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Young Measures
Приєднався 25 бер 2021
Visiting assistant professor at University of Cincinnati (Ohio, USA). Analysis and geometry (on metric space) are my areas of research. My goal is to advertise these subjects to advanced students and experts that are new to the field.
Toics covered so far @YoungMeasures:
(1) Hausdorff measures (5 videos)
(2) Area Formula (13 videos): Lipschitz maps, Rademacher’s theorem, Jacobians
(3) Coarea Formula (14 videos): Fubini’s theorem
(4) Curves in Euclidean and metric space (12 videos): rectifiable curves, arclength parameterizations
(5) Geometric Measure theory (29 videos): Cantorsets and fractals and their various dimension, Densities of Hausdorff measures
(6) Analysis on Metric Spaces (55 videos): Hajlasz-Sobolev and Newtonian-Sobolev functions, upper gradients, Poincare inequality, Modulus of Path Families
(7) Heisenberg, sub-Riemannian manifolds
Future plans: Lipschitz Maps into Metric Spaces: metric differentiability
PLEASE SUBSCRIBE!
Toics covered so far @YoungMeasures:
(1) Hausdorff measures (5 videos)
(2) Area Formula (13 videos): Lipschitz maps, Rademacher’s theorem, Jacobians
(3) Coarea Formula (14 videos): Fubini’s theorem
(4) Curves in Euclidean and metric space (12 videos): rectifiable curves, arclength parameterizations
(5) Geometric Measure theory (29 videos): Cantorsets and fractals and their various dimension, Densities of Hausdorff measures
(6) Analysis on Metric Spaces (55 videos): Hajlasz-Sobolev and Newtonian-Sobolev functions, upper gradients, Poincare inequality, Modulus of Path Families
(7) Heisenberg, sub-Riemannian manifolds
Future plans: Lipschitz Maps into Metric Spaces: metric differentiability
PLEASE SUBSCRIBE!
Відео
Geometric Proof that Linear Maps on Heisenberg Group Preserve Orientation
Переглядів 3921 годину тому
Geometric Proof that Linear Maps on Heisenberg Group Preserve Orientation
Homogeneous Homomorphisms on Heisenberg Group 2/2
Переглядів 1814 днів тому
Homogeneous Homomorphisms on Heisenberg Group 2/2
Homogeneous Homomorphisms of Heisenberg Group 1/2
Переглядів 4414 днів тому
Homogeneous Homomorphisms of Heisenberg Group 1/2
Hausdorff Dimension of Heisenberg Group is 4.
Переглядів 5228 днів тому
Hausdorff Dimension of Heisenberg Group is 4.
Hausdorff Measure and Dimension of Heisenberg Group
Переглядів 94Місяць тому
Does this video show that the Hausdorff dimension of the Heisenberg group is 4?!
9. Normal Vector and Tangent Plane to a Parametric Surface
Переглядів 31Місяць тому
9. Normal Vector and Tangent Plane to a Parametric Surface
2. Triple Integral in Spherical Ccordinates
Переглядів 16Місяць тому
2. Triple Integral in Spherical Ccordinates
10. Integrating Scalars Functions on Surfaces
Переглядів 32Місяць тому
10. Integrating Scalars Functions on Surfaces
5. Gradient Vector Fields, Path Independence
Переглядів 22Місяць тому
5. Gradient Vector Fields, Path Independence
12. Integrating Vectors on Surfaces - Flux
Переглядів 44Місяць тому
12. Integrating Vectors on Surfaces - Flux
Dilations Scale the Carnot Caratheodory Distance: proof
Переглядів 49Місяць тому
Dilations Scale the Carnot Caratheodory Distance: proof
Does the group dilation define a rectifiable path?
Переглядів 33Місяць тому
Does the group dilation define a rectifiable path?
Dilations (=scaling) in Heisenberg Groups as Homomorphisms
Переглядів 56Місяць тому
Dilations (=scaling) in Heisenberg Groups as Homomorphisms
Koranyi Metric On Heisenberg Group, vs Carnot Caratheodory Distance
Переглядів 91Місяць тому
Koranyi Metric On Heisenberg Group, vs Carnot Caratheodory Distance
Carnot Caratheodory Distance is Left Invariant
Переглядів 1112 місяці тому
Carnot Caratheodory Distance is Left Invariant
Left Invariant Vectors in Heisenberg group-proof
Переглядів 862 місяці тому
Left Invariant Vectors in Heisenberg group-proof
Grushin Space - a non-Group Sub-Riemannian Manifold
Переглядів 582 місяці тому
Grushin Space - a non-Group Sub-Riemannian Manifold
Rectifiable Curves and Geodesics in Heisenberg Groups
Переглядів 563 місяці тому
Rectifiable Curves and Geodesics in Heisenberg Groups
Why in 24:17 that inequality holds? Do you assume that \lambda_1<\lambda_2<\lambda_3?
Yes. Lambda is the percentage of "surviving" intervals. We keep increasing them in order to preserve as much as we can. In limit, you remove almost nothing, so lambda is 50% -- remember there are two copies of size lambda. So, in limit, you basically remove nothing. The denominators and log work out to give that inequality.
@@YoungMeasures thanks.
@@YoungMeasures have you made some video on uniform rectifiability? I have been referred to your channel long back by some professors in Finland. Your content is very good. Keep doing this.
@@sushobhandas7752 I have no videos on that particular topic. I would look up Jonas Azzam’s channel for them.
@@YoungMeasures can you share the link of that video or channel here.
Very nice tools, I will follow your advise on creating a website! Another tool I like is research rabbit. It's similar to mathscinet where you can look for paper abstracts and related research but free, and you also have a nice graph view of how papers connect with each other so you can easily find related work.
Very nice exersice! I have one question tho, when you refer to equicontinuity of the family F you are actually referring to uniform equicontinuity, is that right? I ask because, apparently, the continuity modulus delta you choose for the family F holds for every pair of elements in the domain, or maybe im missing something?
Yes yes. I did not know not uniform equicontinuity, i.e., pointwise equicontinuity, was a thing until this comment! LoL!
A second lecture will complete the discussion here.
Apologies for the type in definition of the Hausdorff measure. It is lim of inf of sums not "sup". There is equivalent definition as sup of inf of sums, which I had initially prepared, but when switching to limit I erased the wrong item :(. Obviously that supremum is always infinite, except if your space is discrete or something, LoL!
I was just scrolling and i found this video hopefully i will get motivated to study more math
Where can I find a proof of the theorem that is used as a starting point?
It is called CARATHEODORY^S CRITERION. It is Theorem 5 in first edition of Measure Theory and Fine Properties of Functions, by C. Evans and F. Gariepy. Other advanced books must have a proof too.
@@YoungMeasures Thank you!
Very nice video. Thanks for the illuminating examples. Did you construct them yourself? Is there a reference where notions in GMT are explained using examples?
Thanks. Any GMT book will do. Examples? I do not know of one doing examples as explicit. Maybe they toss them as exercises. I did come up with them for the lecture but to be fair, they are easy to come up with.
I have a video on different books for GMT. Might help to check out.
Congratulations! Looking forward to more
I would love to see more on writing papers. Things like citations and general conventions. Also, 10/10 channel. Love this HQ math content.
Noted!
If we use absolutely continuous curves or Lipschitz curves in the infimum in the definition of d_cc, then the triangle inequality is indeed obvious, as well. But with C1 curves not so -- there is technical details with concatenation of curves.
!!LIVE!! On Sunday May 26, 11:00am EST (USA & Canada) time. Can’t wait to see you there! Any questions I can ponder on already?
Hi, can you please suggest some material for Heisenberg group.
The standard book on the topic is: Capogna, Luca and Danielli, Donatella and Pauls, Scott D. and Tyson, Jeremy T., An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. But depending on you need and focus there may be other (possibly non-book) options. If you give more details I might find better references --Young measures
ua-cam.com/video/VE3YVS4i_Do/v-deo.htmlsi=lcwMHUmva8y--MKQ&t=871
Great videos, very clear explanations. Looking forward to poincare inequalities.
Hello sir! Can you please suggest books related to metric measure space
Lectures on Analysis on Metric Spaces by Juha Heinonen Sobolev Spaces on Metric Measure Spaces An Approach based on Upper Gradients Juha Heinonen Pekka Koskela Nageswari Shanmugalingam Jeremy T. Tyson Topics on Analysis in Metric spaces, Luigi Ambrosio, Paolo Tilli. First published in 2004, approximatly 130 pp. Easy to read? Yes. Exercises? Yes. Nice ones! Focuses on: Lipschitz curves/maps into metric spaces, geodesic problem, Sobolev spaces on metric spaces
@@YoungMeasures Thank you very much sir for suggesting these books. Appreciated!
Great video but your corollary is wrong... The condition is Hs(A)>0 (not Hs(A) finite)!
Yes. I think you are right! The least confusing way to state it is to assume both finite and positive. I will either cut it out or add clarification. Thanks.
thanks for the nice presentation! Are you aware of the generalization to the case where the domain of measurable function U(x) is unbounded?
Unfortunately, I am not aware of any references. But was boundedness used in any essential way?
I want to send you my thesis about area formula, I want your opinion
Sure. Please do.
I don't know why the comment with the link is being deleted ... Could I send it by any other mean?
@@YoungMeasures I have sent it via your email in the description of the channel
Of course. On channel descriptions you can find my emails.
@@YoungMeasures I sent it to your gmail. Thank you
Nice :)
Spent hours trying to understand my professor notes, you saved me!
Love your content, sir! They are really helping me a lot. Could you recommend some textbooks that can help me to better follow your videos? I have only done real analysis before, so a lot of definitions mentioned in your videos are totally unfamiliar, things like rectifiable paths and so on. That'll be much appreciated, thanks!
Thanks for the feedback! I have covered multiple topics, so if you tell me which playlist or topic you like most and also with what math you need to learn, I can find some titles for you.
Thank you I like the concreteness, I'm trying to get a feel for Lie algebra and Lie group from category theory that's rather abstract. Do you have any tips about how to imagine this exponential map itself intuitively in H1?
Around min 23 I explain that we identify G with g via exp map. This then means that in our special setting the exponential map is the identity! It is initially a bit difficult to live with this but you can go over the process to convince yourself that it is true! The real reason is Campbell Hausdorff formula.
Hi! Thank you for that video, I liked it very much. Then It would be interesting for me the further introduction to Newtonian-Sobolev spaces. Also I wander if it is possible to go from Newtonian-Sobolev space back to Sobolev space on a Riemannian manifold ?
Absolutely. Up to precision with a.e. representations, the two classes should coincide. How easy it is to prove depends on how you define the Sobolev functions between manifolds. The Newtonian-Sobolev spaces are also defined for maps into Banack space targets as well. I will refer to the book by Jeremy T. Tyson, Nageswari Shanmugalingam, Juha Heinonen, Pekka Koskela.
Very interesting videos, thank you! Is there a video talking about tangent measures?
I refer to Azzam’s channel for these. He has fantastic videos on GMT. ua-cam.com/play/PLp0TNqYe2DSEA8r9xYJnJCibS1pN-NQq3.html&si=M7aYBvhygsEaGkwt
help me please, i want to know covering lemma on doubling metric space, i can find in the metric space, but i cant find in the doubling metric space, can you help me ? pleaaaase im mathematic
I do not understand the question itself. Do you want a proof of the 5r covering lemma in metric setting?
VERY GOOD VIDEO. THANK YOU SIR
great video, thank you!
Congratulations on the class, very good. I have some questions, I hope you can help me. A Carnot distance was defined in H^1, do you know if at this distance H^1 is a complete space? Then you gave a "norm", Koranyi norm, in fact from your expression it is not homogeneous, but do you know if it is possible to induce a norm in H^1 that makes it Banach?
It is a complete metric space yes. For example because convergence with respect to carnot distance is equivalent to convergence as if in usual R^3. Do you mean if there is a norm on H^1 that is, say, bilipschitz to the Koranyi, and as a 3D vector space over R, H^1 becomes a Bsnach space? I have to say no, but dont have a simple readon why. Maybe the scaling would provide a contradiction. Because scaling that is consistent with Carnot geometry is different from the usual scalar product.
@@YoungMeasures That's right, the only thing we don't have for the Koranyi norm is the property ||ax|| = |a|.||x|| and to define Banach space, it is assumed that a norm satisfies such properties. My question is precisely whether I could either define a norm in H^1 that makes it Banach, or whether the definition of Banach space (but maintaining the known properties) can be relaxed to meet this case in which the norm is not homogeneous. Maybe this isn't trivial, or doesn't make sense, as I haven't found anything about it.
nicely explained! keep making videos please
oh thank you finally, i didn't find it elsewhere
Can you please give a page number with the full theorem referenced at the end of the video, about "if the projection has hausdorff measure zero then E is purely unrectifiable"? Glanced at simons and it isn't obvious which theorem is being referenced. Thank you this is a helpful video.
Will do soon.
Ok, Simon has some notes online, which is basically a newer version of his book's material. There in Remark 3.4 it is stated that: Fix an (orthogonal) basis of R^(n+m). If projection of a set S onto any of the hyperplanes given by span of n-many of these basis vectors has zero Hausdorff-n measure then S is purely H^n-unrectifiable.
If you know or can find a reference for that fact that in any rectifiable set with positive measure you can find a set with positive measure that is bi-Lip to a subset of Euclidean space with biLip constant as close to 1 as we wish, then this should not be difficult to see: Working locally, the projection of a part of the set onto a suitable tangent space (a hyperplane) must have positive measure. Because the set and its projection onto the tangent are (working on even smaller neighborhoods) extremely nearby, the projection of the set to the hyperplanes that are in general position with regard to that tangent plane, is pretty much the same projecting the set onto tangent plane first then projecting onto that hyperplane. Since projection onto the hyperplane does not vanish the area, we are left with a projection of S of positive area. (If you want to draw a picture, first draw a 2D manifold'ish set in 3D, i.e. S, then a tanget plane to it, the tangent plane projects nontrivially to one of xy, or xz or yz planes. )
Thanks for the video
Accoridng to wikipedia, its the Chebyshev ineq that you are talking about at 13:16 , but in the book Concentration inequalities by Boucheron, Lugosi & Massart they call it Markovs ineq in pag 19 Also note that in Boucheron et al, they use Mf(x) as the Median of the function and not the Maximal 🤷
Thank you for very clear references.
can you please make a video on the chapter 8 of Gilberg and trudingers book. I have a difficulty of understanding theorem 8.9, 8.10 and their proofs.
I am not the right person to do so :( Have you tried to see what those theorems claim in the simplest examples? Try increasingly more sophisticated examples and see how significant the claim is and why the proofs of them is not trivial.
Is one of the motivations for the field fact that a GH limit of Riemannian manifolds may fail to be manifold. When people say the limit is “non-collapsed” do they mean the limiting metric supports some sort of Poincare inequality as you described ?
I am really not an expert on RCD. Sorry. Please check online.
thank you for your work it has been very useful to me
Glad to hear that!
Thanks for question. At 21:15 I claim that because f is uniformly continuous on unions of reds it extends uniquely to closure of reds which is whole interval. This is a well known fact that uniform continuity allows extension. Because limits do not depend on particular approximating sequences.
You didn't really define f. How is it defined on the points not in the red parts?
I replied on a separate comment (by mistake:) Good question!
Really looking for it. I think a reference list/bibliography would be awesome, perhaps also something like "An introduction to Metric-Measure theory" or something👍👍
Sobolev Spaces on Metric Measure Spaces An Approach Based on Upper Gradients; Juha Heinonen, Pekka Koskela, University of Jyväskylä, Finland, Nageswari Shanmugalingam, University of Cincinnati, Jeremy T. Tyson, University of Illinois, Urbana-Champaign
so there is funding for people in US and Europe but not developing and poor countries, which are the ones that really need the funding??
I just remember USA being specified. Funding is available to anyone.
@@YoungMeasures oh I see, I must missed read the page then Thanks for the clarification 👍
Gem of a small channel. Thanks for this video, I had no idea this Notices existed. Very interesting, especially the What is...
Thanks for feedback. I always have second thoughts about if I should do such videos. Glad to hear they are useful. Introducing resources for advanced serious math was the objective of the channel from the beginning.
Oh no, you have the same block structure is Finland also for the courses? I hate this kind of structure for mathematics 😅 you never have enough time to really dive and understand the topics (we have the same in Sweden, and in Denmark)