Sam Gunningham
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Registered User
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Student of David Nadler at Northwestern University (currently visiting UC Berkeley).
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May 2 |
accepted | Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles? |
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May 1 |
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Dense Affine Subvarieties of Algebraic Varieties @Dan: Sorry, didn't see your comment in time. |
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May 1 |
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Dense Affine Subvarieties of Algebraic Varieties Can't you just take open affines in each irreducible component which do not intersect any other component? Then the union of such will be disjoint. |
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Apr 30 |
answered | Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles? |
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Apr 30 |
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Quotient of Lie rings and quotient of Lie groups! Lie ring = universal enveloping algebra? |
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Apr 28 |
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Infinity-categories vs Kan complexes See Scott Carnahan's answer below for more about the relationship between the cubical and the simplicial way of thinking. Also, Ronnie Brown points out in his answer that the simplicial approach to higher groupoids is not necessarily "better", at least not in all regards. |
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Apr 27 |
awarded | ● Nice Answer |
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Apr 26 |
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What are some examples of weak ω-categories? I thought the $(\infty, \infty)$-category of bordisms should in fact be an $(\infty, 0)$-category. Naively, wouldn't being ``$\infty$-dualizable'' mean that every morphism is invertible? |
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Apr 26 |
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Infinity-categories vs Kan complexes Just to clarify Zhen's comment: the definition of Kan complex in the question at the moment is not correct. A simplicial set (not space) is Kan if every horn (not just inner horns) can be filled. |
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Apr 26 |
answered | Infinity-categories vs Kan complexes |
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Apr 24 |
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What is a higher derived constructible sheaf @David: I agree about the finiteness issues. I only mention this, as in the question, the $C_\ast(\Omega X)$-version of local systems was talked about. And it happens to be an issue that I am interested in myself! |
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Apr 24 |
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What is a higher derived constructible sheaf Even in the simply connected case, the two versions of $\infty$-local systems are not quite the same. As I commented below, they correspond to $C_\ast(\Omega X)$ vs $C^\ast(X)$ modules. These categories are closely related (by some version of Koszul duality), but not quite the same. |
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Apr 24 |
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What is a higher derived constructible sheaf @David: I think you need to be careful about finiteness issues. For example, local systems on $\mathbb C^\times$ in Dmitry's sense would be representations of $\pi_1 (\mathbb C^\times)$. But you won't be able to see the indecomposible infinite dimensional representations of $\mathbb Z$ using $D$-modules (I think the ``de Rham homotopy type'' should only be able to the pro-algebraic completion of $\pi_1$, or something along those lines...) |
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Apr 24 |
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What is a higher derived constructible sheaf The difference is more clear in the case $X = S^1$; in that case, the usual constructible derived category would decompose in to blocks for each generalized eigenvalue of the monodromy. This sees only the profinite completion of $\pi_1(S^1) = \mathbb Z$, as opposed to all representations. |
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Apr 24 |
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What is a higher derived constructible sheaf @BZ: Good point. I think the answer should be yes in reasonable cases, but you have to be careful about what you mean by the constructible derived category. For example, a constructible sheaf is usually taken to have finite dimensional stalks. The issue occurs even with no stratification (i.e. local systems). If $X$ is simply connected with no stratification (say), then the I would say that the usual constructible derived category is equivalent to $C^\ast(X)$-mod. This is not (quite) the same as $C_\ast (\Omega X)$-modules (though the two are closely related. |
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Apr 23 |
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What is a higher derived constructible sheaf Awesome! . |
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Apr 23 |
accepted | What is a higher derived constructible sheaf |
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Apr 23 |
answered | What is a higher derived constructible sheaf |
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Apr 22 |
awarded | ● Enlightened |
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Apr 22 |
accepted | HIgher Homotopy Groups and Representation Theory |
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Apr 18 |
awarded | ● Nice Answer |
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Apr 17 |
revised |
HIgher Homotopy Groups and Representation Theory added 1395 characters in body |
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Apr 17 |
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Connected groupoids and action groupoids But, I agree, your answer is much clearer. |
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Apr 17 |
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Connected groupoids and action groupoids Sure, I never claimed this was the only way. This was just the first construction that came to mind. |
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Apr 17 |
answered | HIgher Homotopy Groups and Representation Theory |
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Apr 16 |
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Connected groupoids and action groupoids In any case, I think what is written on the wikipedia page is wrong. They seem to want to set $G$ to be $G_0$. But if you have a connected groupoid with no automorphisms, this is not correct (in this case you should just take $G$ to be some group structure on $X$ and use the translation action). |
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Apr 16 |
answered | Connected groupoids and action groupoids |
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Apr 13 |
accepted | Character table of Sn |
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Apr 12 |
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Character table of Sn That means the size of the conjugacy class in $S_n$ corresponding to the partition $\mu ^i$. There is a formula for this, I just didn't want to write it down... |
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Apr 12 |
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Character table of Sn Of course, this isn't really specific to the symmetric group... we have the same formula for every finite group $G$ (enumerating $G$-Galois covers). But the symmetric group case has historical significance. |
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Apr 12 |
answered | Character table of Sn |
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Mar 31 |
accepted | Naive question about the representation theory of algebraic groups and hopf algebras |
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Mar 31 |
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Naive question about the representation theory of algebraic groups and hopf algebras Thanks! I figured something along those lines must be true. |
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Mar 30 |
revised |
Naive question about the representation theory of algebraic groups and hopf algebras deleted 38 characters in body |
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Mar 30 |
answered | Naive question about the representation theory of algebraic groups and hopf algebras |
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Mar 25 |
accepted | D-affine morphisms and composition |
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Mar 17 |
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Mathematicians whose works were criticized by contemporaries but became widely accepted later "...for a decade or so, category theory was derided by other mathematicians as "abstract nonsense"" ...you mean this doesn't still happen? |
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Mar 14 |
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Is there an analog of ‘Directed Graph’ for topological spaces? Perhaps this is not what you are looking for, but your question reminds me of the notion of the exit-path category of a stratified space. This is similar to the notion of the fundamental groupoid of a space $X$: the objects are points of $X$ and the morphisms are paths. However, in the exit path category, we only allow paths that move from lower dimensional strata to higher. Consequently, we get a category rather than a groupoid. |
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Mar 13 |
answered | Computing Homology via Sheaf Cohomology |
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Mar 11 |
accepted | A fact about $t/W$ and the centralizer bundle on $\mathfrak{g}^{\text{reg}}$ |
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Mar 3 |
answered | A fact about $t/W$ and the centralizer bundle on $\mathfrak{g}^{\text{reg}}$ |
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Feb 21 |
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An example where Čech and derived functor cohomologies don’t agree. The question linked to by Steven provides an answer to the question as stated: take $X$ to be the (Zariski) topological space corresponding to the affine plane, and $\cF$ to be the sheaf described. However, it looks like there is still no answer to Q2 of the linked question: is there a (non-paracompact) Hausdorff space $X$... |
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Feb 20 |
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How to see the quaternionic hopf map generates the stable 3-stem? Perhaps this paper on the vanishing of the third spin cobordism group is relevant: maths.ed.ac.uk/~aar/papers/stipsicz.pdf The author has tried to make the proof "as elementary as possible". |
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Feb 12 |
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D-affine morphisms and composition @Victor: You are correct. Sorry, wasn't thinking straight... I had in mind the lack of deformations of projective space, but of course this doesn't imply what I was saying. |
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Feb 12 |
revised |
D-affine morphisms and composition added 9 characters in body |
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Feb 11 |
answered | D-affine morphisms and composition |
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Feb 10 |
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D-affine morphisms and composition Not sure how to answer your question, but I think the first definition is ``correct'', and I don't think you are missing something! |
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Feb 10 |
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D-affine morphisms and composition Is there a reason to expect that the composition of two $D$-affine morphisms should be $D$-affine? For example, is an affine bundle over a projective space $D$-affine? |
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Dec 20 |
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Non-characteristic is to pullback as (blank) is to pushforward. Thanks for pointing that out - I have edited the question. I'll think about your answer. |
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Dec 20 |
revised |
Non-characteristic is to pullback as (blank) is to pushforward. added 7 characters in body |
