Prof. Bocherds: "The most difficult part of the Schwarz lemma is trying to remember that Schwarz doesn't have a t in it." automatically generated subtitles: "Schwartz"😂😂😂
The moments of explanation for intuition on theorems/ideas in a proof are invaluable, these videos are an incredible resource for anyone learning more advanced math!
15:59 : I see how, for every compact subset of the domain, we get a uniform bound on the derivatives of the functions on that compact subset. So, by the Arzela-Ascoli, I guess we get that there is a sub-sequence which converges on that compact set. But then, to get that there is a subsequence that converges overall, I’m not so sure how to get? Oh, I guess if we have a nested sequence of compacts C_1 \subset C_2 \subset \cdots, which exhausts the domain (\bigcup_{i=1}^\infty C_i = D) then if we just take such a subsequence for C_1, then a subsequence of that for C_2, etc., and then take the diagonal of this sequence of subsequences, then I suppose that should converge on each of the compacts in the sequence of compacts. (Hm, if I have a sequence of compacts that exhausts an open set in this way, must any other compact subset of the set, be a subset of some compact in the sequence? Ah, if the compacts are the closure of their interiors, then their interiors form an open cover of any compact subset of the set that the sequence of compacts is exhausting, and taking a finite subcover , gives that yes, some term in the sequence will include it. So, being uniformly convergent on each compact subset from an exhausting sequence of compacts, implies uniform convergence on *any* compact subset of the domain. Cool. Though, still need the “the exhausting sequence of compacts, has its terms each equal to the closure of their interiors”... which, I suppose is not much of an assumption.)
Quantizable functions on Kähler manifolds and non-formal quantization Abstract Applying the Fedosov connections constructed in [7], we find a (dense) subsheaf of smooth functions on a Kähler manifold X which admits a non-formal deformation quantization. When X is prequantizable and the Fedosov connection satisfies an integrality condition, we prove that this subsheaf of functions can be quantized to a sheaf of twisted differential operators (TDO), which is isomorphic to that associated to the prequantum line bundle. We also show that examples of such quantizable functions are given by images of quantum moment maps.
Just wondering if you familiar with this algebra? Noncommutative geometry and deformation quantization in the quantum Hall fluids with inhomogeneous magnetic fields Giandomenico Palumbo(Dublin Inst.) Apr 8, 2023 It is well known that noncommutative geometry naturally emerges in the quantum Hall states due to the presence of strong and constant magnetic fields. Here, we discuss the underlying noncommutative geometry of quantum Hall fluids in which the magnetic fields are spatially inhomogenoeus. We analyze these cases by employing symplectic geometry and Fedosov’s deformation quantization, which rely on symplectic connections and Fedosov’s star-product. Through this formalism, we unveil some new features concerning the static limit of the Haldane’s unimodular metric and the Girvin-MacDonald-Platzman algebra of the density operators, which plays a central role in the fractional quantum Hall effect.
Because he "cheated" a little to make the argument clear. The square root is not defined on the whole unit disk, just on the image of D (or any simply connected open not containing 0, and including the image of D). Therefore the map h is not really a map defined on the unit disk, but only on a subset, and Schwarz lemma can't be applied. On the other hand, the inverse map h^(-1) is defined as composition of maps defined on the unit disk, namely: inverse of last Moebius transformation, then squaring, then inverse of first Moebius transformation. Here you can apply Schwarz's lemma and you can be confident it's not a rotation because sticking in there the squaring (any n-th power will do) makes sure h^(-1) is not injective.
for example instead of geometrical projection you imagine just a bundle of fiberoptic cables of 0 diameter, and the condition for an isomorphism is that any line is preserved, neighbors stay next to each other and so on, and otherwise you can move them any way you like, the only line that gives you trouble is the line around the circle, which now is just mapped to a certain ratio of infinite components for imaginary and real components for each of the threads output. you could definitively define such a thing to also be an isomorphism, but i digress :P it is just mapping a circle to itself with the same connectivity between points in the same direction around the origin, but rotated and lopsided in any way you would like.
any infinite stretching to more than a countable set of directions should fail :P. because of a similar problem as the one you get with uniqueness if you project the shape to the exterior in the complex plane, infinite then sort of needs to have some poles, like if you invert out of the circle and into itself to cover the whole complex plane, with some boundry inside itself dividing the area inside and outside the unit circle, then the center of the circle get projected to all directions form a point. not that familiar with the proofs of isomorphism, but there are several ways to go about it, and if you do a kind of projection there are these sorts of problems, where you must exclude a point or something to prove the shape projects into another without any loss of connectivity between points and including all the points. i can imagine it going wrong for that reason or something like that, but ofc there is always the limit of making the unit disk larger as well, not sure how the difference between any kind of projection vs this kind of stretching. there is certain differences in how you deal with infinities i guess is what i am trying to get at. maybe the failure in the case of the whole plane is down to some artifact of how isomorphism are defined.
When you have a hole /inclusion on the disk and deform it you get one side of the theory which is on the torus, if you remove the hole /inclusion by closing the disk you get the s3 as 2 hemispheres / mapping the same transformations to both topologies. When applying a dehn twist we get a heegaard splitting where the transformations get conserved into spin 1/2 Dirac fermions. What is interesting about this, is that the higher dimensional projections get preserved in this splitting. In the other case you get edge modes which related to your previous lecture. With the s3 you get hydrodynamics in the bulk on the torus the same but on the surface. Thanks again for the great videos.
@@Unidentifying you may be interested to look into symmetry protected topological orders. (Xiao-Gang Wen) I think there is certain topological protection correspond to momentum space, we call these Skyrmions and hopfions. These are not only correspond to knots but remarkably to null fields in relation to maxwell theory. I think 3 spatial dimension is a unique case but that's just my opinion. I suspect lower dimensional topology plays bigger role and we take 3 spatial dimensions for granted. Floer theory also could be something your interest.
There are likely scholars or professionals more adept and capable at applying this than I am. So, what does the transition from theory to realizable application look like? What does realization look like? Consider this question open-ended, there's no 'correct' answer to give.
I wonder what your mental abilities are that make you familiar with all these branches of mathematics!!!! Do you eat? Do you drink? Do you sleep like us?!! Do you go for a walk?!! Do you have a family?! What do you need more than that to solve the Riemann hypothesis?😮❤
Prof. Bocherds: "The most difficult part of the Schwarz lemma is trying to remember that Schwarz doesn't have a t in it."
automatically generated subtitles: "Schwartz"😂😂😂
These videos are such a joy. Thank you for continuing to do this!
Love your teaching. Have a healthy and comfortable life
The moments of explanation for intuition on theorems/ideas in a proof are invaluable, these videos are an incredible resource for anyone learning more advanced math!
My lecturer didn't have time to cover this topic during the semester so I'm very glad to see this video!
I need more Borcherds' opinions on everything.
Same!
Nice, I just finished my thesis on the RMT today!
15:59 : I see how, for every compact subset of the domain, we get a uniform bound on the derivatives of the functions on that compact subset.
So, by the Arzela-Ascoli, I guess we get that there is a sub-sequence which converges on that compact set.
But then, to get that there is a subsequence that converges overall, I’m not so sure how to get?
Oh,
I guess if we have a nested sequence of compacts C_1 \subset C_2 \subset \cdots, which exhausts the domain (\bigcup_{i=1}^\infty C_i = D)
then if we just take such a subsequence for C_1, then a subsequence of that for C_2, etc., and then take the diagonal of this sequence of subsequences, then I suppose that should converge on each of the compacts in the sequence of compacts.
(Hm, if I have a sequence of compacts that exhausts an open set in this way, must any other compact subset of the set, be a subset of some compact in the sequence? Ah, if the compacts are the closure of their interiors, then their interiors form an open cover of any compact subset of the set that the sequence of compacts is exhausting, and taking a finite subcover , gives that yes, some term in the sequence will include it.
So, being uniformly convergent on each compact subset from an exhausting sequence of compacts, implies uniform convergence on *any* compact subset of the domain. Cool.
Though, still need the “the exhausting sequence of compacts, has its terms each equal to the closure of their interiors”... which, I suppose is not much of an assumption.)
Quantizable functions on Kähler manifolds and non-formal quantization
Abstract
Applying the Fedosov connections constructed in [7], we find a (dense) subsheaf of smooth functions on a Kähler manifold X which admits a non-formal deformation quantization. When X is prequantizable and the Fedosov connection satisfies an integrality condition, we prove that this subsheaf of functions can be quantized to a sheaf of twisted differential operators (TDO), which is isomorphic to that associated to the prequantum line bundle. We also show that examples of such quantizable functions are given by images of quantum moment maps.
Just wondering if you familiar with this algebra?
Noncommutative geometry and deformation quantization in the quantum Hall fluids with inhomogeneous magnetic fields
Giandomenico Palumbo(Dublin Inst.)
Apr 8, 2023
It is well known that noncommutative geometry naturally emerges in the quantum Hall states due to the presence of strong and constant magnetic fields. Here, we discuss the underlying noncommutative geometry of quantum Hall fluids in which the magnetic fields are spatially inhomogenoeus. We analyze these cases by employing symplectic geometry and Fedosov’s deformation quantization, which rely on symplectic connections and Fedosov’s star-product. Through this formalism, we unveil some new features concerning the static limit of the Haldane’s unimodular metric and the Girvin-MacDonald-Platzman algebra of the density operators, which plays a central role in the fractional quantum Hall effect.
Thank you very much professor. Please keep uploading videos and lessons
Great video as always! I am curious though, what is the intuition behind trying to maximize the derivative?
26:00 Why can apply Schwarz lemma on h^-1 not h? thk
Because he "cheated" a little to make the argument clear. The square root is not defined on the whole unit disk, just on the image of D (or any simply connected open not containing 0, and including the image of D). Therefore the map h is not really a map defined on the unit disk, but only on a subset, and Schwarz lemma can't be applied. On the other hand, the inverse map h^(-1) is defined as composition of maps defined on the unit disk, namely: inverse of last Moebius transformation, then squaring, then inverse of first Moebius transformation. Here you can apply Schwarz's lemma and you can be confident it's not a rotation because sticking in there the squaring (any n-th power will do) makes sure h^(-1) is not injective.
Professor, please for the next video, show us your "fields medal",❤❤❤
get ‘em Richard
Thank you very much for the excellent lecture.
for example instead of geometrical projection you imagine just a bundle of fiberoptic cables of 0 diameter, and the condition for an isomorphism is that any line is preserved, neighbors stay next to each other and so on, and otherwise you can move them any way you like, the only line that gives you trouble is the line around the circle, which now is just mapped to a certain ratio of infinite components for imaginary and real components for each of the threads output. you could definitively define such a thing to also be an isomorphism, but i digress :P it is just mapping a circle to itself with the same connectivity between points in the same direction around the origin, but rotated and lopsided in any way you would like.
Thank you so much for this awesome video!
any infinite stretching to more than a countable set of directions should fail :P. because of a similar problem as the one you get with uniqueness if you project the shape to the exterior in the complex plane, infinite then sort of needs to have some poles, like if you invert out of the circle and into itself to cover the whole complex plane, with some boundry inside itself dividing the area inside and outside the unit circle, then the center of the circle get projected to all directions form a point. not that familiar with the proofs of isomorphism, but there are several ways to go about it, and if you do a kind of projection there are these sorts of problems, where you must exclude a point or something to prove the shape projects into another without any loss of connectivity between points and including all the points. i can imagine it going wrong for that reason or something like that, but ofc there is always the limit of making the unit disk larger as well, not sure how the difference between any kind of projection vs this kind of stretching. there is certain differences in how you deal with infinities i guess is what i am trying to get at. maybe the failure in the case of the whole plane is down to some artifact of how isomorphism are defined.
very good
When you have a hole /inclusion on the disk and deform it you get one side of the theory which is on the torus, if you remove the hole /inclusion by closing the disk you get the s3 as 2 hemispheres / mapping the same transformations to both topologies. When applying a dehn twist we get a heegaard splitting where the transformations get conserved into spin 1/2 Dirac fermions. What is interesting about this, is that the higher dimensional projections get preserved in this splitting. In the other case you get edge modes which related to your previous lecture. With the s3 you get hydrodynamics in the bulk on the torus the same but on the surface. Thanks again for the great videos.
What do you think of the idea that certain geometrical/ topological spaces are fundamental to space? And the fundamental forces emerging from them?
@@Unidentifying you may be interested to look into symmetry protected topological orders. (Xiao-Gang Wen)
I think there is certain topological protection correspond to momentum space, we call these Skyrmions and hopfions. These are not only correspond to knots but remarkably to null fields in relation to maxwell theory. I think 3 spatial dimension is a unique case but that's just my opinion. I suspect lower dimensional topology plays bigger role and we take 3 spatial dimensions for granted. Floer theory also could be something your interest.
Thank you
beatiful video thank you
What's the reason for your American spellings and saying 'zee'?
1776🇺🇸
Hello professor what you motivate you to teach in this age . I really impressed ❤❤
I don't understand your question. What has age to do with teaching and transferring knowledge?
@@markborz7000 yes I think I ask wrong question here 🙃
There are likely scholars or professionals more adept and capable at applying this than I am. So, what does the transition from theory to realizable application look like? What does realization look like? Consider this question open-ended, there's no 'correct' answer to give.
yeeeeeeeeeeee
I wonder what your mental abilities are that make you familiar with all these branches of mathematics!!!! Do you eat? Do you drink? Do you sleep like us?!! Do you go for a walk?!! Do you have a family?! What do you need more than that to solve the Riemann hypothesis?😮❤
Mathematician generate their energy from the sum.