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Aug
14
comment Can one branch of mathematics be completely learned from the perspective of another branch of mathematics?
@quid: OK, so the problem is just that we understand this answer differently. I think the author should clarify his point. If he will write that OFTEN IT IS NOT SUFFICIENT TO KNOW THE SET-THEORETIC DETAILS..., then I would not object! But up to this moment I would think, that my objections are correct...
Aug
14
comment Can one branch of mathematics be completely learned from the perspective of another branch of mathematics?
@quid: In your example it is helpful for you to know the definition of a group, is it? Goldstern writes that "it is often neither necessary nor even helpful to know the set-theoretic details...". So in this example you disagree with Goldstern, and agree with me, who claims that it is always helpful to know the set-theoretic details, right? If yes, then I don't see controversy between us. If no, then I don't understand your point. (Just in case: the downvote here was not mine.)
Aug
14
comment Can one branch of mathematics be completely learned from the perspective of another branch of mathematics?
I agree with you that differential geometry is a branch where specialists abuse their intuition, ignore the necessity to attach their field closer to the rest of mathematics. In my opinion, this is not because of the lack of will to explain everything directly from rationals, etc. I would say if they used category theory more actively, that would be much more clear. I don't want to scold anybody, but this is indeed a bright example.
Aug
14
comment Can one branch of mathematics be completely learned from the perspective of another branch of mathematics?
@quid: Maybe I misunderstood, but my point is that there are common languages that allow people to understand each other, and it is not wise to ignore this. I used to think that for mathematicians set theory is one of those common laguages (by the way, category theory is another). As a corollary this can't be useless to translate (from time to time) what you say into those common languages.
Aug
14
comment Can one branch of mathematics be completely learned from the perspective of another branch of mathematics?
In my opinion, the question should be changed a little bit, in its current statement it sounds rhetorical: much work in mathematics is carried out exactly for the purpose of finding relations between different branches. Perhaps, it would be better to ask "which examples of unexpected relations between different disciplies do you know?". Or something like this.
Aug
14
comment Can one branch of mathematics be completely learned from the perspective of another branch of mathematics?
"...to understand a high-level concept in some other branch of mathematics... it is often neither necessary nor even helpful to know the set-theoretic details behind it", -- this doesn't sound convincing for me. I would say, on the contrary, usually it is impossible to understand what people in one branch of mathematics have in mind, when explaining something, if you are not a specialist in this field, and you can't guess the set-theoretic backgroud that lies behind their explanations.
Aug
11
comment Bondarchuk, Kaluznin, Kotov, Romov’s Theorem on inclusion of Polymorphism ($Pol \rho \subseteq Pol \sigma$)
Bondarchuk (Бондарчук), actually...
Aug
11
revised when do norm-continuous unitary representations separate points of a group?
english grammar
Aug
11
comment when do norm-continuous unitary representations separate points of a group?
François, actually I'd like to have a theorem for general situation (not only for Lie groups, but a description for this subclass among all locally compact groups)... Is it possible that there is a gap between SIN-groups and those I am interested in?.. (Thank you for the reference anyway!)
Aug
11
revised when do norm-continuous unitary representations separate points of a group?
misprints
Aug
10
revised when do norm-continuous unitary representations separate points of a group?
I added "unitary" in the title
Aug
10
asked when do norm-continuous unitary representations separate points of a group?
Aug
5
comment A fixed point problem
Oh, I missed this.
Aug
5
comment A fixed point problem
@dimo Something is wrong, we do not understand each other. You consider a map from the closed interval $[0,1]$ into the plane $\mathbb R^2$ acting by formula $t\mapsto (tr,1-t)$ -- is this correct? And $A$ is the image of this map. If this is correct (and $[0,1]$ and $\mathbb R^2$ have usual topology), then this map is a homeomorphism between $[0,1]$ and $A$. (By the way, in this case it doesn't matter whether $r$ belongs to $\mathbb Q$ or to $\mathbb R$.)
Aug
5
comment A fixed point problem
Of course, this works only if I understand you correctly: the topology on $\mathbb R^2$ must be the usual topology of direct product of two $\mathbb R$, and each of those $\mathbb R$ is endowed with the usual topology of $\mathbb R$ (i.e. intervals $(a,b)$ form a base of open sets).
Aug
5
comment A fixed point problem
This is a general theorem: every bijective and continuous map from a compact space to a Hausdorff space is a homeomorphism, see R.Engelking, General topology, 3.1.13, see also Wikipedia en.wikipedia.org/wiki/Compact_space#cite_note-10 (the properties of Hausdorff spaces).
Aug
3
comment What are the qualities of a good (math) teacher?
I saw math teachers who are unable to build a logical chain carefully, without gaps and/or mistakes. I think, Jose had in mind that a teacher must have enough qualification for doing this in case when he sees that his pupils don't understand the logic of the exposition. Not that every time he must explain each millimeter of the proof. Am I right, Jose? :)
Jul
31
comment A single paper everyone should read?
I wonder if there is a progress since that time in the directions that Dyson mentioned as what should be done.
Jul
29
comment A fixed point problem
I don't understand the problem. $A$ is homeomorphic to the interval $[0,1]$. So everything follows from the Brouwer fixed-point theorem (it's variant for the dimension 1). The statement of the problem looks strange.
Jul
21
awarded  Revival