Transcript curv-40.ppsx - UW Math Department

The Curvature of Space
Jack Lee
Professor of Mathematics
UW Seattle
Euclid (around 300 BCE)
The Elements
by Euclid
Euclid’s Postulates for Geometry
• Postulate 1:
A straight line segment can be drawn from any point to
any other point.
Euclid’s Postulates for Geometry
• Postulate 1:
A straight line segment can be drawn from any point to
any other point.
Euclid’s Postulates for Geometry
• Postulate 1:
A straight line segment can be drawn from any point to
any other point.
• Postulate 2:
A straight line segment can be extended in either
direction to form a longer straight line segment.
Euclid’s Postulates for Geometry
• Postulate 1:
A straight line segment can be drawn from any point to
any other point.
• Postulate 2:
A straight line segment can be extended in either
direction to form a longer straight line segment.
Euclid’s Postulates for Geometry
• Postulate 1:
A straight line segment can be drawn from any point to
any other point.
• Postulate 2:
A straight line segment can be extended in either
direction to form a longer straight line segment.
• Postulate 3:
A circle can be drawn with any point as its center and
any other point on its circumference.
Euclid’s Postulates for Geometry
• Postulate 1:
A straight line segment can be drawn from any point to
any other point.
• Postulate 2:
A straight line segment can be extended in either
direction to form a longer straight line segment.
• Postulate 3:
A circle can be drawn with any point as its center and
any other point on its circumference.
Euclid’s Postulates for Geometry
• Postulate 1:
A straight line segment can be drawn from any point to
any other point.
• Postulate 2:
A straight line segment can be extended in either
direction to form a longer straight line segment.
• Postulate 3:
A circle can be drawn with any point as its center and
any other point on its circumference.
• Postulate 4:
All right angles are equal.
Euclid’s Postulates for Geometry
• Postulate 1:
A straight line segment can be drawn from any point to
any other point.
• Postulate 2:
A straight line segment can be extended in either
direction to form a longer straight line segment.
• Postulate 3:
A circle can be drawn with any point as its center and
any other point on its circumference.
• Postulate 4:
All right angles are equal.
Euclid’s Postulates for Geometry
• Postulate 1:
A straight line segment can be drawn from any point to
any other point.
• Postulate 2:
A straight line segment can be extended in either
direction to form a longer straight line segment.
• Postulate 3:
A circle can be drawn with any point as its center and
any other point on its circumference.
• Postulate 4:
All right angles are equal.
Euclid’s Postulates for Geometry
• Postulate 1:
A straight line segment can be drawn from any point to
any other point.
• Postulate 2:
A straight line segment can be extended in either
direction to form a longer straight line segment.
• Postulate 3:
A circle can be drawn with any point as its center and
any other point on its circumference.
• Postulate 4:
All right angles are equal.
• Postulate 5:
If a straight line crossing two other straight lines makes two interior angles on
one side that add up to less than two right angles, then the two straight lines,
if extended far enough, meet each other on the same side as the two interior
angles adding up to less than two right angles.
Euclid’s Postulates for Geometry
• Postulate 5:
If a straight line crossing two other straight lines
Euclid’s Postulates for Geometry
• Postulate 5:
If a straight line crossing two other straight lines
makes two interior angles on one side
Euclid’s Postulates for Geometry
• Postulate 5:
If a straight line crossing two other straight lines
makes two interior angles on one side
that add up to less than two right angles,
1 + 2 < 180.
Euclid’s Postulates for Geometry
• Postulate 5:
If a straight line crossing two other straight lines
makes two interior angles on one side
that add up to less than two right angles,
then the two straight lines, if extended far enough,
1 + 2 < 180.
Euclid’s Postulates for Geometry
• Postulate 5:
If a straight line crossing two other straight lines
makes two interior angles on one side
that add up to less than two right angles,
then the two straight lines, if extended far enough,
meet each other on the same side as the two interior
angles adding up to less than two right angles.
1 + 2 < 180.
Euclid’s Postulates for Geometry
Using only these five postulates, Euclid was able to prove all of the facts
about geometry that were known at the time.
For example,
Theorem: The interior angles of every triangle add up to exactly 180.
1 + 2 + 3 = 180.
Trying to Prove the Fifth Postulate
… but that darned fifth postulate doesn’t really seem as “obvious” as the
other four, does it?
Many mathematicians thought it seemed more like something that
should be proved.
Trying to Prove the Fifth Postulate
Around 100 CE (400 years after Euclid):
The great mathematician and astronomer
Ptolemy, living in Egypt, wrote down a
proof of Euclid’s Fifth Postulate, based on
the other four postulates.
So the fifth postulate is not needed, right?
Ptolemy
Trying to Prove the Fifth Postulate
Around 100 CE (400 years after Euclid):
The great mathematician and astronomer
Ptolemy, living in Egypt, wrote down a
proof of Euclid’s Fifth Postulate, based on
the other four postulates.
So the fifth postulate is not needed, right?
Wrong! Ptolemy’s proof had a mistake.
Ptolemy
Trying to Prove the Fifth Postulate
Around 400 CE (700 years after Euclid):
The Greek mathematician Proclus criticized
Ptolemy’s proof.
Proclus
Trying to Prove the Fifth Postulate
Around 400 CE (700 years after Euclid):
The Greek mathematician Proclus criticized
Ptolemy’s proof.
But he still believed that the Fifth Postulate
was a theorem that should be proved, not a
postulate that should be assumed:
“The fifth postulate ought to be struck out
of the postulates altogether; for it is a
theorem involving many difficulties.”
--Proclus
Proclus
Trying to Prove the Fifth Postulate
So Proclus offered his own proof of the fifth
postulate …
Proclus
Trying to Prove the Fifth Postulate
So Proclus offered his own proof of the fifth
postulate …
…which was also wrong!
Proclus
Trying to Prove the Fifth Postulate
Around 1100 CE (1400 years after Euclid):
The great Persian mathematician and poet
Omar Khayyam published a commentary on
Euclid’s Elements, in which he offered his own
proof of the Fifth Postulate …
Omar Khayyam
Trying to Prove the Fifth Postulate
Around 1100 CE (1400 years after Euclid):
The great Persian mathematician and poet
Omar Khayyam published a commentary on
Euclid’s Elements, in which he offered his own
proof of the Fifth Postulate …
… which was also wrong!
Omar Khayyam
Trying to Prove the Fifth Postulate
Around 1700 (2000 years after Euclid):
The Italian mathematician Giovanni Saccheri set
out to “fix” Euclid once and for all. He wrote and
published an entire book devoted to proving
Euclid’s fifth postulate, and triumphantly titled it
“Euclid Freed of Every Flaw.”
Trying to Prove the Fifth Postulate
Around 1700 (2000 years after Euclid):
The Italian mathematician Giovanni Saccheri set
out to “fix” Euclid once and for all. He wrote and
published an entire book devoted to proving
Euclid’s fifth postulate, and triumphantly titled it
“Euclid Freed of Every Flaw.”
Unfortunately, Saccheri forgot to free his own book
of every flaw. He, like every mathematician before
him, made a mistake. His proof didn’t work!
Trying to Prove the Fifth Postulate
Other famous mathematicians who published “proofs” of the
Fifth Postulate:
•Aghanis (Byzantine empire, 400s)
•Simplicius (Byzantine empire, 500s)
•Al-Jawhari (Baghdad, 800s)
•Thabit ibn Qurra (Baghdad, 800s)
•Al-Nayrizi (Persia, 900s)
•Abu Ali Ibn Alhazen (Egypt, 1000s)
•Al-Salar (Persia, 1200s)
•Al-Tusi (Persia, 1200s)
•Al-Abhari (Persia, 1200s)
•Al-Maghribi (Persia, 1200s)
•Vitello (Poland, 1200s)
•Levi Ben-Gerson (France, 1300s)
•Alfonso (Spain, 1300s)
•Christopher Clavius (Germany, 1574)
•Pietro Cataldi (Italy, 1603)
•Giovanni Borelli (Italy, 1658)
•Vitale Giordano (Italy, 1680)
•Johann Lambert (Alsace, 1786)
•Louis Bertand (Switzerland, 1778)
•Adrien-Marie Legendre (France, 1794)
Trying to Prove the Fifth Postulate
Other famous mathematicians who published “proofs” of the
Fifth Postulate:
•Aghanis (Byzantine empire, 400s)
•Simplicius (Byzantine empire, 500s)
•Al-Jawhari (Baghdad, 800s)
•Thabit ibn Qurra (Baghdad, 800s)
•Al-Nayrizi (Persia, 900s)
•Abu Ali Ibn Alhazen (Egypt, 1000s)
•Al-Salar (Persia, 1200s)
•Al-Tusi (Persia, 1200s)
•Al-Abhari (Persia, 1200s)
•Al-Maghribi (Persia, 1200s)
•Vitello (Poland, 1200s)
•Levi Ben-Gerson (France, 1300s)
•Alfonso (Spain, 1300s)
•Christopher Clavius (Germany, 1574)
•Pietro Cataldi (Italy, 1603)
•Giovanni Borelli (Italy, 1658)
•Vitale Giordano (Italy, 1680)
•Johann Lambert (Alsace, 1786)
•Louis Bertand (Switzerland, 1778)
•Adrien-Marie Legendre (France, 1794)
Trying to Prove the Fifth Postulate
Why did so many great mathematicians make mistakes when
trying to prove Euclid’s fifth postulate?
It’s simple, really. Every one of these failed proofs had to use some
properties about parallel lines.
Trying to Prove the Fifth Postulate
By definition, parallel lines are lines that are in the same plane but never
meet, no matter how far you extend them.
Trying to Prove the Fifth Postulate
But most of the failed proofs also accidentally used other properties of
parallel lines, such as:
• Equidistance: If two lines are parallel, they are everywhere the
same distance apart.
Trying to Prove the Fifth Postulate
But most of the failed proofs also accidentally used other properties of
parallel lines, such as:
• Equidistance: If two lines are parallel, they are everywhere the
same distance apart.
Trying to Prove the Fifth Postulate
But most of the failed proofs also accidentally used other properties of
parallel lines, such as:
• Equidistance: If two lines are parallel, they are everywhere the
same distance apart.
• Uniqueness: Given a line and a point not on that line, there is only
one line parallel to the given line through the given point.
Trying to Prove the Fifth Postulate
But most of the failed proofs also accidentally used other properties of
parallel lines, such as:
• Equidistance: If two lines are parallel, they are everywhere the
same distance apart.
• Uniqueness: Given a line and a point not on that line, there is only
one line parallel to the given line through the given point.
Trying to Prove the Fifth Postulate
Equidistance Postulate
Euclid’s Fifth Postulate
Trying to Prove the Fifth Postulate
Equidistance Postulate
Euclid’s Fifth Postulate
The Parallel Postulate
(uniqueness)
Trying to Prove the Fifth Postulate
Equidistance Postulate
Euclid’s Fifth Postulate
The Parallel Postulate
(uniqueness)
A Bold New Idea
In the 1820s (more than 2100 years after Euclid), three
mathematicians in three different countries independently hit upon the
same amazing insight…
Carl Friedrich Gauss,
in Germany
(58 years old)
Nikolai Lobachevsky,
in Russia
(38 years old)
Janos Bolyai,
in Hungary
(18 years old)
A Bold New Idea
Maybe there is a simple explanation for why
nobody had succeeded in proving the Fifth
Postulate based only on the other four:
Maybe it’s logically impossible!
A Bold New Idea
Maybe there is a simple explanation for why
nobody had succeeded in proving the Fifth
Postulate based only on the other four:
Maybe it’s logically impossible!
Euclid thought his postulates were describing the
only conceivable geometry of the physical world
we live in.
But what if other geometries are not only
conceivable, but just as consistent and
mathematically sound as Euclid’s??
A Bold New Idea
HOW CAN THIS BE?
The key is curvature.
What are Dimensions?
To see how to imagine the curvature of space, let’s start by pretending
we live in a 2-dimensional world.
Dimensions = “how many numbers it takes to describe the location
of a point.”
What are Dimensions?
To see how to imagine the curvature of space, let’s start by pretending
we live in a 2-dimensional world.
Dimensions = “how many numbers it takes to describe the location
of a point.”
A line or a curve is 1-dimensional, because it only takes one number
to describe where a point is.

What are Dimensions?
To see how to imagine the curvature of space, let’s start by pretending
we live in a 2-dimensional world.
Dimensions = “how many numbers it takes to describe the location
of a point.”
A plane is 2-dimensional, because it take 2 numbers, x and y, to
describe where a point is.
What are Dimensions?
The surface of a sphere is
also 2-dimensional, because it
takes two numbers—latitude
and longitude—to say where a
point is.
What are Dimensions?
Space is 3-dimensional, because it take 3 numbers, x, y, and z, to
describe where a point is.
Curvature in 2 Dimensions
Positive
curvature
Zero
curvature
Negative
curvature
Curvature in 2 Dimensions
A 2-dimensional being, living in a 2-dimensional world, could not
possibly be aware of anything outside of his two dimensions, because
it would be “out of his world”.
Curvature in 2 Dimensions
A 2-dimensional being, living in a 2-dimensional world, could not
possibly be aware of anything outside of his two dimensions, because
it would be “out of his world”.
Even if his 2-dimensional world is curved, he cannot look down on it
from “outside” to see the curvature…
Curvature in 2 Dimensions
A 2-dimensional being, living in a 2-dimensional world, could not
possibly be aware of anything outside of his two dimensions, because
it would be “out of his world”.
Even if his 2-dimensional world is curved, he cannot look down on it
from “outside” to see the curvature…
And yet he can tell if
he lives in a curved
world!
Curvature in 2 Dimensions
How can Bart tell if his world is curved or not?
ZERO CURVATURE CASE:
• Parallel lines are equidistant.
Curvature in 2 Dimensions
How can Bart tell if his world is curved or not?
ZERO CURVATURE CASE:
• Parallel lines are equidistant.
• Through a point not on a line, there’s
one and only one parallel.
• Triangles have angle sums of 180.
Curvature in 2 Dimensions
How can Bart tell if his world is curved or not?
POSITIVE CURVATURE CASE:
From the “outside,” it looks spherical.
Curvature in 2 Dimensions
How can Bart tell if his world is curved or not?
POSITIVE CURVATURE CASE:
From the “outside,” it looks spherical.
“Lines” are great circles—the path he
would follow if he went as straight as
possible.
Curvature in 2 Dimensions
How can Bart tell if his world is curved or not?
POSITIVE CURVATURE CASE:
• Lines get closer together as you
follow them in either direction.
Curvature in 2 Dimensions
How can Bart tell if his world is curved or not?
POSITIVE CURVATURE CASE:
• Lines get closer together as you
follow them in either direction.
• Through a point not on a line, there
is no parallel, because all lines meet
eventually.
• Triangles have angle sums greater
than 180.
Curvature in 2 Dimensions
How can Bart tell if his world is curved or not?
NEGATIVE CURVATURE CASE:
From the outside, it looks
“saddle-shaped.”
Curvature in 2 Dimensions
How can Bart tell if his world is curved or not?
NEGATIVE CURVATURE CASE:
• Triangles have angle sums less
than 180.
Curvature in 2 Dimensions
How can Bart tell if his world is curved or not?
NEGATIVE CURVATURE CASE:
• Triangles have angle sums less
than 180.
• Parallel lines diverge from each
other.
Curvature in 3 Dimensions
Now here comes the surprising part …
It’s possible for our 3-dimensional universe to be curved!
Curvature in 3 Dimensions
Now here comes the surprising part …
It’s possible for our 3-dimensional universe to be curved!
Just like 2-dimensional Bart, we can’t step “outside the universe” to
see the curvature.
But we can detect it from inside our world by measuring anglesums of triangles.
Curvature in 3 Dimensions
Now here comes the surprising part …
It’s possible for our 3-dimensional universe to be curved!
Just like 2-dimensional Bart, we can’t step “outside the universe” to
see the curvature.
But we can detect it from inside our world by measuring anglesums of triangles.
Gauss tried to determine experimentally if our
world is flat or not—that is, if it follows the laws
of Euclidean geometry or not.
Curvature in 3 Dimensions
Gauss’s experiment:
Gauss set up surveying equipment on the tops of three mountains in
Germany, forming a triangle with light rays.
He then measured the angles of that triangle, and found …
The angles added up to
180, as close as his
measurements could
determine.
But maybe they just
weren’t accurate
enough?
Curvature in 3 Dimensions
Einstein’s Theory:
If there was any remaining doubt about whether it was possible for our
universe to be non-Euclidean, it was dispelled around 1900 by Albert
Einstein.
Curvature in 3 Dimensions
Einstein’s Theory:
In physical space, “straight lines” are paths followed by light rays.
Einstein’s theory predicts that a large cluster of galaxies between us
and a distant galaxy will warp the space between us and the galaxy,
and cause its light to reach us along two different paths, so we see
two images of the same galaxy.
Curvature in 3 Dimensions
Einstein’s Theory:
This has actually been observed.
Curvature in 3 Dimensions
Einstein’s Theory:
Of course, this is a “small” region
of space in the larger scheme of
things. In a small region, space
is “lumpy,” with areas of positive
and negative curvature caused
by the gravitational fields of all
the stars and galaxies floating
around…
Just as the earth is “lumpy” if you
look at it up close.
Curvature in 3 Dimensions
Einstein’s Theory:
But physicists expect that, if you
imagine looking at the universe
from far, far away, so that galaxies
look like dust scattered evenly
throughout the universe, then it will
have a nice smooth “shape.”
Curvature in 3 Dimensions
Einstein’s Theory:
The Big Question:
When the bumps are smoothed out, does the universe have positive
curvature (“spherical”), zero curvature (“flat”), or negative curvature
(“saddle-shaped”)?
Curvature in 3 Dimensions
Einstein’s Theory:
Einstein’s equations tell us how to determine the answer:
Just measure the average density of matter in the universe.
Einstein’s theory predicts that there’s a critical density – about 5 atoms
per cubic yard. (Remember, most of the universe is empty space!)
Curvature in 3 Dimensions
If the average density of the universe is …
• exactly equal to the critical density, then the universe has zero
curvature, is infinitely large, and will go on expanding forever.
Curvature in 3 Dimensions
If the average density of the universe is …
• exactly equal to the critical density, then the universe has zero
curvature, is infinitely large, and will go on expanding forever.
• less than the critical density, then the universe has negative curvature,
is infinitely large, and will go on expanding forever.
Time
Curvature in 3 Dimensions
If the average density of the universe is …
• exactly equal to the critical density, then the universe has zero
curvature, is infinitely large, and will go on expanding forever.
• less than the critical density, then the universe has negative curvature,
is infinitely large, and will go on expanding forever.
• greater than the critical density, then the universe has positive
curvature, is closed up on itself and only finitely large, and will eventually
stop expanding and collapse into a …
BIG CRUNCH!!
Curvature in 3 Dimensions
Einstein’s Theory:
The Big Question:
When the bumps are smoothed out, does the universe have positive
curvature (“spherical”), zero curvature (“flat”), or negative curvature
(“saddle-shaped”)?
The current evidence suggests that it is positively curved, which means
that it is closed, like a sphere.
Curvature in 3 Dimensions
What does it mean for the universe to be closed?
On the surface of an ordinary (2-dimensional) sphere, if you start
traveling in any direction and go far enough, you’ll eventually come back
to the place where you started.
If our 3-dimensional universe is closed,
the same is true: if you start traveling in
any direction and go far enough (about
290,000,000,000,000,000,000,000 miles
probably), you’ll come back to the place
where you began!
The Shape of the Universe
If the universe is closed, what shape is it?
The Shape of the Universe
If the universe is closed, what shape is it?
If the universe were 2-dimensional, we would know what all the
possibilities were (after smoothing out the bumps) …
The Shape of the Universe
What do we mean by “smoothing out the bumps”?
Two surfaces are said to be topologically equivalent if one can be
continuously deformed into the other.
The Shape of the Universe
For example, the surface of a hot dog is topologically equivalent to a
sphere.
The Shape of the Universe
For example, the surface of a hot dog is topologically equivalent to a
sphere.
The Shape of the Universe
The surface of a one-handled coffee cup is topologically equivalent to a
doughnut surface (a torus).
The Shape of the Universe
The surface of a one-handled coffee cup is topologically equivalent to a
doughnut surface (a torus).
The Shape of the Universe
But a sphere cannot be continuously deformed into a torus, because that
would require tearing a hole in the middle, a discontinuous operation.
The Shape of the Universe
The sphere is the simplest closed surface, in a very precise sense…
The Shape of the Universe
Thought Experiment:
Imagine you’re a 2-dimensional being living
on a closed surface.
The Shape of the Universe
Thought Experiment:
Imagine you’re a 2-dimensional being living
on a closed surface.
Start somewhere, and walk as straight as
you can, trailing a string behind you, until
you get back to the place where you
started.
Then try to pull the loop of string back to
where you are.
The Shape of the Universe
Thought Experiment:
Imagine you’re a 2-dimensional being living
on a closed surface.
Start somewhere, and walk as straight as
you can, trailing a string behind you, until
you get back to the place where you
started.
Then try to pull the loop of string back to
where you are.
If you’re on a sphere, there are no holes to
stop the string from shrinking back to you.
We say the sphere is simply connected.
The Shape of the Universe
Thought Experiment:
If you’re on another surface like a 1-holed
doughnut, this doesn’t work, because the
string can’t get across the hole.
The Shape of the Universe
Thought Experiment:
In 3 dimensions, there are many more possibilities.
But the simplest one is a 3-dimensional sphere-like object called a
hypersphere or a 3-sphere.
Fake picture of a
hypersphere
The Shape of the Universe
Thought Experiment:
In 3 dimensions, there are many more possibilities.
But the simplest one is a 3-dimensional sphere-like object called a
hypersphere or a 3-sphere.
There are many other possible three-dimensional closed universes.
But are there any others that are simply connected?
Fake picture of a
hypersphere
The Poincaré Conjecture
Around 1900, the French mathematician Henri
Poincaré tried to figure out if there are any
possible closed 3-dimensional spaces that are
simply connected, other than the 3-sphere.
Henri Poincaré
The Poincaré Conjecture
Around 1900, the French mathematician Henri
Poincaré tried to figure out if there are any
possible closed 3-dimensional spaces that are
simply connected, other than the 3-sphere.
He couldn’t think of any others, and he conjectured
that the hypersphere is the only one.
The Poincaré Conjecture: Every simply
connected closed 3-dimensional space is
topologically equivalent to a 3-sphere.
Henri Poincaré
The Poincaré Conjecture
Around 1900, the French mathematician Henri
Poincaré tried to figure out if there are any
possible closed 3-dimensional spaces that are
simply connected, other than the 3-sphere.
He couldn’t think of any others, and he conjectured
that the hypersphere is the only one.
The Poincaré Conjecture: Every simply
connected closed 3-dimensional space is
topologically equivalent to a 3-sphere.
He thought this would be a simple first step in
determining all closed 3-dimensional spaces.
Henri Poincaré
The Poincaré Conjecture
But it turned out not to be so easy.
The Poincaré Conjecture
Various bits of progress were made, until
In 1984, the American mathematician
Richard Hamilton thought of a
systematic way to “smooth the bumps”
on any simply connected surface, called
the Ricci flow.
Richard Hamilton
The Poincaré Conjecture
Various bits of progress were made, until
In 1984, the American mathematician
Richard Hamilton thought of a
systematic way to “smooth the bumps”
on any simply connected surface, called
the Ricci flow.
Richard Hamilton
The Poincaré Conjecture
Various bits of progress were made, until
In 1984, the American mathematician
Richard Hamilton thought of a
systematic way to “smooth the bumps”
on any simply connected surface, called
the Ricci flow.
It’s like heating up a chocolate bunny
and watching the bumps smooth out.
Richard Hamilton
The Ricci Flow For Surfaces
The Ricci Flow in 3 Dimensions
The Ricci Flow in 3 Dimensions
The Ricci Flow in 3 Dimensions
The Ricci Flow in 3 Dimensions
The Ricci Flow in 3 Dimensions
The Ricci Flow in 3 Dimensions
The Ricci Flow in 3 Dimensions
??
Richard Hamilton
Hamilton got stuck for 20 years …
The Poincaré Conjecture
In 2000, the Clay Mathematics Institute announced its seven Millennium
Problems: unsolved math problems with a $1,000,000 prize to anyone who
solves one of them.
One of the problems was:
Either prove the Poincaré conjecture, or show that it’s false by finding a
counterexample (a simply-connected closed 3-dimensional space that isn’t a
hypersphere when the bumps are smoothed out).
The Poincaré Conjecture
Finally, in 2003, an eccentric Russian
mathematician named Grigori Perelman
figured out how to prove that when
pinching occurs, it always pinches along
a nice cylinder, and he completed the
proof of the Poincaré conjecture.
Grigori Perelman
The Poincaré Conjecture
The Clay Institute announced last March
that he had won the million-dollar prize.
Grigori Perelman
The Poincaré Conjecture
The Clay Institute announced last March
that he had won the million-dollar prize.
Reporter: You just won a million dollars.
Do you have any comment?
Perelman: I don’t need it! It’ll just make
me a target of the Russian mafia.
Grigori Perelman
Much more to do …
There are still many unanswered questions for aspiring
mathematicians to work on.
Check out the remaining six Millennium Prize problems at
www.claymath.org
Besides those problems, there are thousands of other unanswered
questions in mathematics…