Solve This Mathematics Problem and Get 1 Million Dollars
Summary
TLDRThis video script explores the seven Millennium Prize Problems, offering a glimpse into the challenges that could win mathematicians a place in the Hall of Fame and a $1 million reward. It delves into the Riemann Hypothesis's connection to prime numbers, the Navier-Stokes equations' role in fluid dynamics, the P vs NP problem's impact on computer science, the Poincaré conjecture's triumph in topology, the Hodge conjecture's geometric intricacies, the Birch and Swinnerton-Dyer conjecture's focus on elliptic curves, and the Yang-Mills theory's foundational role in particle physics. Each problem is a testament to the depth and complexity of mathematical inquiry.
Takeaways
- 🏆 The speaker introduces the seven Millennium Prize Problems, each offering a $1 million reward for solutions that would significantly advance mathematics.
- 🔍 The Riemann Hypothesis is highlighted as one of the most famous and important problems, with deep implications for the understanding of prime numbers and cryptography.
- 🔢 Prime numbers are numbers only divisible by one and themselves, and their distribution pattern is a mystery that mathematicians are eager to solve.
- 🎶 The Riemann Zeta Function is intricately linked to the distribution of prime numbers and is central to the Riemann Hypothesis.
- 🌊 The Navier-Stokes existence and smoothness problem is the speaker's favorite due to its relevance to fluid dynamics, a field the speaker is passionate about.
- 🌀 The Navier-Stokes equations describe fluid flow and are fundamental to understanding phenomena like air travel, ocean waves, and more.
- 💻 The P versus NP problem is related to the efficiency of problem-solving in computer science, questioning whether problems that are easy to check are also easy to solve.
- 📊 The Poincaré conjecture, the first Millennium Prize Problem to be solved, is about the transformation of shapes in topology without cutting or gluing.
- 🏰 The Hodge conjecture, while complex, involves the decomposition of complicated geometric shapes into simpler, invariant components.
- 🧩 The Birch and Swinnerton-Dyer conjecture is about elliptic curves and the number of rational points they contain, which is key to number theory.
- ⚛️ The Yang-Mills existence and mass gap problem is foundational to elementary particle theory, with the mass gap being a crucial property yet to be theoretically established.
Q & A
What are the Millennium Prize Problems?
-The Millennium Prize Problems are seven mathematical problems established by the Clay Institute in the year 2000, each with a $1 million prize for a correct solution. They are considered some of the most difficult and important unsolved problems in mathematics.
Why is the Riemann Hypothesis considered important in mathematics?
-The Riemann Hypothesis is considered important because it is believed to be true and has implications in various fields such as cryptography. It proposes that the Riemann zeta function is only equal to zero when the real part of the complex number s is equal to a half, and its proof could open up new avenues in the study of prime numbers.
What is the connection between the Riemann zeta function and prime numbers?
-The Riemann zeta function is closely related to the distribution of prime numbers. The function is used to describe the frequency of prime numbers, and the Riemann Hypothesis states a specific condition under which the function equals zero, which is connected to the real part of the complex number s being a half.
What are the Navier-Stokes equations?
-The Navier-Stokes equations are a set of differential equations that describe the motion of fluid substances, like air over an aircraft's wing or water waves. They are fundamental in fluid dynamics, but a complete understanding of these equations, especially in areas like turbulence, remains a challenge.
Why is the Navier-Stokes existence and smoothness problem significant?
-The significance of the Navier-Stokes existence and smoothness problem lies in the need to prove the existence of smooth solutions for fluid flow described by the Navier-Stokes equations under certain conditions, which is currently an open question in mathematics.
What does P versus NP problem refer to in the context of computer science?
-The P versus NP problem refers to the question of whether every problem whose solution can be quickly verified (NP) can also be quickly solved (P). It is a fundamental question in computer science regarding the relationship between the complexity of solving and verifying problems.
What was the first Millennium Prize Problem to be solved?
-The first and only Millennium Prize Problem to be solved so far is the Poincaré conjecture, which was proven by the mathematician Grigori Perelman.
What is the Poincaré conjecture about in the field of topology?
-The Poincaré conjecture is about the possibility of transforming a three-dimensional object without holes into a sphere through continuous deformation, without cutting or gluing. It was extended by Perelman to higher dimensions.
What is the Hodge conjecture and why is it significant?
-The Hodge conjecture is a complex problem in algebraic geometry that deals with the decomposition of geometric shapes into simpler shapes. It is significant because it pertains to understanding invariants in topology, specifically cohomology classes.
What are elliptic curves and why are they important in the Birch and Swinnerton-Dyer conjecture?
-Elliptic curves are mathematical curves represented by equations of the form y^2 = x^3 + ax + b. They are important in the Birch and Swinnerton-Dyer conjecture because the conjecture is concerned with determining the number of rational points on these curves, which has implications in number theory.
What is the Yang-Mills theory and its relevance to the last Millennium Prize Problem?
-The Yang-Mills theory is a framework in physics that describes elementary particles using geometric structures. The last Millennium Prize Problem is about understanding the mathematical foundations of this theory, particularly the existence of a mass gap, which is crucial for the theory's application in particle physics.
Outlines
🔍 The Riemann Hypothesis and Prime Numbers
The first Millennium Prize Problem discussed is the Riemann Hypothesis, which is considered one of the most significant unsolved problems in mathematics. The hypothesis is deeply connected to the distribution of prime numbers and is believed to be true by many mathematicians due to its implications in various fields, including cryptography. The Riemann zeta function is central to this hypothesis, and it is proposed that this function equals zero only when the real part of the complex number 's' is a half. Although the hypothesis has been tested extensively and found to hold true for the first 10 trillion solutions, a formal proof is yet to be established. Solving the Riemann Hypothesis would not directly reveal a pattern in prime numbers but would significantly advance the understanding of prime numbers, potentially leading to the discovery of such a pattern.
🌊 Navier-Stokes Equations and Fluid Dynamics
The second Millennium Prize Problem is the Navier-Stokes existence and smoothness problem, which is particularly fascinating due to its relation to fluid dynamics—a field of study that describes the motion of liquids and gases. The Navier-Stokes equations are fundamental in this area, providing a mathematical model for fluid flow. Despite their importance, a complete understanding of these equations, especially in relation to phenomena like turbulence, remains elusive. The challenge lies in proving the existence of smooth solutions for the equations under all conditions, a task that has not yet been accomplished. The implications of solving this problem are vast, affecting both the theoretical understanding of fluid mechanics and practical applications in engineering and environmental science.
💾 P vs. NP: The Complexity of Computational Problems
The third problem, known as P versus NP, delves into the realm of computer science and computational complexity. The problem questions whether every problem that can be solved quickly (in polynomial time, denoted as P) can also be verified quickly. In essence, it asks if efficiency in solving a problem equates to efficiency in checking its solution. This problem has profound implications for various fields, including cryptography, optimization, and algorithm design. The resolution of P vs. NP would be groundbreaking, potentially reshaping our understanding of what is computationally feasible and could lead to advances in technology and security.
🏺 Perelman's Proof of the Poincaré Conjecture
The Poincaré Conjecture, the first of the Millennium Prize Problems to be solved, is a topic in the field of topology, which deals with the properties of space that are preserved under continuous transformations. The conjecture posits that a simply connected, three-dimensional shape without holes can be transformed into a sphere without any cutting or gluing. Gregori Perelman provided the proof for this conjecture, which was a monumental achievement. Remarkably, Perelman declined both the $1 million prize and the prestigious Fields Medal, citing disinterest in fame and fortune, demonstrating an extraordinary level of modesty and dedication to the pursuit of mathematical truth.
🏛 The Hodge Conjecture and Geometric Shapes
The Hodge Conjecture, the fifth Millennium Prize Problem, is a highly complex problem in algebraic geometry that seeks to understand whether certain geometric shapes can be decomposed into simpler components. The conjecture is concerned with 'invariants'—properties that remain unchanged under certain transformations—and specifically with a type of invariant known as the cohomology class. The conjecture aims to determine whether these classes can be represented by geometric cycles within the shape. This problem is intricate and requires a deep understanding of algebraic cycles and their relationship to the topology of complex algebraic varieties.
📏 The Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture, the sixth problem, is a significant issue in number theory that focuses on elliptic curves, which are curves defined by equations of the form y^2 = x^3 + ax + b. The conjecture is interested in the number of rational points on these curves, specifically whether there are infinitely many or a finite number. The conjecture suggests a method to determine this through the use of the Hasse-Weil L-function, which is a complex analytic function that encodes information about the number of points on the curve. This conjecture is central to our understanding of elliptic curves and their applications in cryptography and other areas of mathematics.
⚛️ The Yang-Mills Theory and the Mass Gap
The final Millennium Prize Problem, the Yang-Mills Existence and Mass Gap, is rooted in theoretical physics and concerns the mathematical foundations of quantum field theory. The Yang-Mills theory, developed by Chen Ning Yang and Robert Mills, provides a framework for describing elementary particles using geometric structures. While the theory has been experimentally verified, a complete mathematical understanding is still lacking. A key aspect of the theory is the 'mass gap,' a property that has been observed in experiments but not yet proven theoretically. Resolving this problem would not only solidify the mathematical underpinnings of the Yang-Mills theory but also contribute to our fundamental understanding of the physical universe.
Mindmap
Keywords
💡Millennium Prize Problems
💡Riemann Hypothesis
💡Prime Numbers
💡Navier-Stokes Equations
💡P vs NP Problem
💡Perelman
💡Hodge Conjecture
💡Elliptic Curves
💡Yang-Mills Theory
💡Brilliant.org
Highlights
Introduction to the seven Millennium Prize Problems, each offering a $1 million reward for solutions.
The Riemann Hypothesis is considered one of the most important problems in mathematics, with implications for prime numbers and cryptography.
The mysterious nature of prime numbers and the lack of a discernible pattern, despite their widespread importance.
The Riemann Zeta Function and its relation to the distribution of prime numbers, as proposed by the German mathematician Riemann.
The statement of the Riemann Hypothesis regarding the zeros of the Zeta function and the real part of the complex number s.
The significance of proving the Riemann Hypothesis for advancing the study of prime numbers, despite not revealing a pattern itself.
The Navier-Stokes existence and smoothness problem, central to understanding fluid dynamics and its applications.
The beauty and complexity of the Navier-Stokes equations in describing fluid flow, from air over wings to ocean waves.
The challenge of proving the existence of smooth solutions for the Navier-Stokes equations in three-dimensional fluid flow.
The P versus NP problem, a fundamental question in computer science about the relationship between solvable and verifiable problems.
The Poincaré conjecture, the first Millennium Prize Problem to be solved, concerning the transformation of shapes in topology.
Grigori Perelman's refusal of the $1 million prize and Fields Medal, highlighting his disinterest in fame and fortune.
The Hodge conjecture, a complex problem in algebraic geometry involving the decomposition of shapes into simpler components.
The Birch and Swinnerton-Dyer conjecture, focusing on the number of rational points on elliptic curves and its importance in number theory.
The Yang-Mills existence and mass gap problem, foundational to elementary particle theory and the search for a theoretical understanding of mass.
The quantum Yang-Mills theory and its experimental validation, contrasting with the lack of a solid mathematical foundation.
Invitation for viewers to share their thoughts on the Millennium Prize Problems and which one they find most intriguing.
Transcripts
there are seven problems in mathematics
that if solved will not only earn you a
place in the Hall of Fame within
mathematics but also $1
[Music]
million in today's video I'm going to be
explaining each of the seven problems
and I'll also be making a separate
videos on each of the seven problems
with a little bit more detail and going
into a bit more depth with those as well
a bit of background on these problems
you may have heard of them already as
the Millennium prize problems and that's
because they were established in the
year 200000 by the clay Institute now
because there's seven of them let's dive
straight into the first one which is the
rean hypothesis the rean hypothesis is
actually one of the most famous out of
the seven Millennium prize problems a
lot of mathematicians say that this is
the most important problem within
mathematics and if we can solve it then
this will open up a huge Gateway into a
very specific branch of mathematics and
that is prime numbers now before I dive
into what the hypothesis is
mathematicians actually believe or
they're almost certain that it's true
that this hypothesis that rean proposed
is true and really we hope that it is
because there are a lot of areas in
different fields like cryptography that
assume that this is true okay so what is
the ran hypothesis and what on Earth
does it have to do with prime numbers so
firstly prime numbers if you're not
aware a number is said to be a prime
number if it is only divisible by itself
and the number one so the number seven
for example you cannot divide that by
any other number other than seven and
one you might think okay well let's
think of different prime numbers that
there are and you might think as with
most things in mathematics and in life
there would be a pattern to it but
interestingly there actually isn't we
don't have a pattern for prime numbers
and if we did it would be absolutely
groundbreaking because it is used in so
many areas you know not within just
mathematics but so many different areas
as well so we as mathematicians would
love if we could find this pattern if
there is one that is so that's an
overview on prime numbers and why
they're important but what has that got
to do with the ran hypothesis so the
very famous German mathematician reman
noticed that the frequency of prime
numbers was very closely related to this
formula here which is Zeta as a function
of s equals the sum from Nal 1 to
Infinity of 1/ n the^ s so this written
out would look like 1 + 1 1^ s + 1/ 2 ^
s+ 1 over 3 ^ s and so on and so forth
now this function is very famous it's
known as the rean zeta function and
there's one slight subtlety with this
the number s here is complex which makes
it a whole lot more complicated in
solving now going on from this function
oer actually discovered a relation to
this this rean zeta function and it says
that Zeta of s is actually the product
of P where p is prime now the product
means it's a multiplication rather than
a sum rather than adding so you multiply
rather than add and it's the product of
1/ 1 - p^ - s where p is a prime number
now the rean hypothesis itself says well
when is this rean zeta function equal to
zero and he proposed that it's only ever
equal to zero when the real part of the
complex number s is equal to a half and
that is Ran's hypothesis now it hasn't
been proved but it has been
experimentally tested for the first I
think 10 trillion Solutions and it's
true it's always a half but obviously
all it would take is one counter example
to say the real part of the complex
number is actually not a half although
10 trillion of the solutions have shown
to be true all it takes is one counter
example to disprove that so we really
need someone to show a proof of this and
to show that yes this function does
equal zero when the real part of s is a
half and for me it's a very beautiful
Millennium prize problem and I can see
why it's a lot of people's favorit now
the final thing to mention before I move
on to the next millennium prize problem
is that proving ran hypothesis is not
going to show this pattern in prime
numbers it's it's not going to do do
that unfortunately and I think a lot of
people get confused with that and they
think that's what it's going to do but
actually if we prove rumors hypothesis
that in itself is going to open a whole
new gateway to further development and
further advancement in prime numbers so
if we prove this it will open up the
chance to then hopefully find a pattern
amongst prime numbers which yeah I I
think is absolutely incredible the
second Millennium prize problem that I'm
going to talk about today is the Navia
Stokes existence and smoothness problem
now I might be slightly biased in saying
that this is my favorite Millennium
prize problem and that's just because I
specialized in fluid dynamics so what is
Navia Stokes what what are they the
Navia Stokes equations essentially
describe fluid flow so that could be air
traveling over an aircraft's wing it
could be honey trickling off a spoon it
could be the waves in the ocean anything
that's a fluid these Navia Stokes
equations describe that I'm going to
show you an image of what these
equations look like and to those of you
that don't know Vector calculus they
might look quite daunting but for me I
just think they are so beautiful that
there are just some very small set of
equations that will describe fluid
anywhere on Earth although they describe
it incredibly well the understanding of
these equations is incomplete there are
certain areas within fluid mechanics
like turbulence Breeze and all sorts of
stuff like that that we don't fully
understand yet and that's a huge part of
what the Navia Stokes existence and
smoothness problem is is getting that
deeper understanding of these equations
and seeing if there's anything else that
remains hidden with these equations now
I personally think it's honestly insane
I remember watching the film gifted if
anyone has seen the film gifted
fantastic film I've done a video
actually explaining some of the matths
in that film and the woman in the film
solved this problem this Millennium
prize problem and I remember watching
this when I was younger and I was just
fascinated and I went to University
wanting to specialize in fluid dynamics
because I was like I really want to go
solve the Millennium prize problems or
be it quite a a stretch yeah I I just I
thought it was a fantastic film and ever
since then I've kind of been locked in
on on how cool fluid dynamics is so yeah
if anyone is interested in fluid
mechanics definitely get involved in it
I really really love it and if you go on
to do research you might end up solving
this problem uh which is the navio
Stokes existence and smoothness problem
now it's not just getting this basic
understanding of the Navia Stokes
equations but some very simple
applications of the Navia Stokes
equations have yet to be proved for a
three-dimensional system with some
initial conditions mathematicians have
actually never proved that a smooth
solution exists and neither have they
found a counter example this is a kind
of a very basic problem is is a
three-dimensional fluid flow and yeah
math Ians haven't been able to prove
that a smooth solution always exists or
found any counter examples to suggest
otherwise so although it seems like
quite a simple problem it's immensely
complex and that will be a huge part to
play in hopefully solving the Millennium
prize problems and being able to
understand more about fluid flow in our
world and within engineering as well
Millennium problem number three this is
known as the P versus NP problem now you
may have heard of this because it's
related to computer systems what P
versus NP means is p is a problem that
is solvable so it can be solved in what
is known as polinomial time and NP is a
problem that can be verified or checked
so it's easy to verify or easy to check
and this is where the debate between if
a problem is easy to check is it
solvable and vice versa now if we think
of P for a moment so a problem that's
solvable everything in P is therefore in
NP because if we can solve a problem we
can also check it we can also verify it
then comes the question well is p equal
to NP so it encompasses everything so a
problem that is easy to solve is always
easy to verify or is there some case
where something that's in NP is not in P
now this might sound incredibly
complicated and that's because it is I
remember watching the series numbers and
there's a part where the kind of
mathematician is trying to solve P
versus NP and I think his professor or
somebody comes in and says you you're
never going to solve it so it's
something in mathematics that although
mathematicians believe P does not equal
NP the proof of it would be absolutely
groundbreaking so it's a really really
cool concept and I personally really
like it because I'm a tech nerd and I
love anything to do with computer
systems so I find this really really
cool Millennium problem number four is
the panare conjecture and apologies if I
pronounced that incorrectly so this is
the first of the seven Millennium prize
problems to have been solved and I think
the story behind the mathematician
himself who solved it is is quite
incredible really and I'll be getting on
to that in just a moment but firstly
what is this conjecture so this problem
is to do with topology now as a
background about topology it's
essentially to do with shapes but it
doesn't concern itself with distances
what panker said was if you have an
object that has no holes in it and it's
finite it can always be made into a
sphere so as an example everyone at home
just imagine you have some sort of
object made out of Play-Doh and you
count the number of holes in it you know
you might have pierced it or it might be
say a teacup that's got a hole with the
handle and what I want you to do is try
and somehow in your head think about how
you could make that into a sphere now
there are conditions with this you
cannot cut so you can't cut your object
you cannot glue together and you cannot
close any holes now the question is can
you make that into a sphere so typically
if you chose okay I'm going to have a
cube like I said you can squish that
down into sphere and that worked whereas
if you have a donut for example the only
way to make a sphere from that donut is
to close that hole to squish it together
or you could cut it and somehow mold it
together but you're not allowed to do
that so therefore because you had a hole
in it you cannot make a sphere and
that's essentially the concept behind
the panare conjecture now panare himself
extended this to higher Dimensions which
when you think about in your head it
hurts your brain quite a bit but you
know we can think of 3D and we can look
at 3D objects and it's easy to visually
see that but when you go to higher
Dimensions it's much more complicated
and essentially panker himself extended
his theory about spheres to higher
Dimensions the mathematician who solved
this Millennium prize problem was called
gregori perelman and he has a a
fantastic quote behind this he actually
didn't accept the $1 million
and he was also awarded a Fields medal
which if you don't know is huge it's
like the Nobel Prize for mathematics he
was awarded this and he rejected it and
he also rejected the million dollars and
he said in a quote I'm not interested in
money or fame I don't want to be on
display like an animal in a zoo I'm not
a hero of mathematics I'm not even that
successful that is why I don't want to
have everybody looking at me which I
just I mean I was just so sad when I
read that because I I understand you
know he he probably just does math cuz
he loves maths he's just proven a
millennium priz problem which is is
absolutely insane talk about modesty at
its highest so yeah that was the pon
conjecture and the only Millennium prize
problem so far to have been solved now
to solve a millennium prize problem you
of course need to know a lot about
mathematics and if you're watching this
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we're continuing with the theme of
topology with Millennium prize number
five which is known as the Hodge
conjecture now this is a very very
complicated conjecture and I'm going to
be explaining it in very very brief
terms because is yeah incredibly
complicated but again like I said
subscribe for a very deep dive on this
Millennium prize problem in very simple
terms the Hodge conjecture concerns
itself with very complicated shapes in
geometry and says well can we decompose
that shape into smaller less complicated
shapes it's kind of like Lego where you
have the building blocks to make
something that's absolutely insane Dr
Tom Crawford mentioned that in his
lecture that he did on this and I
thought that was really nice way of
thinking about the hod conjecture and
and how I personally remember it so in
essence it's saying can we build
complicated things with smaller things
now like I said topology concerns itself
with the shape of something rather than
the distance between two points on an
object so like I said we can have a
square and that can be molded into a
sphere now interestingly I mentioned a
donut but a doughnut can actually be
shaped into a teacup now that's because
a teacup has one hole with the handle
and a donut has one hole in the middle
so the two can be formed together in
topology and therefore they are classed
as the same in topology just like a cube
and a sphere are classed as the same now
when you move between different objects
that are classes the same in topology
you have what is known as an invariant
and that is something that doesn't
change now it's very hard to explain
this conjecture in simple terms because
it is just so so complex but it has a
lot to do with invariance it essentially
has to do with understanding a very
specific type of invariant which is
something that doesn't change that is
known as the cohomology class and it's
way too complicated to explain in two
minutes so I'll be making a deeper dive
video on it and hopefully explain what
it's all about so that was the Hodge
conjecture now we're moving on to
Millennium problem number six and that
is known as the Birch and Swinton d
conjecture in mathematics we would be
right in saying that mathematicians have
long been obsessed with equations you
know all you have to do is look at
Pythagoras's equation and date that back
to when that was first established
mathematicians love equations and this
conjecture itself is focused on what are
known as elliptic equations so elliptic
equations they take the form y^2 = X Cub
+ a x + B elliptic equations produce
some beautiful curves that are known as
elliptic curves and that's basically
what this conjecture is about the
conjecture says okay well how many
rational points are there on this
elliptic curve so how many fractional
points are there for both X and Y when
you take a specific point so we could do
this very monotonously and just take
points everywhere on the curve which
would be very very long infinitely long
actually when you think about it and uh
the question of the conjecture is well
how how many are there how many of them
are fractional points for both X and Y
the trick to understanding this
conjecture is by the hassa feel L
function which is too complicated again
to go into detail on the maths behind it
in this video but what it's used for is
say you have an elliptic curve you can
use this L function to figure out if
there are infinitely many fractional
points or finitely many so that in
itself is very useful but this
conjecture is huge in number Theory and
is known as one of the hardest problems
to solve which makes sense as to why
it's a millennium prize problem so that
was the Millennium problem number six
number seven is known as the Yang Mills
existence and mass Gap almost half a
century ago Yang and Mills discovered a
new remarkable framework to describe
Elementary particles using structures
that also arise in Geometry The quantum
Yang Mills theory is now the foundation
for most of the elementary particle
theory and its predictions have been
tested at many experimental Laboratories
but unfortunately the mathematical
understanding is still unclear and
that's what the Millennium prize problem
is about it's about understanding this
Theory now to successfully use this Yang
Mills Theory there is one thing that is
needed and that is known as the mass Gap
this property has been discovered by
physicists using both experimental work
and through computers but unfortunately
the theoretical side of things is yet to
be established and that is why it's such
a huge problem and why it is a
millennium prize problem arguably one of
the hardest and one of the most
fundamental when it comes to physics
again the Yang Mills theory is
incredibly complicated just like all of
the previous six Millennium prize
problems so if you're interested I'll be
doing a deep dive on it in a separate
video but that was the final Millennium
prize problem and honestly I'd love to
know what everyone's thoughts are on all
of them which one you think is your
favorite comment down below your
favorite thank you so much for watching
and I'll see you all in the next one
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