Rikai Funo · Track 6 · middle

Ode to the Geometric Langlands Program

Robert Langlands's dream carried forward through decades: a deep correspondence between automorphic forms in number theory and the geometry of principal bundles on algebraic curves. In 2024, Gaitsgory, Raskin, and collaborators published a proof spanning nine papers and roughly a thousand pages — the unramified geometric case, resolved. An ode to the bridge that shouldn't exist between number theory and geometry, and the mathematicians who spent lifetimes building it.

Lyrics

Nine of you.
One thousand pages, bound in blue.
The air in the room still smells of chalk from the seminar.

It started with a letter.
1967.
Robert Langlands writes to André Weil.
Just a thought experiment.
"I have a conjectural picture..."
A whisper between two worlds.
Galois representations on one bank.
Automorphic forms on the other.
And a river of silence between.

And you are the bridge that shouldn't exist.
A correspondence held in the dark for fifty years.
Connecting the counting of primes
To the shape of a curve.
Number theory talking to geometry.
A secret language finally proven.

Then came Drinfeld.
And the decades of quiet work.
Building the vocabulary, the tools.
Sheaves and stacks and principal bundles.
Drawing arrows on a blackboard in Princeton,
in Chicago.
Gaitsgory chasing a ghost
across the cohomology of moduli spaces.
A lifetime spent on one side of the equation.

And you are the bridge that shouldn't exist.
A correspondence held in the dark for fifty years.
Connecting the counting of primes
To the shape of a curve.
Number theory talking to geometry.
A secret language finally proven.

And now, 2024.
Gaitsgory and Raskin.
The final thousand pages fall into place.
The unramified case, they call it.
The smooth parts of the shoreline.
Still islands of ramification out there in the water.
But the main span... the main span holds.

I run my hand over the top page.
The ink is still warm.
The bridge is real.
You are real.