# Robert P. Langlands

School of Mathematics

Institute for Advanced Study

1 Einstein Drive, Princeton, New Jersey 08540 USA

**Article**: *Canadian Who Reinvented Mathematics* by Sandro Contenta

ArXiv: About *Langlands Program, Trace Formulas, and their Geometrization* by Edward Frenkel

**CV**: http://publications.ias.edu/rpl/

**Homepage**: https://www.math.ias.edu/people/faculty/rpl

**Wikipedia**: https://en.wikipedia.org/wiki/Robert_Langlands

**Video**: https://video.ias.edu/The-Practice-of-Mathematics

YouTube: A short, but well-done introduction by Sandro Contenta

. . . . . . . Langlands visit at Oxford, July 5, 2014

. . . . . . . . Edward Frenkel did a series of video introductions to the Langlands programs

**Pages within this site that struggle to understand Langlands programs**:

Langlands I, II, II III, IV

Most recent Email: Saturday, April 6, 2019

Dear Prof. Dr. Robert Langlands:

Coming up on two years now since my last email, I thought you might appreciate an update given that there is a link on our homepage to your work ( https://81018.com/e8/#EF ) and to our reference page about your work ( https://81018.com/langlands/ ).

Of course, our most simple outline of the universe must have a place for the Langlands programs and it seems, along with Witten and string theory, it is within Notations 1 to 64. Though an entirely idiosyncratic concept, everything must start “simple” before it becomes “complex.” Thank you.

Best wishes,

Bruce

Second Email: Wed, Apr 26, 2017 at 10:58 AM

Dear Prof. Dr. Robert Langlands:

We may be taking the data from our base-2 chart (starting with the Planck units and going to the Age of the Universe) to create a “spaceapp” this weekend for NASA’s Space App Challenge. Of course, we want to use all 202+ notations within that chart beginning with the so-called singularity at notation #1.

The first second within the life of this universe takes us to just over 143rd notation. Along the Planck Length continuum, the first 67 notations are smaller than the work done at CERN labs so imagination will be a key to creating a visualization of those initial notations.

The epochs defined by the big bang theory should be an excellent guide and level set for this work.

In my simple way, close-packing of equal spheres using the Feigenbaum constant, Wolfram’s computer automaton, and possibly some representation of the Langlands program should “compete” for pointfree vertices. I wonder if perhaps the visual manifestations of the pentagon and the heptagon could represent the Langlands programs.

I sent a note to Stephen Wolfram and Mitchell Feignebaum to see if they have suggestions. Would I be correct to believe that this project would be the first time dimensionless or pointfree geometries vis-a-vis Alfred North Whitehead and topos theory would actually have been visualized?

This morning I have been listening to your *The Practice of Mathematics* (October 26, 1999).

Thank you.

Most sincerely,

Bruce

First email: Mon, Jan 18, 2016 at 4:58 PM

Dear Prof. Dr. Robert Langlands,

In 2011 our high school geometry class divided by 2 the tetrahedron (and octahedron within it) about 105 times until we were at Planck Length & Time, then multiplied by 2 until we were out to the Observable Universe and the Age of the Universe. With just over 202 base-2 exponential notations (doublings), we unwittingly created a shortcut to everything, everywhere for all time. Hardly a theory, it was just simple math and great fun.

When we went looking for the scholarly work behind it all, we could only find Kees Boeke’s 1957 work using base-10. Base-2 had been ignored. Yet, base-10 had no boundaries, no Planck base units, and no simple geometry.

It has taken us four years to begin to believe we actually may have something special here. Of course, it is idiosyncratic to fault. Bring in a quiet expansion of the universe where space and time are finite, discrete, quantized and derivative. I can hear Sir Isaac protesting, “That’s crazy enough.”

The first 67 notations, of course, are the magic. It is here I would wildly guess that the Langlands programs provide the interstitial mathematics and geometries between that which is defined as physical and that which gives rise to the physical.

I have begun playing with the earliest numbers. Might you have a little look? Could we be onto something? Thank you.

Most sincerely,

Bruce

**March 21, 2018**. A mathematician who developed what some consider the “grand unified theory of mathematics” has won one of the most prestigious prizes in mathematics.

Robert Langlands, an emeritus professor at the Institute for Advanced Study at Princeton University, has won the Abel Prize, a prestigious mathematics prize that honors a lifetime of groundbreaking work, organizers of the prize announced yesterday (March 20).

Langlands, 81, won the prize for work in which he found deep connections between two seemingly disparate areas of mathematics: number theory and harmonic analysis, according to a statement from the organizers of the prize.

In the letter, Langlands described a way to extend some of Carl Friedrich Gauss’ pioneering work on prime numbers. Number theorists before Gauss had noticed a hidden relationship among primes: that all the primes that can be formulated as the sum of two squares (for instance, 2^2 + 1^2 = 5 or 3^2+2^2 = 13) have a remainder of 1 when divided by 4, but didn’t know if it held true in all cases Quanta magazine reported. Gauss proved this idea in what’s now known as the quadratic reciprocity law.

Langlands took Gauss’ work and showed that the prime numbers that can be expressed as the sum of numbers raised to the third or fourth power (such as 1^3+2^3+4^3=73) can be tied to the distant mathematical realm of harmonic analysis. (This kind of analysis includes Fourier transforms, a mainstay tool used by scientists and engineers to analyze signals that have a periodic nature, such as sound waves or electromagnetic radiation spectra.)

Langlands showed that these two separate branches of mathematics can be directly related by using a special mathematical approach, a decoder ring of sorts, that became known as functoriality.

Langlands’ work became so critical to math that his findings lured hundreds of other mathematicians into a new field of study that eventually became known as the Langlands program. And in 1995, when Andrew Wiles, a British mathematician, finally proved Fermat’s last theorem, one of the most famous mathematical conjectures in history, he relied on Langlands’ theory for a critical piece of the proof. (That theorem posits that there is no solution to the equation a^n+b^n=c^n for any n greater than 2 if a, b and c are all different numbers.)