Daniel Moskovich
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Registered User
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I am a postdoc working on quantum topology of knots and 3-manifolds, although most of what I'm actually doing is fairly classical. I am currently a postdoc at the Nanyang Technological University. I blog on mathematics that interests me at Low Dimensional Topology.
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objects which can’t be defined without making choices but which end up independent of the choice @Jeremy Hahn: You're right- as a functor, π<sub>1</sub> is indeed a functor from a category of based spaces. Thanks! But despite this, the fundamental group of ONE SINGLE path connected space is always considered only up to automorphism (isomorphic fundamental groups are considered "the same" to the point of placing equal signs between them), and therefore is base-point independent for all practical purposes, isn't it? |
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1d |
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objects which can’t be defined without making choices but which end up independent of the choice @anon and @Jeremy Hahn: "Caring only up to automorphism" arises even with a choice of basepoint. Which based loop in S^1 should map to 1 in Z? "Which based loop in S^1 wedge S^1 corresponds to a given element of Z*Z"? These are all entirely arbitrary choices. So actually, maybe the fundamental group can be looked at as a functor from the homotopy category of path-connected topological spaces to the skeleton of the category of groups... I think; I know it's never phrased like that. And that's basepoint-independent. |
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May 21 |
answered | objects which can’t be defined without making choices but which end up independent of the choice |
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May 18 |
answered | Is there any proof that you feel you do not “understand”? |
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May 12 |
awarded | ● Self-Learner |
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May 10 |
awarded | ● Necromancer |
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May 3 |
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What is the best *general triangle*? Discussion at meta.mathoverflow.net/discussion/1582/… |
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Apr 30 |
awarded | ● Popular Question |
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Apr 25 |
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Diagrammatic proof of unique prime decomposition of knots Thank you for this answer! So the answer, essentially, is that no such proof exists and that any such proof seems out of reach. So, for now anyway, prime factorization of knots and links is essentially a topological rather than a combinatorial fact. |
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Apr 22 |
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Learning roadmap for geometric topology mathoverflow.net/questions/93408/… . See also math.stackexchange.com/questions/128962/… |
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Apr 22 |
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Compelling evidence that two basepoints are better than one Thank you for this answer! This past week, I just finished teaching a course out of "A concise course", following which I am convinced that indeed "the fundamental groupoid is such a natural thing, and so elementary, that it seems a little perverse to try to avoid it!", to the point that I used groupoids arguments to a greater degree that the "concise course" does in some parts of chapter 3, for example. |
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Apr 19 |
answered | Great mathematics books by pre-modern authors |
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Apr 18 |
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Difference between connected vs strongly connected vs complete graphs Comments shouldn't be posted as answers. |
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Apr 18 |
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Difference between connected vs strongly connected vs complete graphs What you have written is a comment rather than an answer. |
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Apr 15 |
revised |
Diagrammatic proof of unique prime decomposition of knots condition added, final paragraph added |
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Apr 15 |
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Diagrammatic proof of unique prime decomposition of knots Thanks for your answer. The statement of unique prime decomposition has purely combinatorial content- one way or reformulating it would be that there is a unique way to partition chords in a Gauss diagram into disjoint connected sets, modulo Reidemeister moves. And we can get a lot of mileage out of Gauss diagrams (e.g. the Homflypt polynomial), so I'm surprised if we strictly need topology and things like crossing numbers. |
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Apr 14 |
revised |
Diagrammatic proof of unique prime decomposition of knots added 32 characters in body |
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Apr 14 |
asked | Diagrammatic proof of unique prime decomposition of knots |
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Apr 4 |
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symmetric L-groups I care about this also, but I suspect that the answer is no, based on here: mathoverflow.net/questions/48328/… My feeling is that it's hard. |
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Apr 2 |
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Can all knot polynomials derived from Skein relations be categorified? I'm not sure I understand the question. "Can" as in "there is no obstruction in principle", or "can" as in "has been"? |
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Mar 28 |
awarded | ● Popular Question |
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Mar 28 |
awarded | ● Popular Question |
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Mar 22 |
accepted | Understanding four manifolds (more details inside) |
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Mar 21 |
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Are isolated knots in tangles the same as ordinary knots? Two long knots are equivalent if and only if their closures are equivalent. This is because any Reidemeister move on the knot can be realized by Reidemeister moves which avoid a single point, which you can take to be the point at infinity. This is a combinatorial proof. I think there's a smooth proof also. |
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Mar 20 |
awarded | ● Necromancer |
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Mar 18 |
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Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds? @Oscar: corrected to make the statement only for $m\leq 3$. Sorry about that. |
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Mar 18 |
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Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds? added 251 characters in body |
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Mar 17 |
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Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds? deleted 2 characters in body; added 5 characters in body; deleted 3 characters in body; deleted 2 characters in body |
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Mar 17 |
answered | Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds? |
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Mar 17 |
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One question about the quandle A stronger question would be whether knot quandles are residually finite (distinguished by homomorphisms onto finite quandles). |
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Mar 14 |
answered | Understanding four manifolds (more details inside) |
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Feb 25 |
awarded | ● Popular Question |
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Feb 25 |
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Is a knotted trivalent graph determined by its set of unzips? Thinking about this some more, I'm confused: We want the boundaries of the pants to give us the same 3-component <i>framed</i> link. How does this argument take framing into account? Isn't the framing different for the two sides? |
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Feb 24 |
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Is a knotted trivalent graph determined by its set of unzips? This is a very nice idea which should answer the question (after carrying out such a computation)- thank you! |
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Feb 20 |
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Is a knotted trivalent graph determined by its set of unzips? Thank you! I will try this- this sounds very promising in fact! |
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Feb 18 |
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Is this result of Spain correct? Not sure this is a question- voting to close. |
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Feb 18 |
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A description of cellular boundary maps in terms of a Morse function I'll write this up as an answer maybe later (or delete it if it turns out to be wrong), but (if I'm not mistaken) I think this is explained in Goresky-MacPherson "Stratified Morse Theory". |
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Feb 16 |
awarded | ● Good Answer |
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Feb 14 |
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Optimal fitting of spheres in a cylinder. I misjudged this question. As asked, it is trivial, but Joseph O'Rourke's answer indicates that there is an interesting question in the background. |
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Feb 14 |
asked | Is a knotted trivalent graph determined by its set of unzips? |
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Feb 14 |
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Mathematicians whose works were criticized by contemporaries but became widely accepted later And Lobachevsky? |
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Feb 14 |
awarded | ● Nice Answer |
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Feb 13 |
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Mathematicians whose works were criticized by contemporaries but became widely accepted later This is an excellent example, I think. |
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Feb 12 |
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Mathematicians whose works were criticized by contemporaries but became widely accepted later @Jonathan Chiche: Hence the word "perhaps" :-). It would be quite a coincidence if "prosodic reasons" were indeed literally the only reasons, given that Lobachevsky was both famously ridiculed, and later famously falsely accused of plagiarism (he was accused of stealing his ideas from Gauss). |
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Feb 12 |
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Mathematicians whose works were criticized by contemporaries but became widely accepted later I strongly feel this question should limit itself to mathematicians who died before, say, 1960. Otherwise it borders on gossip- I can name contemporary mathematicians whose papers were ridiculed but later accepted, but to do so would be entirely gossip. |
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Feb 12 |
answered | Mathematicians whose works were criticized by contemporaries but became widely accepted later |
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Feb 12 |
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“Philosophical” meaning of the Yoneda Lemma Thank you for your comment. I've used it to clarify and expand this answer- does it make more sense now? |
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Feb 12 |
revised |
“Philosophical” meaning of the Yoneda Lemma clarified and extended; deleted 10 characters in body |
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Feb 10 |
answered | “Philosophical” meaning of the Yoneda Lemma |
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Feb 7 |
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Why is Set, and not Rel, so ubiquitous in mathematics? I don't really understand (3)- what kind of analogue to Yoneda's Lemma would you envision? |
