I have the most experience in combinatorics, I like everything I've seen in topology, and I'm working right now to learn the basics of probability theory
Which is a longwinded way of saying that I don't know but I get my letters written by the combo profs :P
Somewhat less flippantly, because my undergrad research was pretty much all combo, they are the ones who can speak best to the work that I've already done, rather than the work they expect me to do
I usually think of Poincaré as the beginning of algebraic topoology, specifically. But I don't know anything that was done before him, now that I think of it
He and I are Facebook friends, so I'm still in touch ... Ah, cool. I originally met him because he interacted a lot with one of my former students who teaches math at Univ West Georgia.
yeah, I advised him to take the Spivak Calculus w/Theory class, and he spent a few weeks in my multivariable math class, before we agreed that he would be happier doing computer science ... Neat kid, though.
@EricS: I have never been good at counting things, but I learned a little bit of combinatorics/probability a year ago teaching probability before I retired.
No, @Forever, he's got a ways to go. He really should have been in computer science from the beginning. We needn't talk everyone into math.
Because that's the fewest possible cases to study, @cap, and it breaks down naturally according to that. You should ask an algebraic number theorist for a definitive answer.
I definitely am not someone who tries to make everyone study math ... even less so for grad school.
But when I've had super-talented students who fall in love with it, I certainly am happy to encourage them.
I'm a bit worried about some of the high school kids I'm working with who're taking precalculus. Some are working hard and learning, but others are absolutely going to get their clocks cleaned when they go off to college. (And I think plenty aren't thinking of college.)
No, @EricS. After some work on my part, I'm volunteering at a charter high school. And I spend 3 1/2 hours a different afternoon working with 4th - 8th graders at a public library.
So yesterday with one of the precalculus classes I explained e and exponential growth, showed them Newton's Law of Cooling, and then we spent 20 minutes using logs to figure out when someone got murdered :P
@EricS: I do best as a teacher when students really want to learn, even if they're struggling. I find it frustrating to see kids who are totally unmotivated and just don't care.
@ForeverMozart: I think we all think about whether or not we're any good. But I think it's best to just enjoy math while we get to do it and let the chips fall where we may. When I'm not busy being distressed anyway that's what I do.
Howdy, @MikeM. @Forever, my two advisers (one official, one less so) were both famous pillars of 20th century mathematics. They were generous, wonderful people, but I never in a million years thought I'd contribute epsilon of what they did in research.
I always wanted to make a difference as a teacher, didn't think my research papers (despite a few in top journals) would matter much. I hope that my textbooks will have a bit of positive impact.
@Forever: It's useless for me to say that you shouldn't focus on things like that, because we all know it's not that easy. But I find that my life has generally gotten better as I have gotten better at ignoring that voice of doubt.
I'm not going to lie, getting into grad school helped a lot :P
But I think that even as a senior I was better than as a sophomore. For me, at least, it seems that I am most anxious about my abilities when I am doing the least actual work. The amount of progress that I make on the work has been more or less inconsequential
I've seen a number of students who've developed and improved markedly as time passes, and as stuff gets much harder. Others start to crumble. ... The laws of educational nature.
Yeah I definitely have felt generally comfortable with the faculty, except in one course. And I can imagine that if the environment was more like that, I'd feel very different about things.
It seems like the sort of question you would ask if you have more excitement about math than experience with it. While I agree that can be annoying, I always feel bad at my annoyance.
@Balarka: It's strictly about dimension 3, which can be made into a good question (I do not intend to do so). The four color theorem is a complete nonsequitur.
Cool proof of contraction mapping principle from D&K: $X$ be bounded, $f : X \to X$ a contraction mapping with Lipschitz constant $c \in (0, 1)$. Set $g : X \to \Bbb R$, $g(x) = \|x - f(x)\|$. Note that $\|g(x) - g(y)\| \leq \|x - y + f(y) - f(x)\| \leq (1 + c) \|x - y\|$, hence $g$ is Lipschitz continuous and thus continuous. If $X$ is bounded, $g$ admits a global minimum on $X$, call it $x_0$. $g(x_0) \leq g(f(x_0)) = \|f(x_0) -f(f(x_0))\| \leq cg(x_0)$.
But $g(x_0) \leq cg(x_0)$ implies $g(x_0) = 0$, as $c < 1$, which gives us the desired fixed point. If $X$ is not bounded, let $X_0 = \{x \in X : g(x) \leq g(x_0)\}$. Note for $x \in X_0$, $\|x - x_0\| \leq \|x - f(x)\| + \|f(x) - f(x_0)\| + \|f(x_0) - x_0\| \leq g(x) + g(x_0) + c\|x - x_0\| \leq 2g(x_0) + c\|x - x_0\|$. Hence, $\|x - x_0\| \leq 2g(x_0)/(1-c)$, hence $X_0$ is bounded anyway. It's easy to see $f$ restricts to a self-map of $X_0$, so proceed as before to find a fixed point of $f$.
Hmm, what I wonder is how does that tell me that the fixed point is unique.
Meh, I guess I can derive uniqueness anyway. If it has two fixed points $x, y$, $\|f(x) - f(y)\| = \|x - y\|$ and $\|f(x) - f(y)\| \leq c\|x - y\| < \|x - y\|$, impossible.
The proof doesn't give me a unique one though, unlike the other proof I know.
$a^2 \equiv 1 \mod{15} \implies a^2-1 \equiv 0 \mod{15} \implies (a-1)(a+1) \equiv 0 \mod{15}.$ This implies $a-1 = 0\mod{15}$ or $a+1 = 0 \mod{15}$, so $a=1,14$; it also implies that $a-1 = 0\mod{5}$ or $a +1 = 0 \mod{5}$, which gives $a=1,4$; and finally, $a-1 = 0 \mod{3}$ or $a+1 = 0 \mod{3}$ which gives $a=1,2$. Therefore $a=1,2,4,14$. This is wrong, but where did I go wrong?
So we're on a four and a half hour's difference. :)
@DanielFischer On a different note, I learned about the following theorem today: If $f : U \subset \Bbb R^n \to \Bbb R^m$ is Lipschitz continuous, then $f$ is differentiable outside a set of measure zero.
I was wondering if you can give me a reference which contains a proof of this fact.
@BalarkaSen Don't know a reference off hand. It's Rademacher's theorem, that name might help you. Perhaps it's proved in some book about geometric measure theory.
@GGG I see an error: $(a-1)(a+1) \equiv 0 \mod{15}$ does not necessarily imply that $a-1\mod{15}$ or $a+1\mod{15}$ is zero. Trivial counter-example: $a=4$
@skullpetrol i m surprised that the guy who is most tendentious to use "ignore" option against whom he calls "trolls" is even the most user of "f" word
I don't even remember anyone from here who calls people troll and then ignores them.
Sanity-check: If $f_n : U \subset \Bbb R^n \to \Bbb R^n$ is a sequence of Lipschitz continuous maps with same (least) Lipschitz constant $c$, $f_n$ converging to $f$ uniformly, then $f$ is also Lipschitz, right?
Because for any $x,y \in U$, $\|f(x) - f(y)\| \leq \|f(x)- f_n(x)\| + \|f_n(x) - f_n(y)\|+\|f_n(y) - f(y)\|$ and I can pick $n$ to be very large so that this thing is $\leq 2\epsilon + c\|x - y\|$ for any $\epsilon$ I choose. But then if I choose $\epsilon$ so that $\epsilon \leq (c' - c)/2\|x - y\|$ for some fixed $c < c' < 1$, my job is done, correct?
I am unsure because of the inelegant and ad-hoc choice of $\epsilon$ I have done there.
Sorry, I meant $c < c'$, not $c < c' < 1$. Typo. Was thinking of contractions.
Actually, my Lipschitz constants needn't be the same for each $f_n$. I think all I need is that $\sup(c_n)$ exists, where $c_n$ is the Lipschitz constant of $f_n$.
Then I can do the same argument to conclude the uniform limit is Lipschitz.
Well you start with $0b^2+0b+1$, then you multiply by $b$ and get an expression of the form $x b^2+ y b +z$, then every multiplying by $b$ gives you another quadratic expression upon use of the formula
