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.

[josephavant]

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.

[xanthirudha]

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

[pythagorasaurusrex]

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 :)

[gentgjonbalaj]

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

[jsanch]

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.

[atanunath]

Wow, nobody explained these things so clearly. Brilliant.

[HJ-tfnw]

Awesome lecture, very clear and well motivated!

[insignia]

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!

[antoniolewis]

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

[TwinDoubleHelix]

These lectures are outstanding. Thank you.

[karimsouidi]

This is one of the best lectures ever !

[lokendrasunar]

Great dedicated professor.Very comprehensive lecture .Lucky
me.

[edithsmith]

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

[rahnumarahman]

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

[BlackEyedGhost]

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?

[xxqq]

What is the prerequisite for this course?

[CGMario]

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

[leanhdung]

Very interesting and inspiring lecture.

[alpistein]

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