the buildings have received prestigious architectural awards over the last decade. The investment of Qatar in its country is amazing, and Al Jazeera is a gift to humanity.

There is even wonder in the Qatar airways “air sickness” bags!



The Wolf prize mathematicians outside my nook

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 useful collaborations 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.

He seems to be a celebrity in his own right. I enjoyed how he connects Solaris to universals of number theory!

If Frenkel is still too abstract for you, then Mitya Boyarchenko suggests that this poem that I include below might be of use in understanding the Langlands talks


A man called Pakhomych, shaking as he rode on the carriage footboards,

Carried a bunch of forget-me-nots.

He got corn on his heels,

And treated them at home with camphor.

Reader! Having discarded the fable’s forget-me-nots,

Which were put here as a joke,

You can arrive at only a single conclusion:

If you get corns

And you want to rid yourself of the pain,

You, like our friend Pakhomych,

Should treat them with camphor.


MathematicsPoliticsSocial Justice

Mathematicians, rock the vote!

Can the resistance inspire a new generation of mathematicians?

Samuel Hansen thinks so. In his recent post on The Aperiodical, he describes how the recent avalanche of math-informed court decisions on gerrymandering in Pennsylvania and elsewhere are putting mathematics in the spotlight.

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!