You know what, $\delta_{L/K}^{-1} \delta_{E/K} \subseteq \delta_{E/L}$ has to be the easier direction because at least its clear that $\delta_{L/K}^{-1} \delta_{E/K} \subset E$.
But I'm stunned by how a differential geometry student cannot know in a second what the geometric meaning of $v\cdot w = c$ is for fixed unit vector $w$. It comes from all these algebraic courses that never talk geometric meaning.
When everyone in the world comes with a different set of knowledge and experiences it is amazing we are ever able to share ideas, using the assumptions the words I use have the same meaning and context to anyone else.
For what it's worth, @Ted, I think the fact that you leave "HINT: Think X or Use Y" as answers on many questions is excellent. I think the format of your tensor product of bundles answer was also like this, which is where I learnt how to think about them.
How do you teach anything in mathematics without using words? are we to let everyone else discover it on their own? Then why would we have schools and not just use libraries? My statement leads right into Math Education and not using this fancy new "Discovery Learning" instead of direct instruction where the meaning of things are explained.
About 30 years ago I taught a point-set topology class (followed by differential topology). Two math ed grad students were in the class, as well as a bunch of talented math major undergraduates. One of the math ed grad students had taken 3 or 4 classes from me before this. I didn't do "discovery" teaching, but I had a lot of participation, where the students would talk about how something should go.
And then I'd write up something more official after they had seen the ideas. Typically, the undergraduates were the propelling force here. One day, the math ed grad students, who always lobby for "discovering learning" in their curriculum, just lost patience and told me they wished I'd just put traditional lectures on the board.
I laughed.
One of them (not the one who was taking her 4th class from me) stopped taking classes from me after that.
Notice as a point of orthography that 'topos' is a French word, formed from 'topologie.' and not a Greek word. In writing, Grothendieck always forms the plural according to the French rule for words ending in 's,' so it is invariant—'les topos.' So the English plural ought to follow the English rule—'toposes.' Freyd. a confessed lover of classical endings and the inventor of cosmoi and logoi among other types of categories, says he heard that Grothendieck spoke of 'topoi' in Buffalo. I regard this as biased hearsay which can not stand against the published record
Also, I boycott terminology that was probably coined by Freyd
Well, that person (whose post on dot product I'd linked) finally got discouraged by my berating. No further comments. If he/she figures it out for him/herself, as would be appropriate, good.
I am of the opinion that Discovery learning was developed by people of privilege. The privilege of time to think. privilege of having people to ask, and bounce ideas off of. I understand that people feel a closer connection to the idea if they came up with it, but we are usually comparing students to having the "game changing" ideas that helped transform the foundations of mathematics to what it is today.
@copper I truly miss my teaching (or else I wouldn't be frustrating myself here), but I think I picked the right time, given the politics of the Regents in GA and COVID.
@TedShifrin The prevalent method to prevent the kinds of abuse you mention has been to remove all questions that show little context. In most cases, this is warranted, However, there are some good questions with little said in the question, and these get swept away with the rest, along with some good answers.
I was tickled yesterday, though. One of my undergrad advisees/students, who went into math thanks to my multivariable math course, just finished his Ph.D. defense (applied math) at UCSD yesterday. He introduced me (on Zoom) to his advisers and one of them thanked me for inspiring him and sending him to them. It was sweet.
Agreed, @robjohn. But the big problem seems always to be the mobs of MSE answerers who want their rep and don't care about standards or actual teaching.
Really interesting stuff on signal processing, compressed sensing, numerics and theory both.
@TedShifrin I started initially in CS and physics I would like to eventually contribute to both fields. I am very interesting in using geometry techniques in machine learning.
I am just saying that not everyone can go home and think for 3 hours about math class and bounce ideas of a group of people with the like minds. Sometimes you have to go to work to help your mom make the house payment and feed your brothers and sister. It leads me toward a idea of fairness, that I tell people what I am looking for, my definitions, my thinking about the problem.
Yes, it was a nice moment. I owe the student a fine dinner at one of Berkeley's great restaurants when next I get up there and can eat in a restaurant.
I was asking a philosophical question about truth and mathematics here math.stackexchange.com/q/4042325/893293 What do you think about math and truth? Is mathematics Truth, or just semantic arguments?
it's impossible to answer these things. everything's circular, everything's convention. In the hope that convention actually reflects the things we encounter in reality
I mean, one can implement the syntactics in a computer. But that in the end, physically, is electronic circuits so it's not so much different from symbols on a paper. just a representation of what we think is true, although it does hold up pretty well
@KarimMansour we do start from axioms, but before we do that we define a language that has infinitely many symbols
I wish I could meet Terence Tao and ask some silly questions about my understanding of the Colaltz Conjecture. I have a model that would extend forever to show the Collatz True, but its just a image in my mind. I don't know how to write it up.
At least in FOL you do the following: You have an ambient set theory at hand (you can argue about collections and such). Then you start with an infinite collection of variables, arbitrary symbols, arbitrary collections of relations (mostly including equality, relations can also be functions), logical constants (assuming you have a specified object like a zero in a group)
You can also assume to have specific function symbols, which ease up the notation immensely, but are not strictly needed
I have spent 6 years looking at the Collatz conjecture, in base 2, 3, 4, mapping its stopping time, programing in Python to create time and processing efficient methods of computing, creating new notation to look at the problem differently, but I am not PHD, and possibly chasing my tail.
Similarly to that we cannot truly implement a TM, we only implement larger automata
And to these recursively defined formulas you then define "it is true in a model" by evaluating truth of the statements, with objects from the universe plugged into them, in the metatheory
That's tarskis famous '"Snow is white" is true iff snow is white'
Although I think I didn't explain that all perfectly. We can work in a very very weak theory which can follow computations, e.g. peano arithmetic, where you can faithfully argue about the truth of statements like "turing machine X halts after Y steps"
Then we can encode any proof in whatsoever reasonable theory via arithmetic statements and we can decide "correctness of proofs" in the system, by just following computational rules
hey, guys. Can someone help me? I have to describe all the equivalent classes generated by the relation in $\mathbb{Z}$ given by $aRb \Longleftrightarrow |a| = |b|$
My solution is based like a function of integers to naturals. Each equivalent class have the number and your opposite. My description to all the equivalent class is: $\left\{ C_{a} \right\}_{a \in \mathbb{N}} = \left\{b = |a| \in \mathbb{Z} : aRb \right\}$
@BigSocks my first time in france, i was really looking forward to a continental breakfast - i grew up in ireland, and an irish breakfast is a pretty solid affair that sets you up for a day of throwing hay cocks around.
so, i was staying in a hostel in normandie and it came with a continental breakfast. i reckoned an irish breakfast is pretty good, but things on the continent are better, so i went down in the morning and got a coffee and a croissant finished it off and asked where is my continental breakfast? they looked puzzled and said you just ate it. luckily the day improved with a fresh baguette.
maybe my daughter can take the eurostar and collect my wine for me. then i got to get it back to the usa from the uk.
https://math.stackexchange.com/questions/2099596/computing-the-cohomology-ring-of-the-orientable-genus-g-surface-by-considering these cup product calculations make me despair
As a graded abelian group $H^*(\Sigma_g)$ is $\Bbb Z[1] \oplus \Bbb Z[\alpha_1, \beta_1, \cdots, \alpha_g, \beta_g] \oplus \Bbb Z[\gamma]$, factors ordered by grade (0, 1, 2 respectively), and fundamental class is denoted as $\gamma$, the generator of grade 2. It is clear that $\gamma \smile 1 = \gamma$; except this $\gamma$ cups with all other generators and gives $0$. It only remains to figure out what $\alpha_i \smile \alpha_j$ and $\alpha_i \smile \beta_j$ are.
To do this, consider the map $f : \Sigma_g \to \bigvee_{i = 1}^{g} T_i^2$, given by collapsing an appropriate subspace (psst: wedge of circles) to a point. Under this map, $\alpha_i, \beta_i$ gets sent to the meridian $a_i$ and longitude $b_i$ of $T_i^2$, for each $1\leq i \leq g$. $f^* : H^* \left (\bigvee_{i = 1}^g T_i^2 \right) \to H^*(\Sigma_g)$ sends $f^*(a_i) = \alpha_i$, $f^*(b_i) = \alpha_j$.
Observe $a_i \smile b_j$, $a_i \smile a_j$ are all zero in the domain for $i \neq j$. This forces $\alpha_i \smile \beta_j$, $\alpha_i \smile \alpha_j$ to be zero in the codomain as well, for all $i \neq j$, because $0$ gets sent to $0$ by the ring homomorphism $f^*$.
What we have so far: $\alpha_i \smile \alpha_j = 0$ for all $i, j$, $\beta_i \smile \beta_j = 0$ for all $i, j$, $\alpha_i \smile \beta_j = 0$ for all $i \neq j$.
Only $\alpha_i \smile \beta_i$ remains to be understood. Finish it from here?
This is how you should write all your computations. Write the graded abelian group, then understand the graded multiplication on the generators using naturality by some map which breaks the space to simpler spaces, as you said, while preserving "intersectionality" (because that's secretly what cup product is anyway) on the generators you'd like to understand for the time being.
The technical explanation in the exercise class was pretty confusing to me since it didn't make obvious how all this came to being and why it is obvious from here. One question, since everything else is pretty clear: We immediately treat the dual classes as the same as the classes themselves? I see that the groups are isomorphic
@user2103480 Oh no, that was just my way of saying what the generators of $H^1(T^2)$ are. You can call "meridian" to be "Rumpelstiltskin" and "longitude" to be "Baba Yaga" if you want.
It is true that that is what they mean geometrically by PD, but you don't need to think like this.
Knowing PD makes life simpler because Rumplestilskin $\smile$ Baba Yaga = ???, but meridian $\smile$ longitude = 1 because that is what you get when the $\smile$ becomes a frown, $\cap$.
So instead of symbolic mathematics now you can make informed guessed based on the meaningful hotkeys you're keeping stored in your head.
you could also draw the fundamental polygon, represent the fundamental class in this fundamental polygon and explicitly evaluate these various cup products on the fundamental class, similar to the torus example we discussed earlier
What's the word or phrase for a structure which is much like a vector space, except the set of "coefficients" isn't a field, it's a commutative ring with unit?
Find the slope of the curve y=x(x+1)(x+2) at the points where it crosses the x-axis. Help me please with solution.
someone help, thank you
