It's called 'public outreach', and it's considered a minor part of the work. In this case, for Borcherds, it's an unbelievable service, because of how obscure commutative algebra and schemes are outside of the small clique of algebraic geometers.
@@annaclarafenyo8185 The typical presentation of essentially reading and explaining what is in the book with drawing pictures can be helpful but still hard to get through, but Borcherds takes explanation to a different level.
It is not public outreach as in: I am promoting myself, or my department or my university. He started it because Covid and the online system for lectures in Bekerley at that time had issues and he decided to put them on youtube (his exact words). What it is not clear is why he decided to carry on and populate a channel with wonderful graduate and advanced undergraduate lectures in topics not available anywhere else (apart of lecture notes in text form). He didn't answer that question yet. He only said: I'm surviving right now like everybody else. And nobody knows what will happen when the campuses are fully open again. These are lectures, and they are a bit less fully detailed than in some classrooms but not so much so. In a British classroom they are absolutely fine. It points you to the main topics without digging too much, it creates the criteria to be able to learn yourself, as now you have the gist about what it is critical to study and what is not. What we don't have is the exercises and the tutorials to go back with our doubts. That will be the full experience.
I just want to express how much I'm loving this lecture sequence. Just as I had decided to brush up on complex analysis by following Freitag's book, someone mentioned that you were doing a series of lectures on complex analysis and I knew I just had to check it out.
I took a DiffEq class 5 years ago, and for linear homogeneous equations with constant coefficients, we were taught to use the characteristic polynomial. I didn't think much of it and just treated it like a magic box. It never crossed my mind that we get it from substituting y = exp(lambda*x) and then dividing the equation by this exp. Seeing it now makes it obvious and makes me feel like an idiot.
Thank you for these videos, you explain it so much better than any other sources I have come across. The extra detail is helpful as I come from a software engineering background and it is difficult to find material that explains advanced topics in an accessible way.
"We can see that it is onto, because we know every number has polar coordinates" Actually, showing every complex number has polar coordinates isn't trivial. It's a really good challenge, since there are multiple ways to proof it.
Hello professor Borcherds, I know that you take suggestions for future courses in polls, but still I'm very much interested (and many others as well) in understanding the mathematical framework for Eric Weinstein's theory of geometric unity. If you are familiar with Weinstein's work then please consider making some videos for us mere mortals to understand it. Kind regards. PD: it's such an honor to have you in the TH-cam maths community.
I'm not familiar with geometric unity, and I was unable to find any published accounts of it beyond a few newspaper articles from 2013, so unless I find something else I will not be able to say anything about it.
@@richarde.borcherds7998 a paper will be published on the first of april according to Eric. I will comment on your latest video when any information comes out. Thank you so much professor Borcherds!
@@leandrocarg physical theories can't really be approached from a pure math perspective. The math in that context is just a tool, and understanding it is not enough to understand the physical significance of the theory, to temper your expectations a bit.
@@leandrocarg The framework for it ought to be quantum field theory, but it is unclear whether the theory is actually well-defined, see the critique by Nguyen and Polya
Hi Dr. Borcherds! You mention in the lecture that the exp map from R+ to R× is injective but not surjective. I was under the impression that this was an isomorphism of groups, because log is a precise inverse of exp so long as the input is nonzero, and the underlying set of R× is R\{0}. Am I wrong? Are there weird properties of exp and log I don't understand? As always, thanks for the lectures; they are a much more productive thing to watch before bed than almost any other content.
I think exp does not reach negative output for any real numbers, so it is not surjective. If you take R^* = R \ {0} that is. However, perhaps there is also the multiplicative group of positive real numbers (which is actually the subgroup of squares in R^*) and exp is an isomorphism between R^+ and that group, and log is it’s inverse.
@@constantijndekker8343 you're absolutely right, exp can't reach negative values. How silly of me! So it is an isomorphism to the posotive half, but only surjective to the whole group. Cool!
I didn't understand what "kernel" means If it means the preimage of the identity element, shouldn't the exponential map from reals under addition to reals under multiplication have kernel {0}? Since e^0 = 1, the identity of reals under multiplication
didn't he mention the kernel was {2nπi | n ∈ Z}? (well, this is equivalent to the statement used in the video) Am I misreading the main comment which seems to assume a one-to-one mapping? 0 is included in this mapping, but it is not the entire kernel due to the non-uniqueness of the inverse
![เปิดใจ "อาจารย์เบียร์ คนตื่นธรรม" อดีตเคยสุดทุกทาง ถ้าใช้หนี้หมดไปบวชแน่ l [Nickynachat]](http://i.ytimg.com/vi/IxeE0xVhy08/mqdefault.jpg)
![The most beautiful equation in math, explained visually [Euler’s Formula]](/img/n.gif)
Making videos such as this should be regarded as legitimate mathematical work. Academia shouldn't be solely about publishing papers.
It's called 'public outreach', and it's considered a minor part of the work. In this case, for Borcherds, it's an unbelievable service, because of how obscure commutative algebra and schemes are outside of the small clique of algebraic geometers.
@@annaclarafenyo8185 The typical presentation of essentially reading and explaining what is in the book with drawing pictures can be helpful but still hard to get through, but Borcherds takes explanation to a different level.
It is not public outreach as in: I am promoting myself, or my department or my university. He started it because Covid and the online system for lectures in Bekerley at that time had issues and he decided to put them on youtube (his exact words). What it is not clear is why he decided to carry on and populate a channel with wonderful graduate and advanced undergraduate lectures in topics not available anywhere else (apart of lecture notes in text form). He didn't answer that question yet. He only said: I'm surviving right now like everybody else. And nobody knows what will happen when the campuses are fully open again.
These are lectures, and they are a bit less fully detailed than in some classrooms but not so much so. In a British classroom they are absolutely fine. It points you to the main topics without digging too much, it creates the criteria to be able to learn yourself, as now you have the gist about what it is critical to study and what is not.
What we don't have is the exercises and the tutorials to go back with our doubts. That will be the full experience.
This channel is just so much fun. It's an island of beauty in this sea of boring videos called TH-cam
Thanks very much Prof. Borcherds, you're an amazing teacher! We're very lucky that you've produced these videos.
I just want to express how much I'm loving this lecture sequence. Just as I had decided to brush up on complex analysis by following Freitag's book, someone mentioned that you were doing a series of lectures on complex analysis and I knew I just had to check it out.
I took a DiffEq class 5 years ago, and for linear homogeneous equations with constant coefficients, we were taught to use the characteristic polynomial. I didn't think much of it and just treated it like a magic box.
It never crossed my mind that we get it from substituting y = exp(lambda*x) and then dividing the equation by this exp.
Seeing it now makes it obvious and makes me feel like an idiot.
This can also be explained with umbral calculus. You get make these differential equations discrete, and even directly turn them into polynomials.
Thank you for these videos, you explain it so much better than any other sources I have come across. The extra detail is helpful as I come from a software engineering background and it is difficult to find material that explains advanced topics in an accessible way.
A pleasure, as always
"We can see that it is onto, because we know every number has polar coordinates"
Actually, showing every complex number has polar coordinates isn't trivial. It's a really good challenge, since there are multiple ways to proof it.
Hello professor Borcherds, I know that you take suggestions for future courses in polls, but still I'm very much interested (and many others as well) in understanding the mathematical framework for Eric Weinstein's theory of geometric unity. If you are familiar with Weinstein's work then please consider making some videos for us mere mortals to understand it. Kind regards.
PD: it's such an honor to have you in the TH-cam maths community.
I'm not familiar with geometric unity, and I was unable to find any published accounts of it beyond a few newspaper articles from 2013, so unless I find something else I will not be able to say anything about it.
@@richarde.borcherds7998 a paper will be published on the first of april according to Eric. I will comment on your latest video when any information comes out. Thank you so much professor Borcherds!
@@leandrocarg physical theories can't really be approached from a pure math perspective. The math in that context is just a tool, and understanding it is not enough to understand the physical significance of the theory, to temper your expectations a bit.
@@leandrocarg The framework for it ought to be quantum field theory, but it is unclear whether the theory is actually well-defined, see the critique by Nguyen and Polya
Hi Dr. Borcherds! You mention in the lecture that the exp map from R+ to R× is injective but not surjective. I was under the impression that this was an isomorphism of groups, because log is a precise inverse of exp so long as the input is nonzero, and the underlying set of R× is R\{0}.
Am I wrong? Are there weird properties of exp and log I don't understand?
As always, thanks for the lectures; they are a much more productive thing to watch before bed than almost any other content.
I think exp does not reach negative output for any real numbers, so it is not surjective.
If you take R^* = R \ {0} that is. However, perhaps there is also the multiplicative group of positive real numbers (which is actually the subgroup of squares in R^*) and exp is an isomorphism between R^+ and that group, and log is it’s inverse.
@@constantijndekker8343 you're absolutely right, exp can't reach negative values. How silly of me! So it is an isomorphism to the posotive half, but only surjective to the whole group. Cool!
Small typo in 23:11; I think the equation should be: a \lambda^2 e^{\lambda x} + b \lambda e^{\lambda x} + c e^{\lambda x} = 0.
Many thanks
Nice one. Thanks.
I didn't understand what "kernel" means
If it means the preimage of the identity element, shouldn't the exponential map from reals under addition to reals under multiplication have kernel {0}?
Since e^0 = 1, the identity of reals under multiplication
Yes. If I said something else it was a slip of the tongue.
@@richarde.borcherds7998 ok, thanks mr. Borcherds
didn't he mention the kernel was {2nπi | n ∈ Z}? (well, this is equivalent to the statement used in the video)
Am I misreading the main comment which seems to assume a one-to-one mapping?
0 is included in this mapping, but it is not the entire kernel due to the non-uniqueness of the inverse
yeeeeeee
Me looking at the thumbnail: explosion, cost tons.
I really hope the person who disliked this video gets professional help
chemists use "log" for logarithm in base 10
smh