Let $G = \Bbb{Q}_{\gt 0}$ or $\Bbb{Q}^{\times}$. Can we write $G$ as a limit or colimit of finite abelian groups? I can't find any information about this by googling.
Limits of finite Abelian groups are finite or uncountable (see if you can construct a surjection to the Cantor set when it's not finite). Colimits of torsion Abelian groups are torsion. Since your group is a countable non-torsion group it is neither a limit not colimit of finite Abelian groups b
@user2103480 well, I have it so that even if I started smoking weed again, I begin (after a week say) even having cleaned out my lungs for 6 months, that I start coughing up dark-green to black mucus
If you don't cough up the stuff, then I guess all the crap is staying inside you!
I can't do any weed whatsoever any more, because people start reading my mind, and I'm not joking or that crazy. Weird paranormal shit just happens, and the weed makes it more uncomfortable.
You'd think oh cool super powers. But I have no use for people reading my random unsorted thoughts. And my brain is permanently out-of-sorts from the tocracco and weed abuse
smoking cannot do serious damage until you have had at least 10 pack-years of smoking, and the damage increases exponentially every pack-years after that. but fwiw i quit after 2 years because it gave me chronic pharyngitis; im about 6 months in.
yes, all variance considered, the scientific consensus is you've done negligible damage after 10 pack-years of smoking. of course you're never going to return to a non-smokers lungs, but that's not what i meant by serious damage
Well, now I'm preaching to the choir, so will shut up ;)
For me, I would get high off of smoking, high off of relaxing calmness. The first hit after a few days sober is this intoxicating nausea and I know I had hit the right note
So much so, that I would just crave breathing in smoke
I smoke everything. What's left of the butts? They all go into a pipe, until I am carpet picking like a tweaker does
It's so strange. I then want cigarettes over weed!
We have an abundance of flower because it's legal to grow here in CA, but I long ago shot my mental neurotransmitter wad and have screwed up (something)
I'm trying to exercise more now.
I also used to be way more proficient at math
I can do crazy abstract things on weed because I forced myself long ago. However, since having quit weed for now about a month ago, my proficiency at concentration / focus / studying power has gone way up again, and I can actually learn again !
You won't notice right away after stopping the smoke. I think also I'm even better at studying off of cigs as well
It's been a pleasure protheletizing to the audience in chat. :D
I've smoked amphetamine before, and didn't get addicted (luckily). Have smoked or snorted it < 10 times in a 20 year span. Stayed up for no more than 2 nights. Had I had access to more, I would probably not be alive today. None the less, Cigarettes - I got addicted to
my step brother struggled with a crack addiction. it was not great. it seemed more like alcohol, where genuinely you do have unpleasant side effects beyond just being pissed off if you don't keep the addiction going
There was also a fellow grad student who smoked during one of my seminar talks, even though it was expressly forbidden. Like I wasn't going to smell the damn thing.
i had tea with lester dubins a few times where he would pose me problems i wish i had written down on something more than napkins, and he had stories about visiting erdos in the hospital
Interestingly, I don't remember a single math faculty member at MIT smoking during my 6 years there. A good friend and French professor from whom I took 6 classes smoked plenty, though :(
Actually, I was in many faculty members' houses numerous times, but never remember being in Wu's. I still laugh at the fights Kobayashi and Chern used to have about who could see more bridges from whose house.
wu lived across the street from marc rieffel and next to rieffel was henry helson. marina ratner almost moved in next door, but eventually didn't. i think that would have been too many mathematicians.
ted you may be right. i took both of them from wu so you can understand my confusion. i did not take the undergrad diff top. it's 214 or something in the grad curriculum, i took that instead.
My topology/geometry qualifying oral was Wu and Kirby. They must have figured out they were passing me, because they asked me various crazy things I actually didn't know well at the time (Thom-Pontryagin and good grief why a manifold with a connection had to be paracompact).
Mike is a strange character, but we became friends because of that course. Because of that, his Calculus book got revised/changed (lots of exercises and some change to the text) because of my influence over the years.
my qual was mostly people asking me about details behind compactness and convexity and stuff i had never really studied. it gave me heartburn although in retrospect they had clearly decided to pass me. i was totally winging it. i think i disappointed marc rieffel. i took the bus home and threw up
Exams were different after 1977 or so. They changed from three 1st year oral qualifyings to a written prelim plus advanced oral. I guess I was on the committee that changed things (so many grad students had total melt-downs with the first-year oral quals, plus there was a serious problem that a student could pass those all and still have no clue who would be an adviser).
"My field of work is algebraic geometry. A significant part of my research is concerned with the study of the geometry and topology of various moduli spaces. Among other things, I am interested in moduli of curves, enumerative geometry, syzygies of algebraic varieties and commutative algebra aspects, vector bundles on curves, abelian varieties and theta functions, Prym varieties, K3 surfaces and Brill-Noether type problems."
first year oral quals sounds like a recipe for disaster. we had the prelim thing. it was better. some people couldn't pass it, but i think it was a meaningful filter. i still do not know why an hour of my qual was spent on generalizations of the krein milman theorem.
In all my years on oral committees at other places subsequently, there was usually a very tight arrangement with the student and the committee of exactly what topics were to be fair game. That doesn't mean that it always worked out well, but usually there were no surprise topics.
i had no idea what it would be going in. there was some stuff i was reasonably prepared for, mostly on operator algebras, and then it went way off track. my outside member wanted to see how much i remembered about CHARACTERISTIC CLASSES for some reason. i will never forgive him.
Um ... that's way too vague. Even when I wasn't on grad students' committees at UGA, I always advised them to be very specific in structuring the oral exam syllabus topics.
we didn't get to submit like a precise list of topics, only families of topics, so the word topology appeared somewhere, and of course you're going to get into characteristic classes
bill arveson was my advisor. he was great and stuck to the script. my outside member did not
The orals on first-year topics which I took definitely gave an advantage to people like me with both teaching experience and confidence. It really was a disaster for a lot of folks.
Well, Chern (who was my adviser) died a long time ago. I was very honored that he came to visit and speak at UGA because I invited him. I got to cook him and his wife another great dinner, too.
Ah, Hutchings probably wouldn't like my answers about characteristic classes (geometry more than topology) :P
when i say this crap about the ring structure on some cohomology crap of the spaces you care about, i'm just parroting what you told me, it's not coming from me
i went back in 2010 and talked to don sarason who now has also passed
And the exercises were hard but often doable and useful for the lecture at the same time. Instructive routine verifications, and the harder exercises were often exactly that level of difficulty where one can solve it after bashing one's head against it long enough
(2) Assume $\mathrm{V}=\mathrm{L}$. Let $C$ be a closed unbounded subset of $\aleph_{1}$ and let $S$ be a stationary subset of $\aleph_{1}$. Prove that there is an $\alpha \in$ Ord with $\left(\mathrm{ZFC}^{-}+" \aleph_{1} \text { exists } "\right)^{\mathrm{L}_{\alpha}}$ and $\aleph_{1}^{\mathrm{L}_{\alpha}} \in C \cap S$
The homotopy group $\pi_3(SO(4))$ classifies oriented $4$-dimensional vector bundles over $S^4$ by the clutching construction. There's an isomorphism $\mathbb{Z}\oplus\mathbb{Z}\rightarrow\pi_3(SO(4))$, mapping $(h,j)\mapsto(x\mapsto(y\mapsto x^hyx^j))$, quaternionic multiplication being understood. What Milnor does is look at the associated sphere bundles $M_{h,j}$, i.e. $S^3$-bundles over $S^4$; explicitly they are given as $D^4\times\mathbb{H}\cup_{f_{h,j}}D^4\times\mathbb{H}$ glued together along $f_{h,j}\colon S^3\times\mathbb{H}\rightarrow S^3\times\mathbb{H},\,(x,y)\mapsto(x,x^hyx^j)…
Thom's result is equivalent to the dual of the rational bordism ring having the Pontryagin numbers as basis. Since signature is bordism-invariant, this implies it is a polynomial combination of Pontryagin numbers. Try CP^4 and CP^2xCP^2 and you will get this formula.
Good writeup, maybe a bit formal for me! I would have phrased it by observing that the formula $\sigma = \frac{1}{45}(7p_2 - p_1^2)$ implies immediately that the invariant of closed 8-manifolds given by $p_1^2/45$ is divisible by 7. Whenever you have a result with a term like that this suggests you tend to get a relative invariant one dimension down in a group where you set this equal to zero
The Milnor invariant lies in Z/7, since the ambiguities resolve to a term in 7Z
On a closed oriented 4-manifold with bundle E and connection A, one can pass from the curvature 2-form F_A to the 4-form tr(F_A^2), then integrate that over M to get a number in 8pi^2 Z (8pi^2 times a Chern class)
Right, since it can be interpreted in terms of characteristic classes
Then on a closed 3-manifold with bundle [meh, don't worry about this too much, just pretend it's the trivial bundle] you can pick a closed 4-manifold it bounds and a connection it bounds --- $(Y, E, A) \to (X, \mathbf E, \mathbb A)$
Then set $\text{cs}(A) = \int_X \text{tr}(F_{\Bbb A}^2)$, and because of the above --- that the integral of something like this on a closed 4-manifold lives in 8pi^2 Z --- it follows that $\text{cs}(A)$ gives a well-defined $\Bbb R/8\pi^2\Bbb Z$-valued function
There are explicit formulae for this that don't involve X
But fundamentally to me it's noticing that whenever you have some set of discrete values for an invariant (which are more than what appears to be all possible values, a priori) you get a relative sort of invariant one dimension down
Physics fun fact of the day: Every real R.V. has a density Proof: $\Bbb P(X \leq t) = \Bbb P(t-X \geq 0) = \Bbb E[1_{(0,\infty)}(t-X)]$ so $$ \frac{\mathrm d}{\mathrm d t}\Bbb P(X \leq t) = \frac{\mathrm d}{\mathrm d t} \Bbb E[1_{(0,\infty)}(t-X)] = \Bbb E[ \frac{\mathrm d}{\mathrm d t} 1_{(0,\infty)}(t-X)] = \Bbb E[\delta(t-X)]$$
it cropped up on twitter not too long ago in some mini-pissing contest (i.e. 50% of math interactions on twitter) in relation to something inscrutable by Shelah, but I do not think this is Shelah
could any please explain why the order of matrices multiplications are same for the following conventions? At least for Z-Y-X, I'm expecting Rzyx= Rx()Ry()Rz()
Let $G = \Bbb{Q}_{\gt 0}$ or $\Bbb{Q}^{\times}$. Can we write $G$ as a limit or colimit of finite abelian groups? I can't find any information about this by googling.
