What kind of curve is uniquely
defined by these two points? A line. How about three points? A
circle. This basic geometry is ancient
history. Euclid proved it more than two thousand years ago in his seminal math treatise "Elements." Now try to figure out the shape of a curve that passes through a million points in a 1,000 dimensions and is super twisty. What would that look like? Although daunting, finding a solution to this problem is fundamental to probing deeper
principles of math. And the solution also holds the
potential to improve digital data storage, the security of
your crypto wallet, and the privacy of electronic
communications. It's called the interpolation problem. And two
young mathematicians, Eric Larson and Isabel Vogt of Brown
University have solved it. The interpolation problem asks,
If you fix some general collection of points, when is
there a curve of some specified type through those points? In search of answers, the pair
chipped away at the problem for years, We started out tackling cases in
small dimensional spaces to try to get our hands dirty and see
the sort of techniques that we would need. And then I think
around 2019 or so we came up with a good idea that we thought
might help us solve the whole problem. Along the way, their
collaboration took on a whole new dimension, Well, we got married in 2017. Much of the groundwork for a
potential solution was laid out 150 years ago by German
mathematicians, Alexander von Brill and Max Noether Before you can ask what
properties does a given type of curve have, you need to first
answer the question of what types of curves are there? The theorem predicts what types
of curves can exist based on three properties. The first is
dimension, You can think about a curve in
actual physical space, that would be a three dimensional
space. But we don't constrain ourselves to two dimensions or
three dimensions. The second quantity is the
degree of a curve. So you should think, here, like
how twisty or wobbly the curve is. The third is the genus or number
of holes a curve has. Curves themselves, have a sort
of topological property. But if you were to look at the
solutions over the complex numbers, then it's going to look
like something that's two dimensional. So it could look
like a sphere. Or it could look like the surface of a doughnut.
Or it could look like the surface of a two-hole doughnut.
In general, it can have any number of holes. It wasn't until the 1980s that
modern math offered a proof of Brill-Noether. And this opened
doors to the interpolation problem. Larson and Vogt, we're
determined to solve the problem for all possible types of
curves. We had to come up with three
different ways of breaking the curve into simpler curves. So we
take some curve, it's complicated, it's very wobbly.
So we take this wobbly curve, and we break it in three
different ways. By breaking down the complex
curve in multiple steps, Larson and Vogt were able to get
something simple enough that they could attack it with their
bare hands. We knew how slippery it was, how
you really had to work hard to get everything to match up, Larson and Vogt's proof
confirmed the interpolation problem in all cases, showing
that curves will always interpolate through the expected
number of points with only four exceptions, which they also
explained. The couple's proof is a powerful tool for exploring a
whole new class of ideas. Bit of a surprise to finally complete it, you know, it's you've just been
thinking, oh, this is what I do is I work on this problem, but
then you're like, wow, it's done. More than 2,000 years ago, the
Greek mathematicians Zenodorus argued that a sphere is the
optimal shape of a single bubble. This may seem obvious,
but it took until the late 19th century to prove it
mathematically. What about a cluster of multiple bubbles? Researchers study bubbles to
both better understand their geometric properties and
potentially improve everything from computer algorithms and
biological cell models to firefighting foams. For decades,
mathematicians have sought the optimal configuration of larger
bubble clusters. But it turns out the solution is one of the
hardest problems in geometry. The bubble problem can be stated
simply: What configuration of bubbles encloses a given number
of volumes while minimizing surface area. And the key point is that the
surfaces are allowed to, like, if I smashed my bubbles together I'm
allowed to save the surface that's between them. I mean,
it's an optimization problem. And that's such a universal kind
of problem with all kinds of applications. From our own experience of
blowing bubbles, we know that larger clusters of bubbles tend
to form tight clumps rather than a long chain. But how can we
know whether these formations are actually optimal? In the 1990s, John Sullivan, now
of the Technical University of Berlin, formed a guiding
conjecture, He described what we now call a
standard bubble or a standard bubble cluster. They're spheres.
They meet at 120 degree angles. They all touch. There could be
lots of different ways to achieve that. It turns out that
that's not the case. Sullivan's conjecture states, if
the number of volumes you want to enclose is at most one
greater than the dimension, there's one cluster
configuration, that's the best. There's basically one way to
achieve this and his conjecture was that this way is the optimal
way. To visualize Sullivan's optimal
cluster place four points equidistant on a sphere. Inflate
the points into bubbles or disks on the surface until they bump
into each other. Keep inflating them until the entire sphere is
covered. We end up with a symmetric cluster that all touch
and meet at 120 degree angles. Now place the sphere on a flat
plane. Add a light at the north pole to cast a two dimensional
shadow of three bubbles. Rolling the sphere around creates
clusters of Sullivan's shadow bubbles that can enclose any
three areas. By 2002, mathematicians had proved
Sullivan's conjecture for the double bubble. And at this
point, some speculated another 100 years would pass before a
proof for three bubbles could be found. But then Emanuel Milman
of Technion University, and Joe Neeman of the University of
Texas, Austin entered the picture. So what we were working on was
sort of the probabilistic version of this problem. And we
were interested, actually, originally, we were just we
would have been happy to do double bubble at the beginning.
We were like, you know, we have to come up with something new.
Right. And so we came up with some new things. After some success solving the
analogous probabilistic bubble problem, the pair hit a snag
transferring their new techniques to the geometric one. So we were stuck on this for a
long time. And definitely a breakthrough came when we
actually, we kind of gave up to be honest. Milman and Neeman realized that
if you give bubbles, one extra dimension of space, you get a
bonus: The best bubble cluster will have mirror symmetry across
a central plane. We were like, okay, let's give
up on what we were trying to do and see what we can get from the
symmetry. And then as soon as we had that symmetry that, you
know, that symmetry combined with what we had been working on
for the previous three years, all of a sudden, like then we
were able to move on from there. But Milman and Neeman's proof
of Sullivan's conjecture for triple bubbles in dimensions
three and up is not the end of the story. We're stuck at a maximum of five
bubbles right now. But I feel like it's just like a tiny
little thing that we are not seeing and then that problem
will go away. Take a set of points and start
randomly connecting them with lines. At what point will an
interesting pattern emerge? Like a triangle or a Hamiltonian
cycle, a chain of lines that passes through each point
exactly once. In a branch of math called combinatorics,
graphs like these are valuable tools for probing the inner
workings of complex systems. A graph is just another name for
a network. Networks are everywhere like
Facebook network. You can think about traffic networks, you
know, or business networks and things like that. But real life networks can be
huge and hard to study. So mathematicians use random graphs
to model their properties. In the random graph theory,
finding thresholds is a central subject. People worked on
finding threshold for each of interesting properties. Interesting properties can
emerge as you increase the density of lines in a random
graph. The sudden moment of change is defined by a
threshold. In nature thresholds are responsible for abrupt
transformations and the emergence of complex patterns. But it turns out that
determining threshold is extremely difficult. Instead, mathematicians use a
method to arrive at a ballpark figure. There is a very natural lower
bound to the threshold that we call the expectation threshold.
The expectation threshold turns out to be much easier to
estimate or determine. In 2006, mathematicians Jeff
Kahn and Gil Kalai posed their expectation threshold
conjecture. It states that the gap between the expectation
threshold and the real threshold is at most a logarithmic factor,
but sometimes smaller, or even nonexistent. The Kahn-Kalai conjecture just
gives one simple solution to find out the location of the
threshold for many, many interesting properties. So it's
just so powerful, and it was hard to believe that this
conjecture is true. That was until two
mathematicians at Stanford, Jinyoung Park and Huy Pham,
stumbled onto an elegant six page solution after a long
sleepless night. It really came about by really
not working or aiming directly at the Kahn-Kalai conjecture. It was a surprise that we found
out another way. The pair were instead working on
several related conjectures. Like, you know, you're like
taking a walk that you aim at a seemingly very different and
distant destination, but on the way, you are suddenly able to
find some very surprising hidden beauty. A key breakthrough was the
realization that a method developed in the pursuit of one
conjecture could be used in a different manner to solve
Kahn-Kalai. The solution utilized a mathematical object
called a cover. You can think about a cover as
essentially a kind of necessary condition, a witness to the fact
that the network satisfies the properties. Park and Pham employed an
algorithm to sample subsets of a random graph to home in on the
cover. In the end, we verify this cover
is small, in particular, so that is exactly how we prove the
conjecture. Park and Pham's concise proof of
Kahn-Kalai is expected to lead to new breakthroughs. This would allow us to get a
much better handle on complex properties of networks and this
is certainly usefull in many applications not just in math
or in probabilistic combinatorics, because networks
are involved everywhere.
