My friend Gavin…I know, right! I don’t understand paying $150K for an education which is just a piece of paper. You can learn anything you want with the device in your hand right now.
It was after seeing your comment and reading the channel name that I realised that this is really the man, the myth, the legend, lord Richard. Hands down the greatest discovery of today, it made my day.
Thank you so much for doing this professor!!! I believe that maybe not everyone who pursues mathematics ends up becoming a great mathematician, but a great mathematician can come from anywhere. Through this platform, you're reaching almost every part of the world, I hope somebody who needs this finds it!!
I am a third year university student, I live in Vietnam. Thank you so much professor for making it easy for me to understand symmetry groups and properties of homomorphisms.
I'm just here to learn as a hobby. This lecture helped to clarify some things about groups (and representations) for me, it was really well presented. A few pennies dropped into place.
Greetings, would you suggest starting to learn group theory from these series? I would also like to learn as a hobby, didn't start yet; but I wonder if things may get too complicated.
@@ErhanTezcan Hi Erhan. This series is wonderful, but highly advanced. If learning as a hobby, and not much mathematics background, I'd first go for popular / laymans books on the topic, which sometimes have titles with "Symmetry" them. Maybe others here might be able to advise too. Also, John Conway has a nice video where he discusses groups. e.g. ua-cam.com/video/lbN8EMcOH5o/v-deo.html
@@ErhanTezcan Hi, a bit late here, but I would suggest having rigorous understandings of basic group theory before attempting this playlist. I talk from experience, because I have started here in the past, and couldn't learn anything. The reason is that the way he presents group theory is very abstract, and intuitive, but also because of this, it can become confusing.
I am proud to say I understand like 80% of this. Going to continue with this series and like rigorously concentrate. I want to understand this as deeply as possible. (12th grader)
Depends on what one means by axiom(atisation)... In a naïve set theoretical sense, yes, one should stipulate closure of multiplication, but the minimal formalism would be via model-theoretic via first order predicate logic, in which case function symbols are interpreted in structures to be n-ary functions on the underlying set of the structure. To be precise, multiplication is a binary function in the language of groups, and in any group G (a structure in the language of groups) the interpretation *_G is a function G x G -> G. The advantage of the model-theoretic approach vis-à-vis closure properties is that it simplifies axiomatic theories greatly and since one does not usually speak of an algebraic structure in which a basic operation is not closed or can be interpreted by a closed operation, we don't lose anything.
These lectures are great! Really appreciate it. And also I would like to share a math study message here for anyone who might be interested: I would like to read by D. L. Johnson. And I would like to ask if anyone is interested to read together? Because having a reading partner will make study more fun, especially when go through detail proof together. You are also welcome to share this message to anyone who is interested. Thank you so much!
I’m a undergraduate math student.can somebody help me with math？I‘m interested in Math.But I‘m not good at learn it by myself.Just like choose what lesson and choose what book. I come from China，I’d like to meet some friends who love math！❤
Visual Group Theory by Nathan Carter is a beginners level (one can do it with a 5 year old offspring and yet it still goes all the way to Sylow theorems and Galois groups. M.A. Armstrong's Groups and Symmetry is much more concise and demands some background. William Paulsen's Abstract Algebra advantage is a set of programmes in SageMath, which let's you explore more complicated groups without writing away a couple of pencils. At last Serge Lang's Abstract Algebra is a text flying in an altogether more rarefied atmosphere, but were I back to 14-19 years old, I would jump for this one and read around as questions arise. The latter seems to be an apt advice anyway.
I have watched this over and over again, but I can't figure out what he means by a group being the symmetries of an object. I know what symmetry is(and what a group is)...
are you asking about Cayley’s theorem, every group is isomorphic to a symmetry (permutation) group ? for example given a group G, let G be “the object”. Then the group G, its elements g are symmetries of G, via the symmetry operation G -> gG. (P.S. a symmetry is a mapping of a set to itself, in this case the set is G)
INTRO TO GROUP THEORY Groups are collections of symmetries More precise then the half symmetries of the body More ordered than the buzzing symmetries of a body of people. People muck about stationarily, rooted, up- oriented Flat and few in comparison to insects - Indeed, there is no bijection from the symmetries of people To the symmetries of insects and their flight formations - they are few and their voices Are a moderate dissonance with small amplitude. In the study of groups We search for isomorphisms and meaning How one group can be perfectly the same As another, but with more than one interpretation. Is it human to look at facets of one thing Set them carefully into an incomplete set And believe in a renaming that makes The world digestible? Or do we mathematicians know that there is something That makes the grouping of (groups of symmetrical thoughts (of unsymmetrical beings)) Wholesome in itself? Out of all this I made absolutely nothing.
It is truly astounding that anybody in the world can take a course taught by a fields medalist for free! What a time to be alive.
Amen
My friend Gavin…I know, right! I don’t understand paying $150K for an education which is just a piece of paper. You can learn anything you want with the device in your hand right now.
@@jmw1500 I know. It really is sad. The hoops one must jump through before you are deemed worthy to join the conversation.
It was after seeing your comment and reading the channel name that I realised that this is really the man, the myth, the legend, lord Richard. Hands down the greatest discovery of today, it made my day.
I stumbled upon this channel last night and couldn't believe my eyes. Thank you very much professor!
This is one of the greatest things I've found lately, great mathematician
s make more, great mathematician
s
Thank you so much for doing this professor!!! I believe that maybe not everyone who pursues mathematics ends up becoming a great mathematician, but a great mathematician can come from anywhere. Through this platform, you're reaching almost every part of the world, I hope somebody who needs this finds it!!
This is one of the best thing I have found in you tube. Thank you ❤️
It is so truly nice to be taught group theory by a famed expert in group theory, Richard Borcherds himself!
Even tho I’m already familiar with basic group theory, this „lecture“ gives a nice perspective. Excellent „lecture“!
I am a third year university student, I live in Vietnam. Thank you so much professor for making it easy for me to understand symmetry groups and properties of homomorphisms.
How have your studies been going?
How I wish I could see your excellent videos when I was learning group theory.
Moments like this when I'm glad to be living in the age of UA-cam..
I'm just here to learn as a hobby. This lecture helped to clarify some things about groups (and representations) for me, it was really well presented. A few pennies dropped into place.
Greetings, would you suggest starting to learn group theory from these series? I would also like to learn as a hobby, didn't start yet; but I wonder if things may get too complicated.
@@ErhanTezcan Hi Erhan. This series is wonderful, but highly advanced. If learning as a hobby, and not much mathematics background, I'd first go for popular / laymans books on the topic, which sometimes have titles with "Symmetry" them. Maybe others here might be able to advise too. Also, John Conway has a nice video where he discusses groups. e.g.
ua-cam.com/video/lbN8EMcOH5o/v-deo.html
@@ErhanTezcan if you want to learn it deeply
I suggest you taking an algebra course in the college or grabbing a textbook of modern algebra
@@ErhanTezcan Hi, a bit late here, but I would suggest having rigorous understandings of basic group theory before attempting this playlist. I talk from experience, because I have started here in the past, and couldn't learn anything. The reason is that the way he presents group theory is very abstract, and intuitive, but also because of this, it can become confusing.
thanks dear professor for opening my eyes on math .....
Sir it is truly a great privilege to watch your lectures
This is awesome, thank you so so much for taking the time to allow us to learn from one of the greatest minds alive.,
finding these lectures made my day
This playlist is a really nice *group* of videos 👍
But seriously, I think the approach of using more examples than proofs is an outstanding one!
Thanks for sharing your incredible lecture Prof. Borcherds!
So symmetries are structure preserving maps where structure can be a algebraic or geometric in nature
I am proud to say I understand like 80% of this. Going to continue with this series and like rigorously concentrate. I want to understand this as deeply as possible. (12th grader)
I relate man
7:34 The figure of symmetric group is helpful to write down an element of symmetric group without annoying notations.
I’m not a mathematician, but this is an excellent presentation!
Many thanks from an enthusiastic 1.year student
Thank teacher, this is awesome video for now I understand ALGEBRA ABSTRACT, thanks
thanks professor!
This is great. Thank you very much!
Thank you for this amazing contribution.
It blows my mind that GL(n, field with one element) is S(n). Every time I hear "set" and "vector space" I now have a link.
Thank you professor
super neat
20:00 technically, the closure of the group composition is a fourth property you need
Depends on what one means by axiom(atisation)... In a naïve set theoretical sense, yes, one should stipulate closure of multiplication, but the minimal formalism would be via model-theoretic via first order predicate logic, in which case function symbols are interpreted in structures to be n-ary functions on the underlying set of the structure. To be precise, multiplication is a binary function in the language of groups, and in any group G (a structure in the language of groups) the interpretation *_G is a function G x G -> G.
The advantage of the model-theoretic approach vis-à-vis closure properties is that it simplifies axiomatic theories greatly and since one does not usually speak of an algebraic structure in which a basic operation is not closed or can be interpreted by a closed operation, we don't lose anything.
Yeah, maybe he just forgot. Was about to write the same thing but found your comment.
it is not necessary to give an explicit axiom of closure if one defines the term "binary operation" to be a function G^2 -> G, as is often done.
this series for undergrad or grad? first time taking group theory. should i watch?
You should, just like everybody should read Dickens or Shakespeare, even without a degree of literature.
At least, I enjoy a lot when watching it :D
lol I'm too excited I found this channel
These lectures are great! Really appreciate it.
And also I would like to share a math study message here for anyone who might be interested: I would like to read by D. L. Johnson. And I would like to ask if anyone is interested to read together? Because having a reading partner will make study more fun, especially when go through detail proof together. You are also welcome to share this message to anyone who is interested. Thank you so much!
Student from malaysia here! :)
based
I’m a undergraduate math student.can somebody help me with math？I‘m interested in Math.But I‘m not good at learn it by myself.Just like choose what lesson and choose what book. I come from China，I’d like to meet some friends who love math！❤
Is this idea of symmetries just for intuition?
Can anyone recommend a good book to accompany this course?
Visual Group Theory by Nathan Carter is a beginners level (one can do it with a 5 year old offspring and yet it still goes all the way to Sylow theorems and Galois groups. M.A. Armstrong's Groups and Symmetry is much more concise and demands some background. William Paulsen's Abstract Algebra advantage is a set of programmes in SageMath, which let's you explore more complicated groups without writing away a couple of pencils. At last Serge Lang's Abstract Algebra is a text flying in an altogether more rarefied atmosphere, but were I back to 14-19 years old, I would jump for this one and read around as questions arise. The latter seems to be an apt advice anyway.
@@Suav58 thanks for the comprehensive answer :-)
I'd recommend Group Theory in a Nutshell by A. Zee if you have a physics bent
I have watched this over and over again, but I can't figure out what he means by a group being the symmetries of an object. I know what symmetry is(and what a group is)...
are you asking about Cayley’s theorem, every group is isomorphic to a symmetry (permutation) group ? for example given a group G, let G be “the object”. Then the group G, its elements g are symmetries of G, via the symmetry operation G -> gG. (P.S. a symmetry is a mapping of a set to itself, in this case the set is G)
perfect cliffhanger
INTRO TO GROUP THEORY
Groups are collections of symmetries
More precise then the half symmetries of the body
More ordered than the buzzing symmetries of a body of people.
People muck about stationarily, rooted, up- oriented
Flat and few in comparison to insects -
Indeed, there is no bijection from the symmetries of people
To the symmetries of insects and their flight formations -
they are few and their voices
Are a moderate dissonance with small amplitude.
In the study of groups
We search for isomorphisms and meaning
How one group can be perfectly the same
As another, but with more than one interpretation.
Is it human to look at facets of one thing
Set them carefully into an incomplete set
And believe in a renaming that makes
The world digestible?
Or do we mathematicians know that there is something
That makes the grouping of
(groups of symmetrical thoughts (of unsymmetrical beings))
Wholesome in itself?
Out of all this
I made absolutely nothing.
i really wish to give 100 likes below this video, but youtube only allow me to give 1.
Ok. When he moved to numbers I lost a grasp xd
I am so lucky. :D
I am trying to get a fields medal do you have any tips sir?
Doing actual math and not aiming for fields medal.
@@Nnm26 it kinda reminds me of the young youtubers that goes for glory seeing others youtubers who did go on ytb with passion
where my Royal Holloway people at?