so definatly dont think u should of done worse

its a hard subject

@MatheinBoulomenos mathein, we talked for quite a while now:D, do you think am ready to do rep theory ?

at least imo

@TedShifrin Exercise 2-40 asks us to redo exercise 2-15 using implicit function. Basically we need to show that if you have a square matrix $A(t)$ where each in entry is a function of $t \in \mathbb{R} $ and we assume that $ det [A(t)] \neq 0 $ for all $t$. You also assume that you have differentiable functions $b_i: \mathbb{R} \rightarrow \mathbb{R}$, and that you have

it is hard in the sense that each question seems like new topic

i never had this on previous courses

@Tedfunctions $s_i:\mathbb{R} \rightarrow \mathbb{R} $ which satisfy $ \sum_{j=1}^n a_{ji}(t)s_j(t) = b_i(t) $

the first knock out was intro to quotient group

@CryinShame: He just means to use implicit differentiation.

from that point I did not understand anythng my teacher said on class

@KasmirKhaan I think you might want to spend some more time on groups and so on before delving into reps

(I don't know much rep theory, though.)

@Antonios-AlexandrosRobotis thanks ><

i dont even know what representation theory is

Don't let that stop you, though. Just maybe have both books open at the same time @KasmirKhaan

well because the name is not full

@MatheinBoulomenos thanks for the AA lesson much appreciated

@Kasmir I don't know about the pace or difficulty of your rep theory course. I think it's going to be really though for you if you're not comfortable with concepts like homomorphisms, quotients etc. But I'd try it if I were you

@Faust you're welcome

@Antonios-AlexandrosRobotis Am gonna take the course because the teacher that will be a good one:D

Don't forget the linear algebra needed!

@KasmirKhaan thats harvard lecture set Ted gave me is really good

Oh yes, I wanted to say that @Ted

@TedShifrin Hmmm, I was thinking we let $s=(s_1,\ldots,s_n)$ and define $ f(t, s) = A^T(t)s-b$ we could use implicit and go from there.

@MatheinBoulomenos so true mathein :D but ill try to make sense of iso theorems :D

I think I told Kasmir this already twice or so

Yes yes :D

@EricSilva I'll be honored to have my contributions in your book

I know guys, its just that am super stressed i say things more than once =p

lmao

lol

@Faust Did he now ? what lectures ? :D

@TedShifrin Wouldn't we have to know that $D_1f(t,s) $ is continuous though, if we want to use implicit differentiation?

Don't forget to acknowledge, otherwise I'll sue you for stealing my memes

"This book is dedicated to @BalarkaSen, who contributed the bulk of the memes found herein."

^ This

wait a sec

Hmm, I have no idea why Spivak transposed his matrix there, @CryinShame.

antonio are you a moderator?

@TedShifrin I know, I was a little thrown off by that too when I originally did the exercise.

@Faust he dont go into the stuff I need the most, like no iso thems

its alittle fast paced i wouldnt reccomend it as intro

@BalarkaSen no worries i would never claim ownership over any memes

ever

@Faust representation theory is the study of how groups act on vector spaces. Formally, you study group homomorphism from groups $G$ (most likely assumed finite in a first course) to $\operatorname{GL}(V)$ for vector spaces $V$ over a field (most likely $\Bbb C$ in a first course)

we gonna use serre

mathein

I know

sounds like a linear algerba class

Yes, you're right, @CryinShame. He needs to assume the entries of $A$ are $C^1$.

@Faust in less complicated terms, you pretend that some matrices are the elements of your group

it's more like linear algebra meets group theory

so that the multiplication makes sense

@TedShifrin as well as the $b_i$'s right? In the problem he only tells us that they are differentiable, not necessarily $C^1$

and everyone knows that linear algebra is the best

Yup.

@TedShifrin Okay! thank you so much. You're lectures online have been a big help by the way.

@MatheinBoulomenos my linear algerbra class is pretty wierd lol we prove everything over an arbitrary field or C and the first homework had a question using mxn matrieces as our vectors

Antonios knows it so well he accidentally posted it twice

lul

i tried to edit and something wacky happened

But, it's stoopid to use implicit differentiation when we can prove directly that $A^{-1}$ is differentiable when $A$ is differentiable (and invertible). @CryinShame ... Oh, and I'm glad.

yeah that happens sometimes

its odd

i think it's a connectivity/server issue

yeah perhaps

@Faust that doesn't sound too weird to me

@TedShifrin how do you actually differentiate a matrix?

i dunno it feels like an easy redo of the abstract algerbra i have already done up to chapter 4 in our textbook im reasonable confident i could prove any theorem

up to that point

@Leaky ... A matrix of functions, of course.

yeah, if you have done abstract algebra, then linear algebra will be really easy

and we wont get to chapter 4 until mid or late febuary

I see

@BalarkaSen it tends to happen for me on mobile

here in Germany we usually do linear algebra before abstract algebra

Hmmm, I'm not sure I see the path from knowing $A^{-1}$ is differentiable.

You can also think of functions on the space of matrices and ask about differentiability.

in the US too @MatheinBoulomenos

though I did it in the opposite order lol

lol at uchicago until like last year it was algebra before linear algebra

For example, $f(A)=A^{-1}$ is a differentiable function on $GL(n)$.

@MatheinBoulomenos i think its intended for one to do it that way but i find abstract algerbra really intresting

@EricSilva: That's cuz that's how g-d Herstein wrote his epic book.

@EricSilva do you guys do an "elementary course" first? We had the crappy "row-reduction" course before algebra

there's no row reduction computational type course in the math dept

It should be a lot more than just "crappy row reduction."

it's not @TedShifrin

LOL

that'd be high school linear algebra here

I actually took a graduate algebra course before linear algebra

@TedShifrin trying to comprehend that

you wouldn't see it as a full course, but as an element of some other course

Im super proud i got an A+ in that class the prof is notorious for almost never giving A+'s when i took 212 with him he didnt give anyone an A+

@TedShifrin I think I can understand that

@Leaky: You need to know the definition of differentiability of a map $\Bbb R^k\to\Bbb R^\ell$.

@TedShifrin never read hersteins book

(Or more generally with Banach spaces ...)

I'm just saying that's the UC history, Eric.

ah i see

i dont like the old system

Where are you in school, @CryinShame?

@TedShifrin sure, using the definition that there is a linear transformation for which [f(x+h)-f(x)-L(h)]/h -> 0, right

Almost right, @Leaky. What you wrote down doesn't make sense.

there was some weird circularity of prerequisites if you decided to take the honors classes @Ted, it was very strange

Nothing surprises me, Eric :)

@TedShifrin I'm finished for now. I finished my MS a year ago. Waiting for my other half to finish her MA and I'll make the decision about PhD. Finances and technical ability seem prohibitive right now though.

lol a curriculum that only works for weirdos tbh

Aha, @CryinShame. You get supported for a Ph.D., or you don't do it. But it's not a lavish lifestyle.

@TedShifrin right, I'll look it up later

@TedShifrin what does supported mean?

so what's the derivative?

@TedShifrin Yeah I was hearing about that. Actually I was more worried about the tax situation. Being taxed at tuition prices on a graduate's stipend sounds ruinous.

being paid to TA, typically, and having tuition paid.

That got trounced, @CryinShame. Idiots running the asylum nevertheless.

there's still some other stuff in there that might play out poorly for higher ed institutions

so rip

Well, we want the population as ignorant as possible so they'll possibly vote R.

We actually had classes that were impossible to take if you always do the recommended prerequisties for and you finish your bachelors and masters in time

@TedShifrin Yeah, I was shocked to say the least. I'm still ambivalent about going. I enjoy studying but I'm not sure if I'd make the cut.

@CryinShame: i don't routinely encourage people to go do PhD's. Far from it.

@MatheinBoulomenos wow at our university pre-reqs are almost completely ignored if your smart the instructor will waive any pre-req

@TedShifrin One of my thesis advisors mentioned that I shouldn't do it unless I was going to study applied math.

So you never did any sort of multivariable analysis course as an undergrad or MA student, @CryinShame?

if the tuition for every student is waied, then why does the school still stipulate the tuition for PhD programs?

@MatheinBoulomenos we had a situation where there was a pre req for class A that was only taught in class B for which class A was a pre req

@TedShifrin Nope. Shocking, I know. I actually mentioned it in the chat a few days ago. We went from Rudin to Folland. Just straight to measure theory and complex. No multivariable analysis.

@CaptainBohemian taxes

There's more push toward applied math because academic jobs are getting more difficult. I would say make sure you have computer science skills, regardless of undergraduate/graduate status. A PhD in math will still get you various real-world jobs, whether pure or applied.

@CryinShame: Not unusual. Most of UGA's grad students had no idea what the inverse function theorem is.

@Faust note that I wrote "recommended prerequisites". Formal prerequisites don't even exist. You can take "étale cohomology 4" in your first semester.

wait really? @TedShifrin

thats wierd im doing mulitvariable analysis right now and its required for anyone in the honours program

@CaptainBohemian: Not waived for everyone. There are (e.g., foreign) students who aren't supported.

rofl @MatheinBoulomenos

Yes, really, @Antonios.

@TedShifrin what's the derivative of $A \mapsto A^{-1}$?

That seems like a serious omission on the part of the undergrad program at their schools...

@Leaky: You should be able to guess the answer. Can you figure out how to do that?

@Antonios-AlexandrosRobotis I know someone who's a PhD now student who took grad algebraic geometry in his 3rd bachelor semester

@TedShifrin I didn't either until I decided to start doing the material myself. Analysis was always difficult for me so I'm trying to fill in the gaps. Yeah they tried to convince me to go back to do a PhD in some computational program but I don't have plans of staying in town for it.

of undergrad, I presume @MatheinBoulomenos

yes

@Antonios: Unless students come through my multivariable math course as undergrads at UGA, they don't learn it. Our multivariable analysis course has not run in decades.

thats really wierd

@TedShifrin well I would guess $-1/A^2$ lol

that's such a shame @TedShifrin. I found it such a pleasant topic.

Um, leaky, that makes no sense.

but if that's wrong, I'll give it a think later

@MatheinBoulomenos lol I met a 14 year old in graduate algebra during undergrad. That was really something.

This is a linear map on the space of matrices?

but what really need support are foreign students, I think. Foreign students seldom have that high economic ability to study PhD there if not given financial support.

@Ted I don't think we did any analysis specifically in one dimension (except in the complex case of course)

Oh, that's quite unusual, @Mathein.

So is it really that hard to find a job with a phd in pure math?

Academic jobs are getting harder to find, @Faust. Yes.

Universities are hiring part-time people more than tenure-track people.

I know about this

hmm

:(

okay, time for me to vanish and get some more work done

happy math-ing

@TedShifrin well technically we defined the Lebesgue measure (at least in our second encounter) first in one dimension and then we used product measures to define it in n dimensions. But other than that, I think Fourier series is the only subject where we restricted ourselves to 1D

thats too bad i really like math

Have fun! I should do it too

@TedShifrin yeah

hence why i want a job in the real world

There are non-university careers.

No fight from me, Semiclassic, as you'll recall.

right

I don't think I'm useful outside of academy

i left working for a big corporation cause i hated it

@MatheinBoulomenos ME TOO.

I think people with exceptional talent should still pursue it. Just have eyes open.

There are still small college teaching jobs in the US, too, of course.

I feel any nonacademic jobs look so dismal.

I would be happy if I can teach abstract algebra and number theory courses. I want to write a book, too

yeah i dont think i fall in that category the only reason why i do well is cause i work harder than everyone else

I don't really care about making money or building a family

Well, I was always motivated by teaching and was — ostensibly — talented at it. But plenty of people in academia don't care about teaching. That doesn't help.

Europe is different, of course, @Mathein, but there aren't so many academic jobs there.

@MatheinBoulomenos neither do I.

@Ted i always have some anxiety about this. my back up to math at this point is probably to go do farm work or smth.

You can't escape anxiety, @EricSilva.

yup

But I would encourage you to keep going.

that's the plan

My back up plan is to take some CS courses and try to become a programmer. It's nothing compared to math, but I don't hate it

I think even pure mathematicians should take CS seriously. There's so much computer investigation even in pure algebra research.

I do take CS seriously. I already took 3 CS classes and I'll probably take some more

Cool.

I did some computer investigation for my thesis.

but I've skipped stuff like "software development" so far

That was stated for general consumption, @Mathein. Not just for you.

nothing fancy but it was basically necessary.

maybe if math doesnt work out i can be a park ranger or something

o.o

More likely a line cook, Eric.

I studied film in california for a year straight out of high school

go for the park ranger

That's cool, @CryinShame.

A few of my former classmates do much better than I do now too.

line cook sounds terribru

@Ted i think cooking professionally would be waaaaaaaay too stressful

cooking is my relaxation time

I don't disagree, Eric.

idt i would be able to handle that

I feel cooking is a drudgery.

Some of us love cooking and food. :)

@MatheinBoulomenos mathein i think you would be super good prof of those subjects :D like really really suits u :D

I think astronomy is my relaxation.

@KasmirKhaan thank you

Just saying what I find to be true :D was not meant to be nice =p

my classmates sometimes say that I'm bound to become a mathematician etc., but I think that builds a lot of external pressure

anyway I hope you all achive your dreams :D

We have many profs in our uni that only does the job to get paid

u can tell they have no love for the subject

They may have love for research and no interest in teaching. That is quite common, and part of my complaint.

@TedShifrin so I need to find a linear function of matrices $L$ such that $\|(A+h)^{-1} - A^{-1} - L(h)\|/\|h\| \to 0$ where $h$ is a small enough matrix and $\|\|$ is the square root of the sum of the entries of the matrix

they should have chose other area of work imo because teaching is something you need to have passion for

lol my intor to analysis prof said in the first lecture "i really hate this course"

prime example is Ted :D

When thinking about my future prospects, I find a lot of comfort in ecclesiastes 9:11.

watch his lectures and you will see that he has fun teaching

:D

But hes great to take an advanced level course with

@Faust dont they chose what they teach ?

i think he just finds it boring

Right, @Leaky. If you want to make an intelligent guess, you can do algebra with that expression (I assigned that as homework in my class) or you can pretend it's differentiable and differentiate $AA^{-1}=I$. But since we're doing linear maps, you need the derivative in some direction $B$.

@KasmirKhaan they have an influence on what they teach i think its that no one wants to teach intor to analysis

so someone has to do it

aha ><

the wierd thing is a buncha profs fight over who gets to teach the next one down the line

then maybe should have those courses for first year profs

No.

That leaves the people who have less experience always teaching the classes that no one wants to teach

i mean despite not liking the material he did do a good job of teaching it...

i dont think a new prof could of done a better job

I like explaining things (as long as they fall into the subjects I like) and I would love to give a course my personal touch, but I'm afraid I'm not that good at teaching. I'm fine at doing it one to one, but having a lot of people at once makes it more difficult

but when taking upper level classes with him its WAAAY better

It takes practice, @Mathein. You might start by giving some seminar talks.

I already gave 3 seminar talks

OK, that's a start on the practice.

two went fine, one went not so well

i really like my topology class for that

i think im kind of quiet and reserved generally but lecturing never felt like a problem weirdly

we have to present our solution to the class on the board and explain them

i think im ok at it

Energy and enthusiasm help, but are not an absolute necessity.

explaining a solution to 15 people when you know almost all of them personally is alot easier imo

Humor is a major plus I think

math profs are not funny people

in my experience

math profs are normal people usually

there are exceptions

some of them are funny and some of them arent

i think its strongly tipped in the not direction

Typically dry humor.

i mean, i think this is true in general for people

Interesting that at least one of the commenters on my videos says I'm an asshole. I guess he didn't get my humorous interplay with my students.

difficult to gauge from a video maybe

there were defintly some funny things in your videos

@TedShifrin Really? I picked up on it. It seems obvious, given that it looks like you knew most of your class by name.

Oh, I learned everyone's names within a week or two. Always do (did).

The advantage of classes with <40 students.

one of them when your student made some kind of ridiculous statement thats was unfathomably untrue the look on your face was absolutely hilarious

@Ted there also exists some generically assish people on the internet who likes to trigger people arbitrarily

it's hard to tell which comment is genuine and which isn't

Oh, I didn't take it personally, Balarka. I just threw that comment in because it fit the discussion.

the internet was a mistake

I'm covering a geometry class today, Balarka, for another instructor. Some of that stuff is sneaky. (We're doing rectangles!)

Aha

On a Saturday?

@TedShifrin aops geometry?

My usual class is Sunday mornings, @CryinShame. This is a class for AoPS (art of problem solving).

hi @TedShifrin

Yup, @EricSilva.

that stuff is scary

@Faust Ted's multivariable calculus classes are the source of great memes

I spent more time thinking about this substitute lecture than I have for most of mine put together, Eric.

good @TedShifrin your a perfect example of a teacher :)

@TedShifrin Oh wow, that actually sounds quite nice. A change of pace. I never had any weekend classes.

competition geometry always struck me as super hard

@Ted do you prove that it's impossible to divide a square into an odd number of triangles of equal area?

everyone should learn from you @TedShifrin

@EricSilva it's interesting nevertheless

These are optional, outside their usual high school curriculum, @CryinShame.

No, no, @Mathei.

@BalarkaSen i agree, i did some of it in hs and liked it

This isn't so competition-ish.

competition math is beyond me. I can do grad algebraic number theory, but I can't do competition number theory

but i dont like going to competitions so gg

oh i was responding to your internet comment

ohhh lmao

i forgot i made that

i am scared shitless of competition geometry

competition any math really

We do better with differential geometry, Balarka :P

@MatheinBoulomenos no

I love competition math

I don't.

@TedShifrin Hahah

Although I learned to write some good questions, eventually, for the UGA high school math competitions

one thing I do in my free time is reverse engineer games and do competition math problems

It seems to come down to guessing more often than I'd like

@MatheinBoulomenos

I did the Putnam one year. They didn't even bother to give me back my score.

i like some of the math but i dont enjoy participating

Well I have gotten better at trick math in the past few months

I never took the Putnam in college. I was teaching high school kids on Saturdays instead.

@Adeek yes?

people here dont generally care about putnam

What's a putnam?

@CryinShame The median score is 0 usually

LOL

Heya everyone

@Mathein: Google Putnam Math exam.

Oh, book came yesterday @Ted

hides from @Cookie

@TedShifrin btw did I tell you that I got best teaching award

Yes, Karim. I told you I was proud of you. :)

nice :D

I am gonna do vector calculus this semester

Hard stuff to teach well!

I will introduce maybe differential forms on $R^n$ at the end

Don't get too theoretical.

yeah

NOOOO.

I should send you my lecture notes from teaching the big multivariable course at MIT. Your students are not like my multivariable math students.

Ah I see, it's a math competition for undegrads

@AlessandroCodenotti Yeah, I don't feel bad or anything. I just did it for fun. I generally consider competitions and awards with disrepute.

yeah that would be nice

do you still have my email ?

@Adeek why not on smooth manifolds, though?

@MatheinBoulomenos I wish

all vector calculus ideas can be transfered to smooth manifolds

Still at Alberta, Karim?

yeah

Found it.

nice

I like difficult exercises in the subjects I enjoy. I spend a lot of time doing past grad school admission tests on abstract algebra (and I found a mistake, lol), because they were a bit more difficult than most exercises in introductory books

But I just can't do competition problems

I mostly can't, either, @Mathein.

im not bad a t physic competition problems

@MatheinBoulomenos It is nice way of spending free time improving problem solving capability.

Here's a piece of trick math. There's a badminton tournament going on between a bunch of people, and every pair of player plays exactly one game together. Now each player makes a list consisting of other players he defeated, as well as players defeated by the players defeated by him (think of this list as a "list of formal inferiors" of that player). Prove that there exists one dude who lists every other player involved in the tournament. Also find an algorithm to determine the person.

they require a unique insight that im very good at

This ain't hard, just cool.

@MatheinBoulomenos eventually once I get really good in geometry and algebra. I would like to start working on my physics.

I have a problem similar to that in my algebra book, Balarka. But for my problem you have to have beaten the person below you on the list.

I don't enjoy competition problems all that much, but it can be fun spending an afternoon leafing through a Martin Gardner book.

Karim: Sent.

yeah got it thanks @TedShifrin

ima get sushi and then read eliashdjldfbasjhfbawhejrg mishdfjadfbkjhsdfkjhbKJHBECV

Just keep things concrete and not too rigorous, Karim.

LOL Eric.

yeah

@Eric chap 9?

yup

lemme quickly finish dinner then

OK, I'm outta here. Have to think a bit more about this geometry class and get going.

okay cya @TedShifrin

Bubye.

@TedShifrin Thanks again for the help! Take it easy.

Bye @Ted

@Eric Im done with din

@MatheinBoulomenos I should do that as well with geometry/algebra/and analysis

solve problems and read books in free time

i have to go grab lunch before i start

ill read it after i eat

fer sure

im going to skim around the symplectic linear algebra bits

mhmm

Hey @Alessandro do you know any really good Italian math books?

Uhm, do you have a particular topic in mind?

I'd prefer algebra or number theory or maybe algebraic topology

or algebraic geometry

I don't know of any outstanding one, we usually use books by English authors

I see

@Alessandro This is random, but I remember recommending you the album "Tilt" by Scott Walker. Did you ever have a look at it?

Just got reminded of it arbitrarily because I have been listening to it's predecessor album again

There's something wrong with the following statement "Therefore, this function is differentiable but not of class C1", right?

No, there is not: it may just be confusing 'cause ambiguous to which "this" refers to". If the derivative is not continuous, then "this" function does not belong to $C^1$.

That is, the existence of the derivative is not sufficient but necessary for a function to belong to the $C^1$ class.

Hence, this terminology is used in contexts where people talk about "smoothness", as the title of that Wikipedia title.

'to which "this" refers to'

Observe and admire the rouf symmetry.