I was a bit surprised to see in the description that this was your first math video, considering how well you managed to use a mix of edited photos, graphs, and colors to get across (not only the motivation but) a substantial amount of the content of a subject that can be very unapproachable due to both jargon and notation - amazing work!
@@worldequation Superb video. I was slightly confused when the Eisenstein series equation appeared on the screen and the voiceover mentioned coefficients which did not appear to match what was written in the equation. t instead of m, etc. Whilst I think that I realised what was happening other people might be more confused than me.
Great video! Another cool application of modular forms is a proof of the sum of two squares theorem. The theta function at 8:25, can be rewritten as θ(τ) = sum q^(2n^2), n = -inf to inf. Then the q^(2m) coefficient of θ(τ)^2 will be the number of ways to write m as the sum of two squares. Lucky for us, θ(τ)^2 happens to be a modular form of weight 1 over the congruence subgroup Γ_1(4) (this is a certain subgroup of SL_2(Z) of index 4). By comparing to a suitable Eisenstein series, we not only classify which numbers of sums of two squares, we also get Jacobi's formula for the number of ways to write a given number m as the sum of two squares, namely 4(d_1 - d_3) where d_i is the number of divisors of m which are i (mod 4).
Thank you for all the useful materials --- all even unimodular theta function is a modular form of the "level 1." Did you explain what is level of a modular form? at 22:04
1:38 "2d spheres ... which is circles" its a great video and you know way more than me, and i don't want to nitpick but i want to correct this. from a topological standpoint circles are 1-d spheres. a sphere is just the boundary of a ball, where a sphere is the set of all points equidistant from an origin and a ball is the set of all points with distance less than the circle. a circle is a 1d sphere, and it's interior is a 2d ball. a sphere in 3d space is a 2d surface so it's a 2d sphere, and its interior is a 3d ball
You’re very right, and I’m honestly surprised I worded it this way in retrospect. I believe I was trying to suggest that the largest dimension of sphere in this 2d space would be a 1d sphere, also known as a circle.
i love and hate this video. i love it because it's so interesting and let's me get a glimpse at a really beautiful and surprising connection in math, but i hate it because it makes me realize i'm not as smart as i think i am
In fairness, this video was ambitious. I tackled two pretty big subjects, so there’s a lot jammed into this video. If I was making this video now, I think I would have made it two videos, one about lattices up to the E8 and Leech lattice and another about how some modular forms are important generating functions related to these lattices. Truthfully, I could make a whole series on modular forms (and might down the line). It’s such a deep topic with rich connections to other fields that it’s tough for a single video to do all that I wanted this one to do. I’m glad you loved it, though. It’s so beautiful once you dive into it, but it’s such an overwhelming topic. I knew zero about modular forms even four months prior to making this video, and I had the same dumbfounded feeling when I was trying to understand them for the first time. I hope this video chipped away at the mysticism around modular forms for some people, but I plan to make better content about this subject in the future.
You know how in a 2 dimensional lattice where nearby points are equidistant can be triangular or square? I know that there are more types of lattices in higher dimensions. I wonder if there is a way to specify them. For example, since I can't visualize a 7-dimentional lattice in my head much less draw one, is there a standard notation I could use to tell someone which lattice I meant?
You write down its basis. Trying to classify them by the "nearest point shape" (called the voronoi cell) is somewhat hard, as it can be quite complex. In dimension n, it is the convex hull of up to 2(2^n-1) vertices iirc. You might also be thinking of something called the *genus* of the lattice. These are also mostly hopeless to try to explicitly specify in high dimensions (say >9). The number of different lattices grows very quickly as a function of the dimension.
Might be a dumb question, but do lattices define an algebra? Or am I understanding the concept wrong? I can see how I'd do addition, subtraction, multiplication, and division using them (and by extension exponentiation and square roots).
Not a dumb question at all. They do, but the algebra does not have multiplication between two elements or division. Lattices act more like vector spaces, where you can add or subtract vectors. You can also scale any point in the lattice. The technical description is that a lattice is an additive subgroup of the vector space it inhabits.
@@worldequation Thanks for the reply! By saying "a lattice is an additive subgroup of the vector space it inhabits" what do we mean by "additive subgroup" exactly?
@TheRealist Basically, a vector space under the operation of addition (ignoring scalar multiplication) forms a group because it is closed under addition and has an identity (0 vector) and additive inverses (the negative of a vector). We can take the subset of the vectors in the space which are on some lattice. These vectors also can be added and also form a group (they meet the same requirements as before). Because the lattice's elements are a subset of the vector space and are also a group, they are considered a subgroup of the vector space. By additive, I meant the group operation that the subgroup "inherits" is addition.
@@worldequation That was incredibly insightful! (One note: when you say "ignoring scalar multiplication" are we talking about multiplication of integers with vectors or rational numbers with vectors? I assume it's the latter but I just want to make sure. Or I might be misreading again. Apologies...
@@therealist9052 That's a good observation. If we ignore scalar multiplication and just treat the vector space and lattice as groups, which only have one operation by definition, everything is fine. If we add scalar multiplication, the situation is slightly more complicated. We actually can only define multiplication by an integer scalar. In fact, the same way that a vector space is all combinations of real multiples of some basis vectors, an integral lattice is all combinations of integer multiples of some basis vectors. I believe the term for it is a Z-module. Modules are basically vector spaces but the scalar multiplication on them can be a ring instead of a field. The definitions get pretty hairy at some point, but that final distinction between rings and fields is multiplicative inverses. Real/complex numbers have them, so they're fields. Integers do not (1/n isn't an integer), so they're rings.
Yeah, the even integers are apart of their own dual lattice. The dual has to include all points that give an integer product with every point from the original. So you're right that the even integers should be a part of this, but the product of any integer with an even one is even. There's also half integers like 3/2 that multiply with any even integer and the denominator will cancel to produce an integer. Hope that helps
@@worldequation, got it. I somehow missed that you’ve said ‘*all* of the points’. So if I understand correctly, this could have been written on the previous screen as Λ* = {y ∈ R^n : ∀x∈Λ x⋅y ∈ Z}?
Sorry, but this went right over my head :( And I have a math PhD in a non-related field. I can't imagine that a non-mathematician can get very far with this. However, I susbcribe nevertheless, because I still learned something new and I like videos that are not too basic.
I'm sorry to hear that. As I put in the description, this was my first time making a math video, so I know there are things I will do differently next time. If you have any specific suggestions, please let me know
I was a bit surprised to see in the description that this was your first math video, considering how well you managed to use a mix of edited photos, graphs, and colors to get across (not only the motivation but) a substantial amount of the content of a subject that can be very unapproachable due to both jargon and notation - amazing work!
Thanks so much! It means a lot to hear you think so.
I’ve see *a ton of* 1-video channels that did #SoME2
It’s surprising how many people decided to make a math channel to participate.
@@fullfungo I totally agree. It's fun to see all the new people making math content
@@worldequation
Superb video.
I was slightly confused when the Eisenstein series equation appeared on the screen and the voiceover mentioned coefficients which did not appear to match what was written in the equation.
t instead of m, etc.
Whilst I think that I realised what was happening other people might be more confused than me.
Great video! Another cool application of modular forms is a proof of the sum of two squares theorem. The theta function at 8:25, can be rewritten as θ(τ) = sum q^(2n^2), n = -inf to inf. Then the q^(2m) coefficient of θ(τ)^2 will be the number of ways to write m as the sum of two squares. Lucky for us, θ(τ)^2 happens to be a modular form of weight 1 over the congruence subgroup Γ_1(4) (this is a certain subgroup of SL_2(Z) of index 4). By comparing to a suitable Eisenstein series, we not only classify which numbers of sums of two squares, we also get Jacobi's formula for the number of ways to write a given number m as the sum of two squares, namely 4(d_1 - d_3) where d_i is the number of divisors of m which are i (mod 4).
This was awesome. Very reachable for non-experts compared to other modular form introductions
Some2 has blessed us with a video on modular forms! Great video!
Brilliant videos thank you a lot , I hope you can make more videos on the application of modular like its relation with elliptic curve
Thanks! I will most likely make a follow up video on modular forms, but I haven't worked out the details yet.
That was so informative! Who knew there was such a great application for this higher level maths? Thanks for posting the video!
This is the missing puzzle in all the videos I've watched regarding Fermat's last theorem, difficult but fascinating
Thank you from my heart🥰
Thank you so much for the kind words! It means a lot to hear that
@@worldequation ❤️
I agree with the .missing puzzle metaphor because I would still be wandering in the dark.
Thank you. This is a good appetizer for crazy stuff.
Thank you for all the useful materials --- all even unimodular theta function is a modular form of the "level 1." Did you explain what is level of a modular form? at 22:04
1:38 "2d spheres ... which is circles"
its a great video and you know way more than me, and i don't want to nitpick but i want to correct this. from a topological standpoint circles are 1-d spheres. a sphere is just the boundary of a ball, where a sphere is the set of all points equidistant from an origin and a ball is the set of all points with distance less than the circle. a circle is a 1d sphere, and it's interior is a 2d ball. a sphere in 3d space is a 2d surface so it's a 2d sphere, and its interior is a 3d ball
You’re very right, and I’m honestly surprised I worded it this way in retrospect. I believe I was trying to suggest that the largest dimension of sphere in this 2d space would be a 1d sphere, also known as a circle.
Absolutely amazing video! Subscribed.
This was so great, was writing and experimenting on my paper while watching
i love and hate this video. i love it because it's so interesting and let's me get a glimpse at a really beautiful and surprising connection in math, but i hate it because it makes me realize i'm not as smart as i think i am
In fairness, this video was ambitious. I tackled two pretty big subjects, so there’s a lot jammed into this video. If I was making this video now, I think I would have made it two videos, one about lattices up to the E8 and Leech lattice and another about how some modular forms are important generating functions related to these lattices. Truthfully, I could make a whole series on modular forms (and might down the line). It’s such a deep topic with rich connections to other fields that it’s tough for a single video to do all that I wanted this one to do. I’m glad you loved it, though. It’s so beautiful once you dive into it, but it’s such an overwhelming topic. I knew zero about modular forms even four months prior to making this video, and I had the same dumbfounded feeling when I was trying to understand them for the first time. I hope this video chipped away at the mysticism around modular forms for some people, but I plan to make better content about this subject in the future.
Hell yeah it's math video season
Absolutely wonderful video!
I would watch your videos, please post more!!
Thanks for the video -- any pedagogical summary reference for deriving equations at 20:12?
awesomeeee!! keep up the good work buddy
Thanks, will do!
The only thing I understood from the title was Counting points, yet I still enjoyed the video
Amazing explanations!
9:14 is it a graph? Is it 3d graph? Where additional marks like 0, x, y to help understand whats this is.
You know how in a 2 dimensional lattice where nearby points are equidistant can be triangular or square? I know that there are more types of lattices in higher dimensions. I wonder if there is a way to specify them. For example, since I can't visualize a 7-dimentional lattice in my head much less draw one, is there a standard notation I could use to tell someone which lattice I meant?
You write down its basis. Trying to classify them by the "nearest point shape" (called the voronoi cell) is somewhat hard, as it can be quite complex. In dimension n, it is the convex hull of up to 2(2^n-1) vertices iirc.
You might also be thinking of something called the *genus* of the lattice. These are also mostly hopeless to try to explicitly specify in high dimensions (say >9). The number of different lattices grows very quickly as a function of the dimension.
This was great. Thanks
Might be a dumb question, but do lattices define an algebra? Or am I understanding the concept wrong? I can see how I'd do addition, subtraction, multiplication, and division using them (and by extension exponentiation and square roots).
Not a dumb question at all. They do, but the algebra does not have multiplication between two elements or division. Lattices act more like vector spaces, where you can add or subtract vectors. You can also scale any point in the lattice. The technical description is that a lattice is an additive subgroup of the vector space it inhabits.
@@worldequation Thanks for the reply! By saying "a lattice is an additive subgroup of the vector space it inhabits" what do we mean by "additive subgroup" exactly?
@TheRealist Basically, a vector space under the operation of addition (ignoring scalar multiplication) forms a group because it is closed under addition and has an identity (0 vector) and additive inverses (the negative of a vector). We can take the subset of the vectors in the space which are on some lattice. These vectors also can be added and also form a group (they meet the same requirements as before). Because the lattice's elements are a subset of the vector space and are also a group, they are considered a subgroup of the vector space. By additive, I meant the group operation that the subgroup "inherits" is addition.
@@worldequation That was incredibly insightful! (One note: when you say "ignoring scalar multiplication" are we talking about multiplication of integers with vectors or rational numbers with vectors? I assume it's the latter but I just want to make sure. Or I might be misreading again. Apologies...
@@therealist9052 That's a good observation. If we ignore scalar multiplication and just treat the vector space and lattice as groups, which only have one operation by definition, everything is fine. If we add scalar multiplication, the situation is slightly more complicated. We actually can only define multiplication by an integer scalar. In fact, the same way that a vector space is all combinations of real multiples of some basis vectors, an integral lattice is all combinations of integer multiples of some basis vectors. I believe the term for it is a Z-module. Modules are basically vector spaces but the scalar multiplication on them can be a ring instead of a field.
The definitions get pretty hairy at some point, but that final distinction between rings and fields is multiplicative inverses. Real/complex numbers have them, so they're fields. Integers do not (1/n isn't an integer), so they're rings.
Very cool video!
3:38 - this makes no sense to me. Two even integers multiplied by each other give an integer. So why aren't even integers unimodular?
Yeah, the even integers are apart of their own dual lattice. The dual has to include all points that give an integer product with every point from the original. So you're right that the even integers should be a part of this, but the product of any integer with an even one is even. There's also half integers like 3/2 that multiply with any even integer and the denominator will cancel to produce an integer. Hope that helps
@@worldequation, got it. I somehow missed that you’ve said ‘*all* of the points’. So if I understand correctly, this could have been written on the previous screen as Λ* = {y ∈ R^n : ∀x∈Λ x⋅y ∈ Z}?
@@mina86 yes, absolutely
Isn't the Norm also called the Quadrance?
Yeah, from what I can tell, the norm of the vector is the quandrance between it and the zero vector, so like distance vs. magnitude
Sorry, but this went right over my head :( And I have a math PhD in a non-related field. I can't imagine that a non-mathematician can get very far with this. However, I susbcribe nevertheless, because I still learned something new and I like videos that are not too basic.
I'm sorry to hear that. As I put in the description, this was my first time making a math video, so I know there are things I will do differently next time. If you have any specific suggestions, please let me know
E8 Lettuce
This is all I will be able to think of in my upcoming seminar about lattices… thanks 😆