Grigory M
Reputation
11,615
365/400 score
 Jan 4 comment Special Cases of Quadratic Reciprocity and Counting Fixed Points Nice indeed! But it's not immediately clear how to generalize this: it's easy to find 3-torsion in $PSL_2(\mathbb Z)\cong\mathbb Z_2\ast\mathbb Z_3$ but not, say, 5-torsion... Jan 3 comment A combinatorial identity: $\sum_{k=m}^n \frac{\binom{1/2}{k-m}}{k \binom{-1/2}{k}}=\frac{\binom{-1/2}{n-m}}{m \binom{-1/2}{n}}$ Looks amusing. What's the source? Jan 3 comment Show $\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$ by block-walking interpretation of Pascal's triangle Related: math.stackexchange.com/q/534202 Jan 3 comment Is there anything to be learned from the spectrum of a cohomology ring? AFAIK this POV is more useful in equivariant cohomology. For example, in the context of localization theorems it's natural (at least) to view $H_G(X)$ as a sheaf over $\operatorname{Spec}H_G(pt)$. See also Quillen. The Spectrum of an Equivariant Cohomology Ring and Goresky, MacPherson. On the Spectrum of the Equivariant Cohomology Ring... Jan 3 comment Loop space and $K$-theory possible duplicate of Yoneda's lemma and $K$-theory. Jan 2 comment Understanding the relationship between $Sp(n)$ and $Sp(2n,\mathbb{C})$ Have you tried Wikipedia? Jan 2 comment Explicit formula for Fermat's 4k+1 theorem Shouldn’t this ‘Weyl’ be ‘Weil’? Jan 2 comment Can the equivalence between principle bundles and maps to classifying spaces be turned into an adjunction. I'm not sure I understand your question. You want adjunction between what what and what? If you're talking about categories $Bun_G(X)$ and $[X,BG]$ — sure, any equivalence of categories can be refined to an adjoint equivalence. Jan 2 comment Prove that $\sum_{k=0}^{m}\binom{m}{k}\binom{n+k}{m}=\sum_{k=0}^{m}\binom{n}{k}\binom{m}{k}2^k$ (Cf. [shifted] Legendre polynomials, btw!) Jan 2 comment The importance of generating series I don't think this has anything to do with algebraic geometry. Generating functions is a useful machine to work with any large collection of numbers satisfying some recurrence relation(s). Jan 1 comment Semialgebraic conditions that convey properties of Galois group The word is semi-algebraic (if I understand OP correctly) Jan 1 comment Proof of $0! = 1$ Basically, this is just a convention. But notice how nicely this agrees with general formula $n!=n\cdot(n-1)!$ (take $n=1$). Dec 31 comment Contracting a contractible set in $\mathbb R^2$ Dec 31 comment Contracting a contractible set in $\mathbb R^2$ @M314 Complement to the horned ball is not simply connected. So in $\mathbb R^3/A$ we have a point $A/A$ with non-simply-connected complement. Dec 30 comment Mapping cylinder is a CW complex Have you seen section 'Topology of Cell Complexes' in Hatcher? At least product of CW-complexes is discussed in detail there. Dec 30 comment Cohomology Ring of Projective Space Dec 30 comment Cohomology ring of $\mathbb R P^\infty$ with $\mathbb Z_{2k}$ coefficients Related: the same question for $k=2$ Dec 30 comment Contracting a contractible set in $\mathbb R^2$ I'm not even sure if this is true — for example for $\mathbb R^3$ the statement seems to be false (take $A$ to be the horned ball...). Dec 29 comment Contracting a contractible set in $\mathbb R^2$ $S^1\subset\mathbb R^2$ is compact and connected but not contractible. Dec 29 comment Concrete non trivial computation of Morse homology [Let's wait for answers but] I've always thought it's mainly useful for (co)homology of loop space and such [and not in finite-dimensional situation]...