Understanding the mathematics of modular group, fundamental domain and modular symmetry

In this video, I explain modular groups, modular symmetry and how the fundamental domain for the modular group is defined, in a mathematically rigorous manner. However, since these are pretty advanced topics in mathematics, I have discussed everything relevant so that even a layperson or a beginner can make some sense of the video. That's the reason behind this video being so long, but if you are interested in this topic, I believe you'll find this video helpful. This video is not meant only for mathematics students or researchers, modular symmetry plays an important role in theoretical high energy physics as well. I learned all of this as part of my summer project on theoretical high energy physics. But yeah, a good knowledge of group theory, complex analysis, linear algebra etc. will help. Please note that I am obviously not an expert on this topic, having learned the basics myself in the past three months. But I've tried to share all I know in an accessible manner in this video, because I found there was no single material on this topic on the internet that's suitable for beginners. I had to do a lot of research to make this video, and I hope this will be helpful to whoever is watching. Thanks!
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[@mattm8314]

Phenomenal video, I think one of your best yet. A very beautiful way to explain such dense material. Even forgetting almost all of my linear algebra from college, a layperson like me is able to follow the concepts to a manageable (though very slow) degree. I originally thought that the essay-format you used might be a bit confusing, but as I progressed I realized how valuable it was to be able to pause on an entire chunk of text to understand the content a bit deeper. Only an hour in so far but hope you keeping making videos like this, this will definitely be one that I return to repeatedly to continue digesting!

[@ArpanD]

Hey Matt!! This seriously made my day... after discussing stuff with you, I'm certain this topic is not that difficult for someone like you, but thank you for your encouragement and I'm really glad it helped!!

[@ssen4410]

Some topics or thoughts I came across the video while seeing it. (Probably would be updating it as i watch it)
6:19 I am presuming H here extends to infinities as the Complex plane is unbounded above
15:26 Ohh this is n-fold symmetry. 120° results in 2pi/3 fold symmetry and reflection is basically 180° rotation and hence 2 fold symmetry
18:16 Can this be checked through Hamiltonian canonical equations? As in, through the conservation of the conjugate momentum through the generalised coordinate being cyclic ( in case of time, it is trivial but does it hold for position or momenta?)
30:49 So does the general Linear group also needs to satisfy this condition or is it an additional condition? Aside from being a matrix multiplication group, does it need the elements to be satisfying the condition that they musn't lie in a hyperplane or does the property automatically come from the idea that the vectors are linearly independent ( Taking the case of 2d:- if the vectors lie in hyperplane - in this case line, they are linearly independent as one vector is basically a scalar multiple of another)
31:31 I am guessing the Field that is associated with the vector in this case is R (taken arbitrarily) as in the previous case it was defined over R
36:49 Yes, there can be some x,y € G and x,y is not in N , then there is no guarantee that these two will satisfy the condition. However does this mean N is abelian? since for x € N and y € N, but x,y also € G as N is a subgroup of G implying that xy = yx making N an abelian.
37:49 the assumption is because in GLN , the operation is multiplication or is it a specific case?
40:28 This also includes xN or is there any restriction that it must be right coset?
43:53 We can also define the lines as vector segments a,b such that if a,b is parallel then the dot product of them is maximum provided a is translated on top of b
45:32 So for positive infinity and negative infinity, the corresponding point will be the point at the top of the circle right? Isn't this also how we define a riemann sphere which we use to define complex numbers in Complex plane
56:37 So only in case of g = e , is it true... in case of other values of g, if it is true, then it is not faithful

[@ArpanD]

You are right about the first two points.
Coming to the third point, yes of course. If the Hamiltonian is not a function of the generalized coordinate (cyclic coordinate), then the time derivative of the conjugate momentum corresponding to that coordinate is zero, meaning the corresponding momentum is a conserved quantity. In fact, you can intuitively understand that conservation of linear momentum is a consequence of translational symmetry just from Newton's second and third laws. Consider the collision of two bodies, initially separated in space. Applying Newton's laws on both the bodies, we can show that the total momentum of the entire system remains unchanged before and after the collision. Now since you applied the same laws on both the bodies, and one of the bodies was shifted from the other, you assume that the laws of physics don't change if we shift (translate) from one position in space to another. That's translational symmetry. Of course, using Hamilton's canonical equations is the more mathematically rigorous way to show this, but for high school students and laypeople, the above explanation might help.
Now the fourth point. The condition that the points must be in general linear position is not an additional condition, it automatically follows from the linear independence of the vectors defined by the columns (or equivalently, rows) of the matrices that belong to the group, as you rightly said. Both the conditions (the invertibility of the matrices and the general linear position condition) are just different ways of saying the same thing. One is more mathematical while the other is more geometric (kind of).
You are right about the fifth point too. Although of course, you can define the general or special linear group over the field of complex numbers too.
Coming to your sixth point, N would be Abelian iff, as you said, x ∈ N plus y ∈ N (and of course x, y ∈ G), but x ∈ N is not necessarily true in general. We have defined x to be the elements of the parent group G, so if any arbitrary x ∈ N, essentially, we are talking about the parent group itself. In other words, N and G are the same group, and thus y ∈ N implies y ∈ G as well. So, both x and y represent any arbitrary element of the group G (or N), and now if xy = yx for all x, y, then of course G (or equivalently N) is an Abelian group.
As for the seventh point, it is a specific case, just an example.
Coming to point number eight, yes you are right, we could also have written xN instead of Nx. Since we have already defined N to be a normal subgroup of G (and not just any arbitrary subgroup), it automatically implies that xN = Nx.
Point nine, yes true we can do that too.
For point number ten, you are absolutely right. Riemann sphere is actually the projective space for complex numbers. It is kind of an extension of what I discussed, for complex numbers.
For the last point, yes you are again right. This is very important to understand, we will use this later in the video when studying the fundamental domain of the modular group. Thanks for your questions and for watching the video!!