Lecture 1: Topology (International Winter School on Gravity and Light 2015)

As part of the world-wide celebrations of the 100th anniversary of Einstein's theory of general relativity and the International Year of Light 2015, the Scientific Organizing Committee makes available the central 24 lectures by Frederic P Schuller.
Titled "A thorough introduction to the theory of general relativity", the lectures introduce the mathematical and physical foundations of the theory in 24 self-contained lectures. The material is developed step by step from first principles and aims at an audience ranging from ambitious undergraduate students to beginning PhD students in mathematics and physics.
Satellite Lectures (see other videos on this channel) by Bernard F Schutz (Gravitational Waves), Domenico Giulini (Canonical Formulation of Gravity), Marcus C Werner (Gravitational Lensing) and Valeria Pettorino (Cosmic Microwave Background) expand on the topics of this central lecture course and take students to the research frontier.
Access to further material on www.gravity-and-light.org/lectures and www.gravity-and-light.org/tutorials

[@josephavant8250]

This professor is EASILY one of the best I've ever seen - every student should be so lucky to study from such an articulate, patient, and clear instructor at some point in their academic career!

[@addemfrench]

I cannot get over how great his presentation is. The ideas are so crystal clear, the notation and board work so pretty and suggestive of the ideas they represent, all of it organized, and even balanced like a painting.

[@elliotnicholson5117]

He's utterly brilliant. :)

[@addemfrench]

Agreed, his lecture is inspiring.

[@boxerpop82]

His lectures are simply beautiful

[@xanthirudha]

this is amazing,i cant believe virtual learning is this promising

[@pythagorasaurusrex9853]

Great lecture! Wished I had such a competent professor when I studied math. I never really got it, cause lectures were bad. This here is explained easy and one can follow. What I like so much about topology is the fact that you don't need these annoying delta-epsilon-calculations to proove continuity :)

[@gentgjonbalaj8359]

Wow! Easily the best lecture I have ever listened to. Thank you!

[@antoniolewis1016]

This man crafts his lectures from diamonds. He even has board cleaners!

[@jsanch855]

I'm an Electronic Engineer, and I allways want to take a Course where you see Topology, Differential Geometry and Gravity, thenx, by the way, all those asking, what you need to know to understand this course, is just Set Theory and Read and Do Proofs, all the rest is explain.

[@HJ-tf9nw]

Awesome lecture, very clear and well motivated!

[@atanunath]

Wow, nobody explained these things so clearly. Brilliant.

[@insignia201]

I have never heard of the term "chaotic topology", I know I have heard it being referred to as a trivial topology or an indiscrete topology. Great lecture nonetheless!

[@TwinDoubleHelix]

These lectures are outstanding. Thank you.

[@muneer332]

The method of using board is amazing

[@lokendrasunar5457]

Great dedicated professor.Very comprehensive lecture .Lucky me.

[@tuneerchakraborty5836]

German Precision.

[@karimsouidi1]

This is one of the best lectures ever !

[@BlackEyedGhost0]

I learned: a) The power set (P) of a set (M) is the set which contains all subsets of that set. u∈P(M) u⊆M b) A topology (O) can be defined on a set (M) as a subset of the power set -i) a topology must contain the set (M) and the empty set. ∅,M∈O (∴{∅,M}⊆O⊆P(M)) -ii) the intersection of any two members of a topology must also be a member of the topology. (v∩u)∈O | u,v∈O -iii) the union of any number of members of the topology must also result in a member of the topology. Ui(u)∈O | u∈O (is there any reason it needs to be an indexed set rather than simply v∪u like the previous axiom?) c) Members of a topology are called open sets d) A set is closed if it's compliment (relative in M) is an open set e) A map (f) from set M to set N is continuous if the preimage (with respect to f) of every open set in N is an open set in M (obviously requireing a topology in both). ∀V∈O:preim(V)∈O f) If we have 2 maps (f:M->N and g:N->P) and they're both continuous, then the composition of the two is also continuous. g) A subset (S) of a set with a topology can inherit that topology by taking the intersection of the subset and every element in the topology. Os = {u∩S|u∈O} h) If you restrict a continuous map to a specific subset in the domain and inherit the topology, then the restricted map is still continuous. Nice synopsis for such a long video eh?

[@tuneerchakraborty5836]

+BlackEyedGhost Pretty good. In b) iii) The set is indexed because only a finite number of unions can be taken for the resulting set to be an open set. The finiteness of the index set is pertinent due a technicality regarding some peculiar properties of infinite sets. Apparently all the rules that apply to finite sets don't automatically translate to infinite sets. I could point you to a book if you'd want me to.

[@BlackEyedGhost0]

I'd love to be pointed to a book. And thanks for responding to that. I thought about it for a while and couldn't come up with a reason.

[@teriyakichicken1848]

I just finished pre cal and all this math is sooo daunting, I wonder if it will ever end

[@jaredtramontano5249]

+BlackEyedGhost Of Course the index set is needed! What you've written only says the topology contains finite unions of members... It must contain arbitrary unions

[@fortoday04]

I think d) is the reverse?

[@viveknsharma]

Fantastically Well-Planned!

[@rahnumarahman6227]

Can anybody kindly tell me what literature is being followed here.......the lecture is great but It helps having a literature reference that you can look at.

[@edithsmith9257]

Totally brill - and his enthusiasm is making millionaires of the blackboard chalk oligarchs.

[@abstract835]

such clearity=========

[@CGMario]

Thank you for posting this! It's very helpful!

[@xxqq96]

What is the prerequisite for this course?

[@alpistein]

The lecture was great, but I got annoyed very quickly over how many curly braces I had to draw in my notes :P

[@leanhdung9848]

Very interesting and inspiring lecture.

[@phillipwong4283]

Very good... Remind me of college days.

[@drlangattx3dotnet]

Terrific instructor. Thank you sir.

I want him to be my lecturer :(, he is amazing!

[@muneer332]

SUPERB SIR SUPERB

[@CarlosGonzalez-rg6ht]

Wonderful lecture, thank you

[@christerholmsten2060]

Thanks for an excellent lecture, what literature is used during the course if there is any?

[@ErnestYAlumni]

Christer Holmstén So far, I've found that @9400754094) by Norbert Straumann to be the closest in spirit to his lectures, but Schuller's video presentation here is the best and most clear and well-organized (solid?) presentation out there on General Relativity, vs. lecture note, textbook, other media. I really think it's even reference worthy. By the way, I write notes up about these lectures here: drive.google.com/file/d/0B1H1Ygkr4EWJbF9mQXluQVVQTDg/view?usp=sharing and on my wordpress.com blog: ernestyalumni.wordpress.com/2015/05/25/20150524-update-on-gravity_notes-tex-pdf-notes-and-sage-math-implementation-of-lecture-1-tutorial-1-topology-for-the-we-heraeus-international-winter-school-on-gravity-and-light-2015/

[@theleastcreative]

Did anyone attend this and still have the questions from the tutorials?

[@IgorItkin]

Brilliant lecturer! Just brilliant

[@TwoonyHorned]

Point set topology is just a matter of language.

[@ADA4750]

Thanks.Very Interesting.

[@deleogun1717]

thanks

[@吉田新一-l5w]

he is genius

[@rahaal2590]

fantastic!

[@taraspokalchuk7256]

Can such f() be defined that maps M to N and also the chosen topology on M to another topolgy on N?

[@addemfrench]

Sure, we typically define f from M to N and then look for various properties. The most important one is whether the inverse function maps open sets to open sets.

[@taraspokalchuk7256]

addemfrench thanks)

[@RalphDratman]

Superb lecture

[@adamlantos2319]

Anybody knows the prerequisites for these videos?

[@adamlantos2319]

***** yeah but you have to know the actual prerequisites before you start searching the internet

[@NSBassler]

+adam landos It is a good idea to have had some basic university level math courses like basic linear algebra and calculus courses. However they are not strictly required. You should also be able to make due with high school level mathematics with some difficulties.

[@ingifreyr]

+adam landos I'm guessing this is a graduate level course, so a BS in physics or mathematics should suffice.

[@cliliv]

I think a basic knowlegde on sets, differential equations and calculus would be enough. These lectures are preparation for a bigger and more richer course on General Relativity, I think. So, if you want to learn more about GR, it would be a great start :)

[@jonabirdd]

The math part cannot be understood without exposure to variational calculus (just the Euler-Lagrange eqns), multivariate calculus, and vector calculus on the level needed to understand maxwell's eqns, for example. The physics part requires exposure to special relativity, and again, some lagrangian mechanics would help.

[@NothingMaster]

Ja Ja --- a great teacher.

[@taraspokalchuk7256]

what is the difference between U(alpha) and U? Is U(alpha) a set of all UєO?

[@UnforsakenXII]

What mathematics should I know prior to starting this course?

[@NoNamedNobody692]

I would say I have a thorough understanding in both Single and Multivariable Calculus, Differential Equations, Non-Euclidian Geometry and possible a course in Differential Topology. And then comes the physics.... have a grand ol time. Lol.

[@777shadowdragon]

+SFLOVER94 what if the only prerequisites ive taken is youtube? 😂

[@robertwilsoniii2048]

Linear Algebra and Analysis; I recommend Linear Algebra by Levandosky, Principles of Mathematical Analysis by Rudin and Vector Calculus by Marsden and Tromba. You can skip these and go straight to Advanced Calculus by Loomis and Sternberg if you want too, this book will cover the content of this class though.

[@IndianHeathen1982]

Calculus, linear algebra and some exposure to abstract algebra probably.

[@robertwilsoniii2048]

***** Calculus is not necessary at all.

[@MrAkashvj96]

Superb

[@MarkMattingleyScott]

Freddy! Moin Moin!

[@lucasdarianschwendlervieir3714]

Ah, the Eintein's view on gravity (as opposed to the Feynman view).