Topology & Geometry - LECTURE 01 Part 01/02 - by Dr Tadashi Tokieda

What people said

Recommends quapka · 2015-05 · Mathematics Stack Exchange · on “Proofs of theorems, where picture is sufficient”

A while ago I have had the pleasure to come across those lectures [this video] of Topology & Geometry by Dr Tadashi Tokieda (I do recommend watching at least the first lecture, both parts). My question is based primarily on the second part [LINK] . In few words my question is about Proofs done by picture . During my studies, I must say, that I was not ( am not ) very motivated to draw/sketch the problems we faced (face). I think this is related to the fact, that coming from high school students are not used to rigorous proofs/arguments and tend to prove something by giving, although general, but not sufficient example by picture . I remember this happening during the class of geometry, e.g. discu

Recommends quapka · 2015-05 · Mathematics Stack Exchange · on “Textbooks for visual learners”

@Rzeta although you might not be interested in Topology and Algebra and you specifically asked for books, I will still link you to few videos. Maybe it will interest you in the future: Tadashi Tokieda's Topology/Geometry lectures [this video] and Nathan Carter's Visual Group Theory - video [LINK] and also as a book [LINK] .

Recommends iadvd · 2015-08 · Mathematics Stack Exchange · on “If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold?”

I am reading some basic context books about topology (i.e. The Poincaré Conjecture [LINK] , by Donal O'Shea between others) and following this open Topology and Geometry video lectures of the brilliant professor Tadashi Tokieda [this video] in the African Institute for Mathematical Sciences (for beginners in the matter, if you have time I would suggest you to have a look to them!). I would like to ask the following question: Is every $n$-manifold the boundary of an $(n+1)$-manifold? Is every compact $n$-manifold the boundary of a compact $(n+1)$-manifold? Thank you! p.s. This question was rewritten according to the suggestions in the comments and Meta ( here [LINK] ), I hope now will be more accu

Recommends iadvd · 2015-08 · Mathematics Stack Exchange · on “Gluing diagrams: is it possible to glue a surface with itself in the same point? how is the diagram drawn?”

I am adding the answer I received by email from Doctor Tadashi Tokieda, he is the director of studies in mathematics at Trinity Hall, University of Cambridge, some bio here [LINK] and here [LINK] , I was following his Topology and Geometry open lectures at Youtube for the AIMS [this video] , so I dared to send him an email yesterday (I did not expect an answer, just tried) and received his answer!! (thank you very much Doctor Tokieda, it was a honor!) He sent me a very nice easy to follow explanation of the concepts, so I am transcribing exclusively the explanation here as a complement (more adapted to layman terms) to the good answer of @AloizioMacedo. The drawings are mine, so I apologize if th

Recommends iadvd · 2016-09 · Mathematics Stack Exchange · on “Can A Knot Be Tied In Anything Other Than 3 Dimensions?”

A n-dimensional knot can be untied in a n+1-(or more)dimensional space: you have at least one more dimension and the threads can be moved of position through that extra dimension without colliding with the rest of the threads of the knot remaining in the n-dimensional space (since only that part has been moved into the n+1 dimension and the rest has not). Once you move a thread in the n+1-dimension then you can put it back in another position in the original n-dimensional space, untying the knot. And well of course you can "keep" the knot "tied" in a n+1 dimensional space (and still will be tied for "somebody" looking at it in the same n-dimensional space where the knot exists) if you do not

Recommends Clemens Bartholdy · 2021-06 · Mathematics Stack Exchange · on “Advice for which "advanced math" topic for a self-taught student to start with? (number theory, group theory, probability, etc)”

Disclaimer: Most of what I suggest is informal and applied mathematics but I think is still good mathematics because it sets up as a primer for when you will actually do those topics in university (?). Hi! I too found myself in this position many times. There are a few routes you can take once you are done with the school mathematics (I am assuming you have done the NCERT syllabus). I'll try to mention all of them: Finish vector calculus and multivariable calculus with this animated lecture series by professor Greist [LINK] . I suggest doing it with a standard book for solving problems. This will open up a lot of door into 'applied' mathematics books. Visual Complex Analysis by Tristan Needh

Recommends Eden Zane · 2022-06 · Mathematics Stack Exchange · on “Is there a good set of lectures or notes on topology for studying characteristic classes?”

I was watching the lectures of Dr. Tadashi Tokieda on "topology and geometry" [this video] , which was pretty amazing that I finished all the the lectures in 3 days. What particularly helpful was the pictures he drew on the black board. Whenever he describe a homeomorphism, he drew pictures of intermediate stages of the deformation, that helped me a lot. Towards the end of the lecture 13 [LINK] he talked about twistedness of a manifold and how "characteristic classes" can be used to measure how twisted a manifold is. That was exactly what I am looking for quite some time on youtube but failed to get a good video to learn about. But to my unlucky, that was only a mention of "characteristic classes

Recommends Clemens Bartholdy · 2022-06 · Mathematics Stack Exchange · on “Book recommendation for geometric intuition”

Also this playlist on Algebraic topology [this video] maybe of interest (It should mostly be accesible at HS level, some part will be confusing, just skip over it for the time being)

Cites Chill2Macht · 2016-08 · Mathematics Stack Exchange · on “Why does the only non-orientable surface embeddable in $\mathbb{R}^3$ have boundary?”

As far as my understanding goes, if one has a closed (compact) surface and "cuts out" a disk, then the resulting object has a non-empty/non-trivial boundary. It can be shown that any non-orientable surface without boundary cannot be embedded into $\mathbb{R}^3$ (e.g. A non orientable closed surface cannot be embedded into $\mathbb{R}^3$ [LINK] , [LINK] ). However, if we allow surfaces with boundary, then we can find a non-orientable surface which does embed into $\mathbb{R}^3$ , the Möbius strip. As I understand, it can be formed by cutting out a disk from the real projective plane $\mathbb{RP}^2$ . (Which corresponds to the fact that the boundary of both the disk and of the Möbius strip is