Mathematics - 2020-05-15 (page 1 of 3)

Balarka Sen

00:05

@Thorgott Ya that's correct

Thorgott

nice

but I still haven't figured out why this implies your vector field is smooth

restricting the differential to the subbundle gives a bundle map that is a fiberwise isomorphism, but itself not invertible, so it's still awkward

Balarka Sen

It actually gives a smooth bundle isomorphism $F : (\ker df)^\perp \to M \times \Bbb R$, $F(x, v) = (x, df(v))$

As bundles over $M$

Thorgott

ah, that looks like the map I was missing

Balarka Sen

Therefore define a smooth section $s : M \to (\ker df)^\perp$ of the first bundle by $s(p) = (F^{-1} \circ c)(p)$ where $c : M \to M \times \Bbb R$ is the constant section $c(p) = (p, 1)$.

As $(\ker df)^\perp \subset TM$ is a subbundle of the tangent bundle, $s$ defines a section of the tangent bundle too (composing with the subbundle inclusion), which is our vector field $(df)^*(d/dt)$

So in this formalism smoothness of the vector field is automatic.

Thorgott

01:07

$F$ is clearly a bundle map and a fiberwise linear isomorphism. I can also see that $F$ is smooth, but I can't see why $F^{-1}$ is smooth (this is the same thing I was struggling with earlier, essentially).

Balarka Sen

Bundle isomorphisms are diffeomorphisms

Thorgott

But I don't know it's a bundle iso if I don't know the inverse is smooth in the first place

(I'm taking "bundle isomorphism" to mean bundle map with an inverse bundle map)

Balarka Sen

This is a language issue. Bundle isomorphism to me just means a smooth bundle homomorphism which is a fiberwise isomorphism. This is the same as your definition, small exercise

It's just the fact that inverse of an invertible matrix varies smoothly with the matrix, nothing fancy

These things you can sit down and figure out on your own, they're really basic :P

Thorgott

That can't be true, a smooth bundle map that is a fiberwise isomorphism doesn't even need to be bijective

Balarka Sen

Bundle homomorphism fixes the base for me

$E_1, E_2$ two vector bundles over the same base $B$, $f : E_1 \to E_2$ a smooth bundle map over $B$ taking the fiber over $b$ in $E_1$ to the fiber over $b$ in $E_2$ isomorphically for all $b \in B$. Then $f$ is a diffeomorphism

Thorgott

01:14

ok, that sounds useful, I'll try it

Balarka Sen

this was the whole point of the construction of $F$

Thorgott

yeah, that makes sense

but I wasn't aware of the useful fact and struggled accordingly

Balarka Sen

well now u do

Why What Where When How

01:32

math.stackexchange.com/questions/3675476/… Would anyone like to try this?

Thorgott

01:52

@Balarka Ok, let $\pi_1,\pi_2$ be the projections. Let $p,q\in E_1$ s.t. $f(p)=f(q)$. Then $\pi_1(p)=\pi_2(f(p))=\pi_2(f(q))=\pi_1(q)$, so $p,q$ live in the same fiber of $B$ in $E_1$, but $f$ is bijective when restricted to that fiber, hence $p=q$, i.e. $f$ is injective. Surjectivity of $f$ is obvious since each element of $E_2$ lies in some fiber of $B$.
Choose local trivializations $\varphi_i\colon\pi_i^{-1}(U)\rightarrow U\rightarrow\mathbb{R}^n,\,i=1,2$ for some open $U\subset B$ (same $U$ for both WLOG cause otherwise take intersection). Then $\varphi_2\circ f\vert_{\pi_1^{-1}(U)}\cir…

Balarka Sen

Ya sounds good

Thorgott

this is neat

now our vector field is just given by the compositions $M\rightarrow M\times\mathbb{R}\rightarrow V\hookrightarrow TM$

Why What Where When How

What topic of math are you guys discussing? I feel like I am a complete stranger to it.

Thorgott

differential topology, I'd say

Why What Where When How

Well it's out of my range. It's 3rd yr course.

I am reading combinatorial analysis. It asks from a group of 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed? Which is easy but it asks. What If 2 of men are feuding and refuse to serve on the committee together?

What do you think it means by Now suppose that 2 of men refuse to serve together. Because a total of {2\choose 2}{5 \choose 1}=5cout of {7\choose 3}=35 possible groups of men contain both of the feuding men, it follows that there are 35-5=30 possible groups that do not contain both feuding men.

I am confused by the unexplained calculation {2\choose 2} {5\choose 1}.

rschwieb

02:05

I have a quick question: It is easy to see, using the Lagrangian for length in rectangular coordinates in the plane to see that straight lines minimize the length functional. I ran across an exercise which converted the Lagrangian to polar coordinates, and then asked me to write the Euler-Lagrange system in those coordinates.

Let me say the result is horrific. Is that probably because they wanted to make the point that choice of coordinates will dramatically affect the difficulty of solving the Euler-Lagrange equations, and they just wanted you to see two ends of the spectrum?

Why What Where When How

What does it actually means?

Balarka Sen

@rschwieb Haha sounds like physicists being physicists

Thorgott

${2\choose 2}$ are the ways of choosing the two feuding men from the two feuding men (of course, there's only one way to do so), and ${5\choose 1}$ are the ways to choose the remaining third man (there are $3$ men total in a committee) from the $5$ men that are not the two feuding men

Why What Where When How

@Thorgott I thought the same but I think remaining men should be {3\choose 1} since you may choose wamen.

Thorgott

we are not considering the women right now, only the number of groups of $3$ men containing both two feuding men

Why What Where When How

02:13

Oh now it's okay to say Ohhhhhhhh....... I understand. It's so easy to see.

@Thorgott you have fericious mathematical instinct!

Thorgott

@Balarka horrible question time: I have a smooth manifold $M$, then, for each $p\in M$, I can consider the tangent space $T_pM$ and $I_p$, the set of all inner products on $T_pM$ and $I=\coprod_{p\in M}I_p$. This isn't quite a vector bundle, as each $I_p$ is only a positive cone. Regardless, can we put a differentiable structure on $I$ in such a way that the "sections" of $I$ are precisely the Riemannian metrics?

Why What Where When How

Are you all like 3rd yr undergraduate student? And by looking at profile of your guys it looks like you guys are professors in math department.

Or top student in university.

May be geniuses.

Balarka Sen

@Thorgott Yeah :) First let's look at a space which is a vector bundle, namely, space of all smoothly varying pointwise symmetric bilinear forms

Thorgott

@WhyWhatWhereWhenHow this will come naturally to you once you've done a bit more combinatorics

Balarka Sen

A bilinear form on a finite dimensional vector space $V$ is just an element of $(V \otimes V)^* = V^* \otimes V^*$, where $\otimes$ denotes the tensor product. You know this interpretation, yeah?

Thorgott

02:28

a bilinear map on $V$ uniquely factors as linear map through $V\otimes V$, so the first part is clear

Balarka Sen

yeah forget about the second part, the isomorphism $V^* \otimes W^* \to (V \otimes W)^*$ is given by $\phi \otimes \psi \mapsto (\alpha \otimes \beta \mapsto \phi(\alpha)\psi(\beta))$ which you can check is injective and since we're in finite dimensions it's an iso

Edward Evans

@WhyWhatWhereWhenHow Nah I suck at maths, I just try really really hard and get lame, adequate grades

Balarka Sen

That's formal nonsense

Why What Where When How

@Thorgott Everything comes naturally but sometimes I go to wrong direction. My be everyone is like same .They do do mistake. Anyway you can have busy conversation may be I am bit being a distraction in highly mathematical conversation.

Thorgott

we all do a lot of mistakes

Balarka Sen

02:30

this is a rant room just hang around and rant nobody will be bothered by it

thats what we all do

Thorgott

Mike said this a couple days ago in here, but it's by doing mistakes that you learn the most

and yeah, I'm buying that formal nonsense

Why What Where When How

@EdwardEvans It's normal. But I think about student having perfect grades which quiet scares the crap out of me.

Thorgott

(we will start talking about tensor products in my commutative algebra lecture next week, so I'll do enough of that then)

Edward Evans

@WhyWhatWhereWhenHow lol tbf you can't go through life comparing yourself to others else you'll never be happy

cuz Peter Scholze will always be better than you

Balarka Sen

lmfaooo

Edward Evans

02:32

rofl

Thorgott

~~or will he~~ indeed

Edward Evans

hahaha

Balarka Sen

couldnt have put it better

such a great quote i will use and abuse this

Edward Evans

Spraying my wisdom around

Why What Where When How

Gauss will always be greater than peter scholze.

Thorgott

02:34

screams in perfectoid spaces

Balarka Sen

to gauss everyone is a smoothbrain so

Edward Evans

cough Tilted cough

Asvin

Well, when you are stuck in research and getting nowhere, the only reason to keep going is because you love the subject. But if you really love math, then at least personally, comparing myself to others doesn't come up at all

Thorgott

this

Asvin

You just want to figure stuff out, you know?

Balarka Sen

02:35

So $(V \otimes V)^*$ has a decomposition $\text{Sym}^2(V^*) \oplus \Lambda^2(V^*)$ into the subspace of symmetric bilinear forms and alternating bilinear forms, by sending a bilinear form $\langle -, - \rangle$ to the average after permuting coordinates or the difference after permuting coordinates ("symmetrization" and "alternatization")

I don't know why I am doing this to you but since I am doing it anyway I will keep doing it

Edward Evans

fair one

Asvin

Does latex not compile automatically here?

Thorgott

of course, this is just the Eigenspace decomposition with respect to swapping coordinates

Balarka Sen

yep

Why What Where When How

I don't know how can human being reach intelligence of those freakin geniuses like Gauss, Ramanujan, Terry ....

Edward Evans

02:36

right I just spent like 6 hours doing functional analysis and now I actually wanna die, so I'm gonna go to bed

night y'all

Thorgott

night

Why What Where When How

I am reading everything here in plain latex since the tinyur thing doesn't work in android version of chrome.

Balarka Sen

so no i will define a notion of fiberwise tensor product and dualization in bundles so i can push this formalism through

or do u know this already

Thorgott

nah

Why What Where When How

@EdwardEvans That hit hard rolfmao.

Balarka Sen

02:42

Let $E$ be a vector bundle over $B$, choose trivializing neighborhoods $U_\alpha$ of $B$ such that over each $U_\alpha$ you have a trivialization map $\varphi_\alpha : U_\alpha \times \Bbb R^k \to \pi^{-1}(U_\alpha)$. For any pair of indices $\alpha, \beta$, there are these transition functions $\varphi_\beta^{-1} \varphi_\alpha : U_{\alpha\beta} \times \Bbb R^k \to U_{\alpha \beta} \times \Bbb R^k$ defined over the overlaps $U_{\alpha\beta} = U_\alpha \cap U_\beta$

These guys are identity on the first component, so the only interesting this to look at what is going on in the second component. Keeping track of that I get a map $\phi_{\alpha\beta} : U_{\alpha\beta} \to \text{GL}_k(\Bbb R)$, where $\varphi_\beta^{-1}\varphi_\alpha(x, v) = (x, \phi_{\alpha\beta}(x) v)$.

These are called the transition functions for the vector bundle, and they satisfy the identities $\phi_{\alpha\beta}\phi_{\beta\alpha}^{-1} = I$ and $\phi_{\alpha\beta} \phi_{\beta\gamma} \phi_{\gamma \alpha} = I$ on every triple intersection $U_{\alpha\beta\gamma}$, where $I$ means the map which sends everything to the identity matrix, and multiplication means multiplication pointwise in $\text{GL}_k(\Bbb R)$

Why What Where When How

I think everyone who viewed my question in stackmath is self harming by looking at my question. Or just gave up lol. XD

Balarka Sen

Such family of functions are also called Cech 1-cocycles (you can prolly guess why, given our earlier interaction on sheaves)

Thorgott

oh god

I've been tricked into sheaves again

Asvin

sheaves are great! don't be scared!

Balarka Sen

Formalism aside, $\phi_{\alpha\beta}$ completely determines the bundle in the following sense. Consider the topological space $\coprod_\alpha U_\alpha \times \Bbb R^k /\sim$ where $\sim$ is the equivalence relation defined by $(x, v) \sim (x, \phi_{\alpha\beta}(x) v)$ whenever $x \in U_{\alpha\beta}$ and $v \in \Bbb R^k$

This is homeomorphic to $E$, and the projection to $U_\alpha$ defines the bundle projection $E \to B$

So all information in the bundles are contained in the transition cocycles

I am doing this topologically but to get the smooth structure etc back you want $\phi_{\alpha\beta}$ to be smooth maps to $\text{GL}_k(\Bbb R)$, just to get that nonsense out of the way

Let $E, F$ be two vector bundles over the same base $B$ of rank $n$ and $m$ repectively. Choose a common trivializing cover $U_\alpha$ on $B$ for both $E$ and $F$, and let the corresponding transition cocycles by $\phi_{\alpha\beta}, \psi_{\alpha\beta}$.

(1) $E \oplus F$ is defined by the transition cocycles $\phi_{\alpha\beta} \oplus \psi_{\alpha\beta}$, maps to $\text{GL}_n(\Bbb R) \times \text{GL}_m(\Bbb R) \subset \text{GL}_{n+m}(\Bbb R)$ as a block matrix. This is the fiberwise direct sum.

(2) $E \otimes F$ is defined by the transition cocyles $\phi_{\alpha\beta} \otimes \psi_{\alpha \beta}$, where tensor means pointwise tensor product of matrices, maps to $\text{GL}_n(\Bbb R) \times \text{GL}_m(\Bbb R) \to \text{GL}_{nm}(\Bbb R)$. This is the fiberwise tensor product

(3) $E^*$ is defined by the transition cocycle $\phi_{\alpha\beta}^*$, where dualizing a matrix pointwise means just taking transpose $\text{GL}_n(\Bbb R) \to \text{GL}_n(\Bbb R)$. This is the fiberwise dual

The space of bilinear forms (without adjecitives) on $M$ is then $T^* M \otimes T^* M$ because that is the correct notion fiberwise. A smooth section of the bundle $T^*M \otimes T^* M$ gives a smoothly varying bilinear form on the tangent spaces of $M$

You can convince yourself that you can similarly define the symmetrization $\text{Sym}^2(E)$ of $E \otimes E$ for a vector bundle $E$, and that defines $\text{Sym}^2(T^*M)$, the space of symmetric bilinear forms on $M$, a section of which defines a smoothly varying symmetric bilinear form on $M$

The space of Riemannian metric is a subspace of $\text{Sym}^2(T^* M)$ which you rightly pointed out is the fiberwise open cone of the positive definite guys. This is not a vector bundle, but a bundle nonetheless, with cone fibers.

A smooth section of $\text{Sym}^2(T^*M)$ whose image lies in this cone bundle is a Riemannian metric on $M$

Sorry I didn't pause for breath there

Thorgott

02:59

does it make a difference to construct $\mathrm{Sym}^2(T^{\ast}M)$? that is, why don't we immediately consider sections of $T^{\ast}M\otimes T^{\ast}M$ whose image lies in the cone?

Balarka Sen

yeah it doesnt really i was just being really specific

i mean it makes a difference in the sense that the subspace is not an open subspace of T^*M o T^*M anymore

what the formalism tells you eg is that riemannian metrics are open in the space of all symmetric 2-forms on a smooth manifold, because if you wiggle a section of Sym^2 which is inside the open cone it'll always be inside the open cone because its open

Thorgott

ah, that sounds reasonable

how do we recover the smooth structure from the cocycles?

Balarka Sen

also its a fiberwise convex open subset, with the "boundary" being what i would call "degenerate metrics", which are positive semidefinite and vanish at some places

Thorgott

Riemannian pseudometrics?

Balarka Sen

sounds right, but i think you also call indefinite guys are pseudo-Riemannian

detour: this convexity and openness is useful because there's a theorem of gromov which says fiberwise convex subsets of bundles are special, you can approximate sections which lie strictly inside the cone by sections which lie on the boundary of the cones

Thorgott

03:09

no opportunity to mention gromov was missed

Balarka Sen

lmfao yeah totally just for that

@Thorgott ehh to get the smooth structure you also want to the transition functions for $U_\alpha$ considered as manifold charts

then $U_\alpha \times \Bbb R^k$ has a smooth structure, you have transition functions of the manifold and the cocycles of the bundle which glue these to a smooth structure of $E$

well $U_\alpha$ is just placeholder for $\Bbb R^n$, the thing $U_\alpha$ gets mapped to by the chart diffeo, in that case

Thorgott

I'm very worried about quotienting

Balarka Sen

and you glue these multiple copies of $\Bbb R^n \times \Bbb R^k$ using chart transition functions and bundle transition functions

Thorgott

at the very least the equivalence relation seems open, because the transition maps are homeos

and the graph of the relation should be closed by their continuity

Balarka Sen

sure topologically it's a-ok

Thorgott

03:13

so I'm at least buying this gives a top. manifold

Balarka Sen

its a good exercise to write down the smooth structure as well

i'd explain it but i am nearly asleep now having like my 19th smoke

i cant write formulas, only talk in wobbles

Thorgott

yikes

ima go sleep over this

Balarka Sen

cool

night

Thorgott

after sleep, I'll also have to finally read that last paragraph and figure out why we put so much effort into pulling back a plain vector field

a yway, night

Balarka Sen

its because if u have a vector field where nothing happens for a long stretch of time the topology on the region is trivial

:)

Why What Where When How

03:25

How many user are using stack exchange?

Does anyone know meaning of because nothing here now? In the comment of math.stackexchange.com/questions/3675476/…

Or does he mean... Everything I assumed is right.

abhas_RewCie

03:50

@Knight Okay, I won't pretend to be you/

Why What Where When How

I think if no one answers my question and just leaves me with 4 upvotes then I have to say I have mastered real analysis book. (Hurrrrrrayyyyyyy!) Now I can move to Group theory, Onion ring and galois vegetable then Metric unit with spicy measured quantity theory then Topology of Carrot then Quantum Which was hated einstien then general and special relativity then n

Anyway there are endless then

Self studying undergraduate course is really time consuming ! Woosh. I wish there was professor just taught me but they are tooo overpriced.

Ted Shifrin

I cost at least a million bucks, @WhyWhatWhereWhenHow. However, if you're interested, my 112 lectures on YouTube are available for free.

@BalarkaSen I told you to quit smoking. Now you're past wobbling!

Why What Where When How

@TedShifrin Yes I want to watch your million bucks lecture. Show me your demonstration. Wait... You are a professor.Yikes.

I finished multivariable calculus using Khan academy looks like Salman Khan didn't cover whole multivariable calculus. I don't even know wth is fubini's theorem.

Khan academy multivariable calculus is cringy but has intutive approach.

@TedShifrin I searched you in youtube looks like you provide detailed lecture in each of the contents on multivariable calculus so it is 30+ minutes long. BTW I like the mustache on your face it's like Einstien's mustache. It does fit on every face.

I want to grow that mustache on my face too.

Ted Shifrin

04:14

LOL, right now I also am working on Einstein's hair!! :P

My course incorporates linear algebra and is has all the proofs (as well as plenty of computations).

Why What Where When How

Hey and you got lecture on lagrange interpolation. I was looking for this when I was middle schooler all the time! None of the website satisfied me I will watch you lecture now.

Ted Shifrin

It's an application of stuff from linear algebra. If you don't know linear algebra yet, you should be patient and work your way to it.

Why What Where When How

@TedShifrin God bless professor like you who provides Lecture on internet.

2

Ted Shifrin

Well, it was thanks to my hard-working students who taped them. But they may not fit your personality or learning style.

Why What Where When How

@TedShifrin Yep I haven't completed linear algebra yet. I was not satisfied with easy khan academy video so I am gonna buy Gilbert strang linear algebra.

@TedShifrin @TedShifrin I can learn anything given appropriate time. I am sure any style fits except learning rote memorization and blind rules.

Sorry for tagging too much. Hope it doesn't annoy you.

@skullpatrol I know the news.

But isn't lagrange interpolatio learned in numerical analysis? Instead of Multivariable calculus or Linear Algebra?

Ted Shifrin

04:25

@WhyWhatWhereWhenHow I love Strang, but his books are not great if you're trying to learn things step by step with proofs. But he does an excellent job with applications.

Just start at the beginning of my course :P

Why What Where When How

Mr or Prof Shifrin I would like to read your books rather than watching video. Since I can't do learn in very fast with that 480 ppi resolution. I mostly skip video and watch blackboard .

@TedShifrin I swear I can learn that in a week. XD

But I think book will be faster

Oops it's 110 videos.

May be 2 weeks.

Ted Shifrin

LOL, no. This is a very challenging, year-long course taken only by the best college students. :P

But you're welcome to find the books. Because of the world we live in, they're unfortunately not very cheap.

skullpatrol

COVID-19 is changing the world we live in professor

Why What Where When How

Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds is the book. Ok I will accept the challenge! I will try to finish it in 2 weeks. 500 pages.

Ted Shifrin

@skull: definitively ...

skullpatrol

04:39

:-)

Ted Shifrin

@WhyWhatWhereWhenHow ... Working on a book should mean working (and getting correct) at least half the exercises. That alone will take most of a year.

I suspect you've never dealt with a serious college mathematics book before.

Stan Shunpike

@TedShifrin is it bad practice to write $\forall$ and $\exists$ in a proof?

are u supposed to use words?

or are the symbols accepted too

Why What Where When How

@TedShifrin @TedShifrin I have seriously dealt with college Mathematics. I have finished whole exercise of analysis book Numbers and functions. Steps into analysis. And read springer undergraduate books of real analysis. It looks me weeks to solve it without looking answer from books. And the logic were correct. But sometimes I go to wrong direction but I always correct my mistake.

I know it takes lots of time.

Ted Shifrin

@Stan: I was taught (by Munkres) never to write symbols (other than things like $\implies$ and abbreviations like s.t.). Most mathematical journals do not accept things full of formal symbols. They're very hard to read.

Why What Where When How

@TedShifrin I have not only be correct on half but full. I believe true mathematician can solve everything that has been Discovered.

Ted Shifrin

04:45

@WhyWhatWhereWhenHow: I'm not putting you down. I'm just trying to be realistic. But let's see what you can do.

Stan Shunpike

@TedShifrin good to know. didn't know that. i agree, i find them illegible. I thought it was the preferred way

since i see it in class all the time

i didn't realize it was shorthand

Ted Shifrin

I had almost no professors that wrote symbols, @Stan.

Why What Where When How

@TedShifrin I know you are not trying to put me down.I am just too confident XD.

Ted Shifrin

Some write $\forall$ and $\exists$ but plenty of words.

@WhyWhatWhereWhenHow: Well, you can show me a few finished/polished proofs of exercises as you go along, and we'll see :)

skullpatrol

I believe part of the reason why most students can't handle serious college level textbooks is because there are so few serious high school level textbooks @TedShifrin?

Ted Shifrin

04:47

You can ask @Stan: He's submitted a bunch of linear algebra to me and had to sweat.

@skull: All over Europe, Russia, and Asia, the expectations in the US are way more stringent.

Why What Where When How

@TedShifrin Ok I will do your book exercise after I finish enough course that will make me satisfied.

Ted Shifrin

But in the US we sort of expect too many students to get isomorphic educations.

Why What Where When How

After finishing undergraduate course I can finally start to daydream Mathematics lol.

Stan Shunpike

@TedShifrin to be fair, I sweat with almost all math, but at least your book was fun to sweat over

One of my friends majored in math in college and he says he can barely understand the math we are doing cuz of the notation

that we are using this quarter

very strange experience

i'm looking forward to doing real math again this summer

Ted Shifrin

@Stan: So do you feel the little bit we did together helped prepare you?

Stan Shunpike

04:51

@TedShifrin i feel like it not only helped, but was frankly the only decent linear algebra training ive gotten and i think it's essential i do some more

It has definitely made my courses much easier

and made finding information much easier

cuz i know what to google if i get stuck

Knight

@abhas_RewCie You can, after all we are same person using two different accounts :-)

Stan Shunpike

In retrospect, I regret using my Caltech "multivariable and linear algebra" combo course to excuse myself from taking linear algebra at uchicago

because i think I did myself a disservice by not taking a full course devoted entirely to it

so i'm trying to rectify that by doing exercises from your book, which i plan on resuming as soon as this quarter is over

Knight

What I have observed is if your first course or first introduction to Multivariable Calculus is ill-taught then you won’t ever be able to get it right.

Stan Shunpike

It's just not sensible to avoid taking a course in linear algebra when it is so prevalent

but i didn't know that when i started college lol

just trying to control my GPA

Why What Where When How

@TedShifrin Is you book Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds a standard,Special or Optional course in university?

Ted Shifrin

04:54

@Stan: I'm glad it's helping.

Why What Where When How

I think it's year 1 course.

skullpatrol

Are you saying that all over Europe, Russia, and Asia there are serious high school level textbooks professor @TedShifrin?

Ted Shifrin

@WhyWhatWhereWhenHow: It's used in the toughest Honors courses typically taken by first and second year students. It's been used at Stanford, Vanderbilt, Yale, Georgia, and other places.

Absolutely, @skull. If you paid attention in this room, you'd see that the guys from Germany, France, China ... all had an analysis type course even in high school, but universally first year of university.

Knight

@skullpatrol Pal! The focus of Asians is more on being fast and calculative, they pay less attention on “hard thinking”

skullpatrol

Hmm

Stan Shunpike

04:56

@WhyWhatWhereWhenHow i think what distinguishes Ted's book from others books is it's thoroughness combined with its intuitiveness. He doesn't leave any stones unturned. I wish I had found his book earlier so I highly recommend it. Many of my peers in grad school all used his book

Ted Shifrin

@WhyWhatWhereWhenHow: At these places it's 10-20 students who complete the course, typically. Oh yeah, they used it at UCSD two years ago. 100 or so started, and I think 30 finished the third quarter.

Knight

China and Korea are the best examples of ^

Ted sir I’m still unable to find the volume contained by two intersecting cylinders

Why What Where When How

I swear I used intutive approach on Real Analysis. I see a lots of words turning into picture so I am sure I can complete it since I believe I am persistent enough to do that.

Knight

All I need is a visualisation

Why What Where When How

@Knight Wrong.

Stan Shunpike

05:02

@TedShifrin Baklara really helped me understand a few things earlier.

"The Gaussian curvature is defined for surfaces; in an n-dim manifold there are n(n−1)/2 "coordinate surfaces" at any point given by exponentiating coordinate planes as I mentioned. You can think of the Riemann curvature tensor as the tuple of all these Gaussian curvatures"

i didn't realize this

back then i didn't know enough physics to understand what role differential geometry plays in physics

Why What Where When How

Professor Shifrin your Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds is expensive than my weekly salary lol.

skullpatrol

@WhyWhatWhereWhenHow Right. Try the JEE

Knight

@WhyWhatWhereWhenHow Okay, what are your views?

I personally find the courses of Germany to be very very nice.

They are very on-topic. They don’t pretend to teach us “practical science”

Why What Where When How

@Knight They do have intutive approach. They are not just only fast and calculative they are Smart. You can see indian guys lol.

Ted Shifrin

Yes, @Stan. As long as you understand Gaussian curvature of surfaces (abstractly). That's why I said starting with curves/surfaces is a significant step to understanding.

Knight

05:07

@WhyWhatWhereWhenHow I decline the statement that Asians follow more intuitive approach than Americans or Europeans

Ted Shifrin

@WhyWhatWhereWhenHow: Notice that I was pushing the free lectures, not the book.

Knight

Sir is your dinner done ? :-)

Ted Shifrin

@Knight: If you're talking to me, it's after 10 PM.

skullpatrol

You're back home?

Knight

@TedShifrin Okay! So, are you planning to sleep?

@skullpatrol Did you see the movie Casino?

Ted Shifrin

05:09

@skull: I've been back a month now.

Why What Where When How

@Knight I decline your statement. I have my friends in Taiwan who have far more intutive approach than here in America.

skullpatrol

@Knight yup

@TedShifrin nice to hear

Knight

@skullpatrol What was “back home” referred to in that movie?

skullpatrol

nope

Knight

????

Stan Shunpike

05:11

@TedShifrin ohhhhhhhh

you're book makes so much sense now

wow that took me a while

Ted Shifrin

Now I have no idea what you're talking about, @Stan :P

skullpatrol

@Knight The good professor had to leave his home due to COVID-19

Ted Shifrin

Guess it was almost 4 years ago I made my trek to Chicago, @Stan.

Knight

@skullpatrol Okay. I asked what was referred to as “back home” in Casino is same as Ted’s home?

Ted Shifrin

Well, that's not literally correct, @skull. I just chose to do so ... but then realized I'd have to leave for years.

Why What Where When How

05:13

@TedShifrin How long did it took to make that book.

Stan Shunpike

@TedShifrin that's crazy. it feels so short, but i guess hw makes it feel that way

@TedShifrin I didn't get why you started off your differential geometry book with 2d and 3d concepts when I thought you would jump right into higher dimensions. I didn't realize the tools with Gaussian curvature were used to build up notions of higher dimensional curvature

so I found that confusing, but now it makes sense why you put emphasis on that

Ted Shifrin

@Stan: The diff geo course was for undergraduates. It's the right level for that. To do $n$-dimensional Riemannian stuff you really need serious topology, manifolds, bundles, and a lot more sophistication.

Stan Shunpike

Yeah, and that's probably why I find do Carmo to be hard to digest. I get stuck on bundles

skullpatrol

@Knight pardon my misinterpretation pal

Ted Shifrin

In part, I like the course because it makes students actually use linear algebra and multivariable calculus. But I like the concreteness and the depth of the material, without requiring a bunch of prerequisites.

DoCarmo doesn't do bundles, actually, other than tangent bundle.

Stan Shunpike

05:16

Welp, there goes my excuse

Knight

@skullpatrol No problem pal

Ted Shifrin

@WhyWhatWhereWhenHow Probably three years to get it polished.

Stan Shunpike

@TedShifrin my high school physics teacher retired last september to write a book on group theory

skullpatrol

@TedShifrin indeed sir, this could go on for years.

Ted Shifrin

one of my best friends in Athens is a physicist who occasionally teaches a graduate course on group theory for physics. @Stan

Why What Where When How

05:17

@TedShifrin So you single handedly made a book? o_o

Ted Shifrin

Great, great teacher.

@WhyWhatWhereWhenHow, four, in fact (one with a coauthor).

Stan Shunpike

@TedShifrin what made him a great teacher? Anything in particular?

skullpatrol

Have you ever thought of writing a number theory book, sir?

Ted Shifrin

I actually never saw him lecture, but I know him well, and many of my students raved about him and wrote letters of recommendation for him for teaching awards. He is well-prepared, clear, motivates well, cares deeply about students.

No, @skull. My only use for number theory is to use it to teach the concepts of algebra (so I incorporated some of it into my algebra book).

I actually have never taken a number theory course.

Why What Where When How

@TedShifrin Have you meet geniuses? What does it makes them genius.

skullpatrol

05:20

@TedShifrin :O

Stan Shunpike

@TedShifrin those are really great qualities

Ted Shifrin

I don't know what makes a genius. I certainly have taught students who I knew were extremely talented and gifted. Some of them accomplished far more than I ever have. Some perhaps not.

Stan Shunpike

@TedShifrin I also think a good ability to inspire people to want to accomplish things is a great quality too

Why What Where When How

@TedShifrin Did they became great mathematician?

I am really curious about people tagged genius.

Stan Shunpike

what do u define as a genius?

Ted Shifrin

05:24

I don't tag anyone genius, @WhyWhatWhereWhenHow. Some of them have, yes. Others in other fields.

A few of my most incredibly gifted students, sadly, died young.

Why What Where When How

I define them who are able to think outside the box.

Something extraordinary.

It is too broad to define.

Ted Shifrin

I don't see that as "genius."

I don't know what "extraordinary" actually means. But I'm happy to talk about talent and gift.

I know that a lot of students I've taught (and a lot of people I interact with in this chat) think they're a lot better than they actually are. By a significant margin.

Why What Where When How

They do give very unexpected answers (Now seriously don't try to be sarcastic lol).

Stan Shunpike

Ted is never sarcastic

ever

Ted Shifrin

Well, at least I know what it means, unlike our narcissistic emperor.

Why What Where When How

05:28

I'd say genius is someone who sees something that lots of people don't see. Now I am saying this in a sloppy manner.

Stan Shunpike

@TedShifrin omg he's the worst. I think he's taken narcissism to new levels

@WhyWhatWhereWhenHow it's a question psychologists have tried to answer. people have tried to come up with ways to measure "genius" and "intelligence" and, in general, it can be difficult to ascribe a single word like genius much meaning since there are so many forms of ability.

Was Mozart more of a genius than Arnold Schoenberg?

Why What Where When How

@StanShunpike I can't compare them but I can compare them to normal people.

Now don't tell me to define normal people lol.

@StanShunpike From my perspective this is kinda like asking is multiplication better than addition.

We as human beings do have nature of comparing things . But comparing human intelligence is complicated. There is pattern for sure. Someday human intelligence can be rigorous compared.

skullpatrol

05:53

@CalvinKhor were the formulas that valuable that the lectures wanted them back?

Calvin Khor

@skullpatrol sorry what fomulas? and hello

skullpatrol

Hello.

May 11 at 7:29, by Calvin Khor

i returned all my number theory to my lecturers so don't know, sorry. I know $Li_s$ is notation for polylogarithms but they don't seem related

:-)

What does "i returned all my number theory to my lecturers" mean?

Transcript for

$Mathematics$ Mathematics