Solve This Mathematics Problem and Get 1 Million Dollars

Summary

This 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

00:00

🔍 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.

05:02

🌊 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.

10:02

💾 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.

15:04

🏺 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.

Keywords

💡Millennium Prize Problems

The Millennium Prize Problems are seven mathematical problems established by the Clay Mathematics Institute in 2000 with a reward of one million dollars for each correct solution. These problems are central to the video's theme, as they represent some of the most significant unsolved challenges in mathematics. The script discusses each of these problems in detail, emphasizing their importance and the potential impact of solving them on various fields.

💡Riemann Hypothesis

The Riemann Hypothesis is one of the Millennium Prize Problems and is considered one of the most important in mathematics. It deals with the distribution of prime numbers and their relationship with the Riemann zeta function. The hypothesis proposes that the zeta function is zero only when the real part of the complex number 's' is one-half. The video explains this hypothesis as a gateway to further advancements in the understanding of prime numbers, which are crucial for fields like cryptography.

💡Prime Numbers

Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves. The video script discusses the significance of prime numbers in various fields, especially in the context of the Riemann Hypothesis. The lack of a discernible pattern in the distribution of prime numbers is highlighted, and the potential groundbreaking implications of finding such a pattern are mentioned.

💡Navier-Stokes Equations

The Navier-Stokes equations are fundamental in fluid dynamics, describing the motion of fluids. The video script presents the Navier-Stokes existence and smoothness problem as a Millennium Prize Problem, emphasizing the incomplete understanding of these equations, especially in the context of turbulence. The script suggests that a deeper understanding of these equations could have significant implications for various applications.

💡P vs NP Problem

The P vs NP problem is a central question in computer science and concerns the relationship between the complexity of solving a problem and the complexity of verifying its solution. The video script explains this as one of the Millennium Prize Problems, highlighting its relevance to computational processes and the importance of determining whether every problem that is easy to check can also be solved efficiently.

💡Perelman

Grigori Perelman is the mathematician who solved the Poincaré conjecture, one of the Millennium Prize Problems. The video script recounts his story and his rejection of both the monetary reward and the Fields Medal, illustrating his disinterest in fame and fortune, which adds a human interest angle to the narrative of mathematical problem-solving.

💡Hodge Conjecture

The Hodge conjecture is another of the Millennium Prize Problems, focusing on algebraic geometry and the decomposition of complex shapes into simpler components. The video script briefly touches on this conjecture, likening it to building complex structures from smaller, less complicated shapes, and emphasizing its complexity and significance in the field of topology.

💡Elliptic Curves

Elliptic curves are mathematical curves defined by equations of the form y^2 = x^3 + ax + b. The Birch and Swinnerton-Dyer conjecture, one of the Millennium Prize Problems, is related to these curves and concerns the number of rational points on an elliptic curve. The video script explains the importance of this conjecture in number theory and its connection to the Hasse-Weil L-function.

💡Yang-Mills Theory

The Yang-Mills existence and mass gap problem is the final Millennium Prize Problem discussed in the video script. It pertains to quantum field theory and the mathematical understanding of elementary particle interactions. The script explains the importance of the mass gap in the theoretical validation of the Yang-Mills theory, which is foundational to modern particle physics.

💡Brilliant.org

Brilliant.org is mentioned in the video script as a platform for learning mathematics and related fields through hands-on problem-solving. The script promotes the platform as a way to enhance critical thinking and build knowledge in mathematics, which is essential for understanding and potentially solving the Millennium Prize Problems.

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