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For his entire career,
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this researcher has labored to solve a
mind-boggling math problem called the
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geometric Langlands conjecture.
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This particular thing happens to be very,
very tasty.
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You just start salivating, you want to go and solve it.
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The geometric Langlands conjecture is an
important component of a vast effort to
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develop what's been called a 'Grand
Unified Theory' of mathematics.
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And now after three decades
of work, Dennis Gaitsgory,
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along with his former grad student,
Sam Raskin, and seven others
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have produced a monumental 800-page
proof of the conjecture.
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It took a long time to figure out what's
the right edifice to build and what
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materials are available.
They've built a whole world.
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For Gaitsgory
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the epic pursuit began in 1994 when he
first heard about the geometric Langlands
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program as a graduate student.
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I guess understood about 15% if that,
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but I was kind of completely
awestruck with the way
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mathematical objects that you know about,
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the way they combine and lead to this
conjectures. There's something extremely
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appealing about it in Langlands program,
especially in geometric Langlands.
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What is now known as the Langlands
program began in 1967 when the
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mathematician Robert Langlands wrote a
letter to the French number theorist,
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André Weil, describing a plan to connect
far reaching branches of math.
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So a lot of parts of number
theory, parts of physics,
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and it has kind of different compartments
and different kind of corners that
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operate in parallel.
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The Langlands program takes its inspiration
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from another part of mathematics, Fourier analysis,
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which splits complex signals into simpler components.
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The Fourier transform is one of these
kind of basic building blocks of much of
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math. We simplify, we try to think
of everything in terms of these basic,
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basic patterns.
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Consider the complex sounds
of the world around us.
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Comprising each of these very different
sounds is a fundamental collection:
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the pure tones.
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Each tone is a single frequency.
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Mathematically, the pure tones are sine waves.
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These simple oscillations are the
building blocks of all complex sound waves
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from radio static to symphonies.
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In 1822, the mathematician Joseph Fourier
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showed that any wave can be broken down into an
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infinite sum of sine waves
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using a technique now called the
Fourier transform.
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The Fourier transform
is like a recipe generator,
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you input a complicated wave and
you get back its ingredients,
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the amplitude and frequency
of each component sine wave.
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Fourier analysis is an
essential part of modern technology.
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Its applications range from jpeg compression
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and image recognition
to quantum physics and MRIs.
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It has also opened a revolutionary
new framework in pure mathematics.
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Our experience with Fourier theory
guides many of the ways that we think
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about the Langlands program,
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the way the subject develops and
the sort of phenomena that we're seeing.
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Fourier theory has two components,
basic building blocks and labels.
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Imagine a child's toy castle.
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This castle can be disassembled
into individual building blocks,
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and these building blocks can then be
sorted by color into bins and labeled.
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Similarly, the Fourier transform
disassembles a complex wave
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into individual sine waves.
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These are like the building blocks.
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Each sine wave can be labeled with its
frequency or how quickly it oscillates
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per second. The labels on the bins are
more than just a way to organize things,
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they can be used to rebuild the original
complex wave and as an efficient
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shorthand for communicating
information.
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For example, when you send a voice message,
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your phone doesn't transmit an
entire complex sound wave.
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Instead, it breaks it down and sends
just the labels
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or frequencies of the component
sine waves.
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The receiver's phone
then reverses this process,
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converting the labels back
into the contents of the bins,
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to reconstruct the message's
original sound wave.
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To open up new connections between other
distant mathematical worlds
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Langlands researchers look for analogies
of Fourier theory in other contexts.
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Within the vast world of mathematical
objects, spheres, functions, prime numbers...
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are there other basic building blocks
that can fit into the bins?
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And if so, what are the labels?
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In 1994, the mathematician Andrew Wiles
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proved the famous Fermat's Last Theorem
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by developing just one small
corner of the Langlands program.
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This demonstrated the power and
possibility of Langlands' vision.
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When Langlands began his search
for analogs of Fourier theory,
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he first looked for
labels in number theory,
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a branch of mathematics
that studies arithmetic.
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And what Langlands predicted
is that the labels in this game
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are certain objects of
deep arithmetic interest.
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In the original number theory
formulation of the Langlands program,
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the building blocks are functions.
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But functions can be replaced by a
complex abstraction called sheaves,
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so named because mathematicians visualize
them like sheaves of wheat
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growing on top of other mathematical objects.
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Initially, each sheaf
itself was a way of labeling,
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here's a kind of function
I like, continuous functions,
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differentiable functions,
special class of functions.
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But then something kind of
strange happened over the decades,
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people started thinking about the
collection of all these collections.
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Building on the connections
between functions and sheaves,
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mathematicians shifted many parts of
the Langlands program
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to a new geometric setting.
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In the geometric Langlands program,
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a special subset of sheaves called eigensheaves
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are the building blocks in the bins
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akin to sine waves.
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The labels on the bins are something called
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representations of the fundamental group,
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descriptions of the loops that can be drawn
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on spheres, donuts, and other shapes.
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Similarly to how the Fourier transform
breaks down complicated waves
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and labels them by their frequencies,
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the geometric Langlands program breaks
down sheaves into eigensheaves,
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each with a label.
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Mathematicians can study these labels and
translate that new information back to
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eigensheaves, and then
sheaves themselves.
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In the mid two-thousands,
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amid a flurry of interest in the geometric
Langlands program
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by both physicists and mathematicians,
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Gaitsgory finally began to see a way forward.
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Until that moment, it was in some sense,
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like walking in the dark in the woods.
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From that moment on, I just saw the framework.
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Gaitsgory drew, what he
called the fundamental diagram,
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a particular way of establishing the
correspondence between sheaves
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and their labels.
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But his diagram was missing one piece.
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He needed to know that every single eigensheaf,
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is contained within a special composite sheaf
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known as the Poincaré sheaf.
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Raskin also became hooked on the problem.
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Sam became a grad student year
exactly after this revelation
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until 2006.
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After finishing his PhD, Raskin
continued to study the Poincaré sheaf.
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A Poincaré sheaf is like white light,
just as white light contains every color,
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mathematicians expected the Poincaré
sheaf to contain every eigensheaf.
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Even though it's hard to write
down an individual eigensheaf,
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it turns out that you can write down
very directly what happens when you
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amalgamate all of them.
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If Raskin could prove that the composite
Poincaré sheaf
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contained all eigensheaves,
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then he could use this as a tool to
access the individual eigensheaves like a
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prism splitting white light
into a rainbow.
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In 2022, Raskin and his own graduate
student finally proved this.
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Completing Gaitsgory's
fundamental diagram.
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They cracked this mystery and after
which
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basically, the shape of the solution became clearer.
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Over the course of the next two years,
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gates Gaitsgory and Raskin led a team
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that wrote five papers that proved the geometric Langlands conjecture.
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There is a set of questions that have
so much symmetry, that symmetry
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completely determines the
solution.
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And this is kind of,
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I would say the deepest,
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the richest kind of statement of
that kind that's been established.
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Of course, mathematics is
infinite, and once you solve these,
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some new paradigms appear.
