I cook mostly French, Italian, some Asian, and eclectic stuff I make up, @Stan.
Will evidently do less cooking now that most of my friends go out so much and I have an inferior kitchen to what I used to have. But so much better produce ...
@MikeM: You'll be amused. The woman I played bridge with last night turns out to have got a math Ph.D. from Ralph Fox in geometric topology in the 70s. She long since left math and worked in the computer/defense industry. But ... small world. She just sent me a paper she wrote on knot/link concordance.
@TedShifrin: Studying flat connections is natural. I certainly wouldn't have thought to ask about $F_A^+$. Maybe it would have been more obvious if I had known $d^+$ fits into an elliptic complex or whatever.
I think Perelman would object to being called an idiot, since he's far from one.
Feel free to call him an idealogue.
The Yang-Mills millenium problem is different from the one mathematicians usually look at. The standard mathematical case is to have a specific, special $G$ - $SU(2)$ and $SO(3)$ are popular (I think $U(2)$ is too? and $SL_2(\Bbb C)$ is hip nowadays), and work on closed manifolds. $\Bbb R^4$ is far from closed, and this is for any $G$.
Don't ask me anything about what's going on in the statement of th eproblem beyond that.
I think Perelman is clinically shy, on top of all the ideology. ... I think trying to minimize any energy functional is totally natural, @MikeM ... Self-dual or anti-self-dual is natural enough, too.
A few downvotes I've gotten I could understand — I was giving hints rather than complete solutions. But most of the ones I've gotten seem to be political in nature.
I only downvote when I complain to the author that something's wrong and he/she ignores me.
So, @Stan, when do we get back to discussing math instead of metamath?
Really, really fun. I love it. Easily my most favorite job so far, despite the 55-60 hrs/week. I managed to dig around and find a way for me to learn web development through this company, and my boss is enthusiastic about that idea, since no one usually pursues that route in my role :)
Right ... I still remember the fascination I had learning (in my math adulthood) about the Borromean rings.
I'm settling in in CA, @Clarinet, thanks. Went back and played bridge last night (rather poorly, but ...). Baby steps.
I know basic probability (I hope) after teaching it. Don't know as much stat as I should, although I directed a MA thesis on it.
Didn't I tell you about the guy that did a MA thesis in math on the geometric/linear-algebraic stuff all through stat? He claimed that the stat folks never teach it geometrically at all.
@TedShifrin If you know what a hypothesis test is, you should be able to handle what's on my blog. Basic probability... mmm, I'd say if you know that expected value is additive, you should be fine
Not the manifolds one, the Calculus one. I'd like to actually see how math majors are supposed to learn Calculus these days, since I hate the usual Stewart method
@TedShifrin: Last week I had troubl ebecause they talked about very little (derived distributions, and that's about it). Now there's too much: conditional expectations against other random variables, law of total expectation, moment-generating functions. Convergence in probability.
MGFs are kinda useless, IMO. I'm not sure why they don't just teach characteristic functions (these are not the ones you know about in measure theory).
yes, @Studentmath and I used to have conversations about indicator functions ... I tried to incorporate them more into my course because of those conversations.
LOL ... @MikeM, I know you'll think I'm bullshitting, but I actually did some of my best teaching/learning early in my career when I was super busy. The teaching energized me and I spent a lot more efficient time doing research.
@MikeM: You're very different from me. But if it weren't for the reward I felt from teaching (only taught 2 1/3 years in grad school), I would have quit. It kept me going.
I made a fuss at Berkeley and insisted on teaching my own multivariable/linear algebra class for mechanical engineers. Everyone else just TAed. I acted like a postdoc my 4th year when I taught. But I loved it.
I was going to try to make beer brats yesterday (we're only in the capital of Wisconsin...), but gf said we'd never drink the leftover beer (which is true), so I tried simmering brats in water, cinnamon, black pepper, and italian herbs
I guess the point is that to teach probability right at an advanced level one really needs measure theory and Lebesgue ... which is why one does that for the grad course. :)
That's weird, @Clarinet. You could just buy an individual bottle of beer at the liquor store.
So is your governor still bent on destroying the university system?
@TedShifrin Yeah, I was wondering about that, but gf says you can't buy beer without having to resort to a 6-pack. I'll have to check it out. I've never actually bought beer for myself before heh
If the people polling for him listened to what he says other than the racism they'd be voting for someone else - he has a lot of frankly centrist beliefs.
I know nothing about the Democrat candidates other than Clinton and some Sanders guy who is apparently not accepting any sort of... how do I describe it, corporate funding? [at least that's my impression]
No there aren't. Gore has denied it, Biden hasn't publicly denied it but every rumor stems from Maureen Dowd, NYT columnist who hates Hillary and would love for an establishment candidate to run against her.
Some people are worried about Hillary actually winning and are desperately reaching out to every established name they know.
I won't try to sell my preferred candidate here because I've given other people crap for doing that in the past. Math chat room and all that.
Well, I told @Stan it was time to get back to real math. I'm done hassling Balarka for trying to understand the geometry of cup product when he doesn't have the background for it.
In any case, I've talked more about this than I should, given my previous opinion on other people doing so. Email me if you want to talk more. I'm a political junkie.
I mean, look at Scott Walker and Wisconsin. It's absurd what happened there and I remember my professor, despite what the university told her, telling us to not go to class and protest our views
back when they removed collective bargaining (2011?)
I'm the first person in my family with a 4-year degree. At times, it's difficult to relate to them, but things have definitely gotten better with the complete career change
I'm amazed that people like Terry Tao take the time to maintain a serious blog. I don't know how he manages to do that on top of research, teaching, and family.
Well, I'm outta here for now. Have a good evening.
How do I rigorously say that if I have a 2-dimensional Lie algebra, $\mathfrak{g}$ with basis $\{x,y\}$ and bracket $[x,y]=x$ that an ideal $\langle x\rangle$ $\ne \mathfrak{g}$? I can see that $\langle x\rangle$ can only generate 'half of the basis'. Do I just say that since $x,y$ are linearly independent, then $y\not\in\langle x \rangle$ and hence $\langle x\rangle \ne \mathfrak{g}$
I.e. $y\not\in \langle x\rangle$ since $x,y$ linearly independent, $y\in \mathfrak{g} \implies \langle x\rangle \ne \mathfrak{g}$
I mean, this is the definition of an ideal. It's $\mathfrak g \cdot \text{span}(x)$. $[ax+by,x] = a[x,x]+b[y,x] = -bx \in \text{span}(x)$ as desired.
Sorry, to be more rigorous. The smallest ideal containing some subspace $R$ is the union $R \cup [\mathfrak g, R] \cup [\mathfrak g,[\mathfrak g,R]] \cup \dots$. We call this the ideal generated by $R$. We've shown that this stabilizes immediately in this case.
@BalarkaSen Turns out my combinatorics idea was wrong. You can apply Yoneda's to get natural isomorphisms alright, but I noticed that my reasoning would have also applied equally to the subcategory of just {1}, but the symmetry groups in that case are both trivial instead of S3 and C2 so something must have been wrong. Figured out the issue: you can't apply Yoneda's lemma to restricted hom functors.
Not really sure what to do here. I guess it's pretty unsatisfying to just drop a reference, but Scott explains it better than I could in a couple paragraphs. I guess I could just quote the relevant sections.
@TedShifrin Hey Ted! I decided to tackle one of the problems on Neumann series in Chapter 6
I got to the point where I used the properties of geometric series to show $\frac{1-H^{k+1}}{1-H}$ converges to $\frac{1}{1-H}$ as $k \rightarrow \infty$. Is $\frac{1}{1-H}$ the inverse of $1-H$? I know this is true for numbers. And I guess if we are using norms, then it is true too. But I got a bit confused whether I should be working with $H$ or $||H||$. Should I send you the proof?
Unfortunately, that still happens. No, you need norms to establish convergence of the series of matrices (absolute convergence implies convergence). But you need to prove that you actually have the inverse matrix when you're done.
I keep saying that what you're writing makes no sense, @Stan. What's the definition of the inverse of a matrix $A$? It's a matrix $B$ so that $AB=BA=I$. So show that your candidate inverse satisfies the equation(s).
I am trying to show that the normaliser of a vector subspace of a Lie algebra is a subalgebra of the Lie algebra
I.e. I am trying to show $[N_\mathfrak{g}(Y),N_\mathfrak{g}(Y)]\subseteq N_\mathfrak{g}(Y)$ for $Y$ a sub vector space of $\mathfrak{g}$
I.e.(2.0) I am trying to show that $\forall n_1,n_2\in N_\mathfrak{g}(Y)$ that $[[n_1,n_2,Y]\subseteq Y$
After anti-commute + Jacobi identity etc I get:
Down to wanting to show that $[n_1,[n_2,Y]]+ [n_2,[Y,n_1]]$. Do I write then:
$$[n_1,[n_2,Y]]+ [n_2,-[n_1,Y]]=[n_1,A]+[n_2,B]$$ where $A,B\subseteq Y$ and then since these are subsets of $Y$, it holds that $[n_1,A]+[n_2,B]= C+D\subseteq Y$
I.e. those lie brackets with $Y$ become subsets of $Y$
So my question is, am I in the right direction from the "Down to wanting to show that" onwards
Down to wanting to show that $[n_1,[n_2,Y]]+ [n_2,[Y,n_1]]$. Do I write then:
$$[n_1,[n_2,Y]]+ [n_2,-[n_1,Y]]=[n_1,A]+[n_2,B]$$ where $A,B\subseteq Y$ and then since these are subsets of $Y$, it holds that $[n_1,A]+[n_2,B]= C+D\subseteq Y$
I need to know more language for decribing relationships between vectors.
I have noticed that for a particular collection of sets of points, the point closes to the centroid is very rarely the point closest to all the vectors in the set.
I feel like there should be words to decribe this.
and I feel like it it indicated the sets of points are themselves weird.
