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curious little things - aggregated feedsenMath Overflow Recent Questions: Embedding of $CP^2/CP^1$ into euclidean space
https://mathoverflow.net/questions/315425/embedding-of-cp2-cp1-into-euclidean-space
<p>It is a standard exercise in embedding theory to show that <span class="math-container">$S^3 \to \mathbb{R}^4$</span> given by <span class="math-container">$(x,y,z) \mapsto (x^2-y^2,xy,xz,yz)$</span> induces an embedding <span class="math-container">$\mathbb{R}P^2 \to \mathbb{R}^4$</span>. Since <span class="math-container">$\mathbb{R}P^2/\,\mathbb{R}P^1 \cong \mathbb{R}P^2$</span>, the previous map gives an embedding of <span class="math-container">$\mathbb{R}P^2/\,\mathbb{R}P^1$</span> into <span class="math-container">$\mathbb{R}^4$</span>.</p>
<p>Is there a nice embedding of <span class="math-container">$\mathbb{C}P^2/\,\mathbb{C}P^1$</span> into <span class="math-container">$\mathbb{R}^8$</span>?</p>Thu, 15 Nov 2018 21:06:32 -0700Math Overflow Recent Questions: Graph with at most 2 degrees of separation between every node, but minimal average degree
https://mathoverflow.net/questions/315424/graph-with-at-most-2-degrees-of-separation-between-every-node-but-minimal-avera
<p>Is there a simple way to construct such a graph? For example a fully connected graph obviously has degree of separation between every vertex of 1 but has maximal total degree. If we only wanted to minimise the total degree then I think the answer would be a star graph. But I want the average degree to be smallest rather than just relying on a single high degree vertex to be the common neighbour for all vertices. I can sort of see an algorithm starting with a cycle5 graph and adding nodes until the degree of separation between each pair of nodes is <= 2, but not sure if this would be optimal.</p>Thu, 15 Nov 2018 20:44:02 -0700Math Overflow Recent Questions: Given symmetric semidefinite matrix A and B, prove AB = 0 if and only if tr(AB)=0 [on hold]
https://mathoverflow.net/questions/315416/given-symmetric-semidefinite-matrix-a-and-b-prove-ab-0-if-and-only-if-trab
<p>Let A, B <span class="math-container">$\in S^{n}_{+}$</span>, means that A and B are symmetric semidefinite matrix. Can we prove that <span class="math-container">$tr(AB) = 0$</span> if and only if <span class="math-container">$AB = 0$</span> ?</p>Thu, 15 Nov 2018 18:06:22 -0700Math Overflow Recent Questions: Applications of E8 manifold
https://mathoverflow.net/questions/315414/applications-of-e8-manifold
<p>The <span class="math-container">$E_8$</span> Cartan matrix is given by,
<span class="math-container">$$
K_{E_8}=\begin{pmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 2 & -1 & 0 & -1 \\
0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 2
\end{pmatrix}.
$$</span>
There is one famous application in physics. Which is that a symmetric bilinear <span class="math-container">$K_{E_8}$</span>-Chern-Simons theory describes the low energy physics of a so-called <span class="math-container">$E_8$</span>-quantum Hall state (with a <span class="math-container">$U(1)^8$</span> gauge group). The field theory partition function <span class="math-container">$Z$</span> is given by
<span class="math-container">$$
Z= \int[DA]\exp(\frac{(K_{E_8})_{IJ}}{2 \pi}\int A_I dA_J).
$$</span>
This <span class="math-container">$E_8$</span>-quantum Hall state occurs in a 2-dimensional spatial condensed matter system.</p>
<p>The above is what I am familiar already. Now I am asking a different question about the <strong>application of <span class="math-container">$E_8$</span> manifold</strong>.</p>
<blockquote>
<p>My question here is that: Is there some <strong>real-world application of <span class="math-container">$E_8$</span> manifold</strong>, such as in physics or in any branch of science, or in the engineer?</p>
</blockquote>
<p>The <span class="math-container">$E_8$</span> manifold is the unique compact, simply connected topological 4-manifold with intersection form the <span class="math-container">$E_8$</span> lattice. </p>
<ul>
<li><p>The <span class="math-container">$E_8$</span> manifold has no smooth structure.</p></li>
<li><p>The <span class="math-container">$E_8$</span> manifold is not triangulable as a simplicial complex.</p></li>
<li><p>The <span class="math-container">$E_8$</span> manifold is constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to the Dynkin diagram for <span class="math-container">$E_8$</span>. This results in P<span class="math-container">$E_8$</span>, a 4-manifold with boundary equal to the Poincaré homology sphere. Freedman's theorem on fake 4-balls then says we can cap off this homology sphere with a fake 4-ball to obtain the <span class="math-container">$E_8$</span> manifold.</p></li>
</ul>
<p>Some refs and introductory level of explanations are welcome! Thanks.</p>Thu, 15 Nov 2018 17:43:26 -0700Math Overflow Recent Questions: Mathematical phantoms, specifically but not exclusively in applied mathematics
https://mathoverflow.net/questions/315412/mathematical-phantoms-specifically-but-not-exclusively-in-applied-mathematics
<p>A while ago over at our sister site, there was <a href="https://math.stackexchange.com/questions/2627755/mathematical-phantoms">an interesting question</a> [not by me] with next to no answers which I feel is, fleshed out in a more precise fashion, appropriate for MathOverflow.</p>
<p><strong>The question.</strong> Which objects and concepts would you regard as <em>mathematical phantoms</em>? I'm especially interested in examples from applied mathematics, since I already know a couple of examples from pure mathematics (listed below).</p>
<p><strong>Mathematical phantoms.</strong> Following <a href="http://math.ucr.edu/home/baez/week259.html" rel="nofollow noreferrer">John Baez</a>, <a href="http://www.wra1th.plus.com/gcw/math/MathPhant.html" rel="nofollow noreferrer">Gavin Wraith</a> and surely others, a <em>mathematical phantom</em> is "an object that doesn't exist within a given mathematical framework, but nonetheless 'obtrudes its effects so convincingly that one is forced to concede a broader notion of existence'.
Like a genie that talks its way out of a bottle, a sufficiently powerful mathematical phantom can talk us into letting it exist by promising to work wonders for us. Great examples include the number zero, irrational numbers, negative numbers, imaginary numbers, and quaternions. At one point all these were considered highly dubious entities. Now they're widely accepted. They 'exist'."</p>
<p><strong>Being more precise.</strong> The demarcation from what are merely very fruitful abstractions is obviously a bit blurry; perhaps useful criteria are:</p>
<ul>
<li>There should be a statement which was held to be obviously true before the discovery of the phantom, but which is false in view of the new concept. (Such as the statement "obviously no number squares to <span class="math-container">$-1$</span>".)</li>
<li>It should have required a nontrivial effort to make it precise (to "help it come into being").</li>
<li>It should have great explanatory power and vast consequences.</li>
</ul>
<p><strong>Examples.</strong> Phantoms in this stricter sense could include:</p>
<ul>
<li>The irrational numbers. (Running counter to the basic tenet "all is number" by the Pythagorean school, where by "number" they meant "rational number".)</li>
<li>The complex numbers.</li>
<li>The <span class="math-container">$p$</span>-adic numbers.</li>
<li>Actual infinity, together with the flexible notion of sets we have nowadays (vastly surpassing recursive subsets of <span class="math-container">$\mathbb{N}$</span>) and the axiom of choice.</li>
<li>Sobolev function spaces.</li>
<li>Infinitesimal numbers (as in the hyperreal numbers, where <span class="math-container">$\varepsilon$</span> is invertible, or as in synthetic differential geometry, where <span class="math-container">$\varepsilon^2 = 0$</span>).</li>
</ul>
<p>Phantoms in a broader sense (where I can't think of any held-to-be-obviously-true statement falsified by them) could include:</p>
<ul>
<li>Symmetries of zeros of polynomials, or more generally groups.</li>
<li>The field with one element.</li>
<li>Ideals in number theory.</li>
<li>Motives.</li>
<li>∞-categories.</li>
<li>Toposes. (Generalizing and unifying various cohomology theories.)</li>
<li>Nonclassical logics. (Born in the foundational crisis, nowadays with applications in mainstream mathematics.)</li>
</ul>
<p>For the purposes of this question, I'm interested in both kinds of phantoms.</p>Thu, 15 Nov 2018 17:12:04 -0700Math Overflow Recent Questions: $n$-fold tensor products of $D(A)$ for finite dimensional algebras
https://mathoverflow.net/questions/315409/n-fold-tensor-products-of-da-for-finite-dimensional-algebras
<p>Let <span class="math-container">$A$</span> be a finite dimensional quiver algebra over a field <span class="math-container">$K$</span> and let <span class="math-container">$D(-):=Hom_K(-,K)$</span> denote the natural duality (assume algebras are connected).</p>
<p>Define <span class="math-container">$\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes n} \neq 0 \}$</span> and <span class="math-container">$W_A:=D(A)^{\otimes \psi_A}$</span>.</p>
<p>It seems horribly complicated, but in special cases those definitions might have nice properties. Were they studied before?</p>
<p>Note that Nakayama algebras with a linear quiver and <span class="math-container">$n$</span> simple modules are counted by the Catalan nubmers <span class="math-container">$C_{n-1}$</span> and thus are in bijection with 321-avoiding permutations on <span class="math-container">$n-1$</span>-symbols.
(Nakayama algebras with a linear quiver are exactly the quotient algebras of the ring of upper triangular matrices over the field <span class="math-container">$K$</span> by an admissible ideal)</p>
<p>Here some observations/guesses with the computer for Nakayama algebras with a linear quiver:</p>
<ol>
<li><p><span class="math-container">$W_A$</span> is injective.</p></li>
<li><p>There are exactly <span class="math-container">$2^{n-2}$</span> algebras with <span class="math-container">$\psi_A = 2$</span> and all of those algebras seem to have global dimension at most 3.</p></li>
<li><p>The generating function of the statistic <span class="math-container">$A \rightarrow \psi_A$</span> (<a href="http://www.findstat.org/StatisticsDatabase/St001290/" rel="nofollow noreferrer">http://www.findstat.org/StatisticsDatabase/St001290/</a>) seems to coincide with the generating function on 321-avoiding permutations given by <span class="math-container">$ \pi \rightarrow f(g(\pi))$</span>, where <span class="math-container">$g$</span> is the first fundamental transformation on permutations (<a href="http://www.findstat.org/MapsDatabase/Mp00086" rel="nofollow noreferrer">http://www.findstat.org/MapsDatabase/Mp00086</a>) and <span class="math-container">$f$</span> number of right-to-left minima of a permutation (<a href="http://www.findstat.org/StatisticsDatabase/St000991" rel="nofollow noreferrer">http://www.findstat.org/StatisticsDatabase/St000991</a>).</p></li>
</ol>
<p>Now it looks rather horribly to try to prove those things by direct computations but maybe there is a trick or a nice interpretation of <span class="math-container">$\psi_A$</span> and <span class="math-container">$W_A$</span> for Nakayama algebras and maybe even more general algebras (maybe QF-3 algebras)?</p>Thu, 15 Nov 2018 15:47:44 -0700Math Overflow Recent Questions: Free symmetric monoidal category of compactly generated category is compactly generated
https://mathoverflow.net/questions/315407/free-symmetric-monoidal-category-of-compactly-generated-category-is-compactly-ge
<p>Let <span class="math-container">$k$</span> be a field and let <span class="math-container">$\mathcal{C}=\mathbf{StLin}_k$</span> be the <span class="math-container">$\infty$</span>-category of stable infinity categories enriched over the <span class="math-container">$\infty$</span>-category <span class="math-container">$\mathbf{Vect}_k$</span>, regarded as a symmetric monoidal <span class="math-container">$\infty$</span>-category with unit object <span class="math-container">$\mathbf{Vect}_k$</span>. </p>
<p>The forgetful functor <span class="math-container">$\operatorname{CAlg}(\mathcal{C}) \to \mathcal{C}$</span> admits a left adjoint <span class="math-container">$Sym^*: \mathcal{C} \to \operatorname{CAlg}(\mathcal{C})$</span>. Now let <span class="math-container">$\mathcal{C}[z]=Sym^*(\mathbf{Vect}_k)$</span> - i.e. <span class="math-container">$\mathcal{C}[z]$</span> is the free stable symmetric monoidal category generated by <span class="math-container">$\mathbf{Vect}_k$</span>. </p>
<p>My question is: Is the <span class="math-container">$\infty$</span>-category <span class="math-container">$\mathcal{C}[z]$</span> compactly generated in <span class="math-container">$\mathcal{C}$</span>? (Recall that a category <span class="math-container">$\mathcal{A}$</span> is compactly generated if there exists a subcategory <span class="math-container">$\mathcal{A}_0 \subseteq \mathcal{A}$</span> and an equivalence <span class="math-container">$\mathcal{A} \simeq Ind(\mathcal{A}_0)$</span>.)</p>
<p>More generally: If <span class="math-container">$\mathcal{D} \in \mathcal{C}$</span> is compactly generated, is <span class="math-container">$Sym^*(\mathcal{D})$</span> also compactly generated?</p>
<p>My idea was to show that the map <span class="math-container">$\mathbf{Vect}_k \to \mathcal{C}[z]$</span> corresponding to the identity:<span class="math-container">$ \mathcal{C}[z] \to \mathcal{C}[z]$</span> under the adjunction above identifies the image of <span class="math-container">$k$</span> with a compact generator of <span class="math-container">$\mathcal{C}[z]$</span>, but I'm not sure if this would work.</p>Thu, 15 Nov 2018 15:27:01 -0700Math Overflow Recent Questions: How to determine the unramified character corresponding to an unramified Langlands parameter?
https://mathoverflow.net/questions/315400/how-to-determine-the-unramified-character-corresponding-to-an-unramified-langlan
<p>Let <span class="math-container">$F$</span> be a p-adic field with ring of integers <span class="math-container">$\mathcal{O}$</span>. Let <span class="math-container">$\textbf{G}$</span> be a connected split reductive algebraic group over <span class="math-container">$F$</span>. For simplicity, we assume that <span class="math-container">$\textbf{G}$</span> is a Chevalley group. Let <span class="math-container">$G=\textbf{G}(F)$</span> and <span class="math-container">$K=\textbf{G}(\mathcal{O})$</span>. Let <span class="math-container">$B=TU$</span> be a fixed Borel subgroup with torus <span class="math-container">$T$</span>. Let <span class="math-container">$\hat G$</span> be the Langlands dual group of <span class="math-container">$G$</span> and <span class="math-container">$\hat T$</span> be the dual torus. Let <span class="math-container">$W_F$</span> be the Weil group of <span class="math-container">$F$</span>. Let <span class="math-container">$\varphi: W_F\rightarrow \hat T\rightarrow \hat G$</span> be an unramified Langlands parameter. Then associated with <span class="math-container">$\varphi$</span>, by Satake isomorphism, there is an irreducible spherical representation <span class="math-container">$\pi(\varphi)$</span> of <span class="math-container">$G$</span>. It is known that <span class="math-container">$\pi(\varphi)$</span> is a sub-quotient of a principal series representation <span class="math-container">$\textrm{Ind}_{TU}^G(\mu)$</span>. </p>
<p><span class="math-container">$\textbf{Question}$</span>: How to determine <span class="math-container">$\mu$</span> in terms of <span class="math-container">$\varphi$</span>?</p>
<p>If <span class="math-container">$\textbf{G}=\textrm{GL}_n$</span>, my understanding is as follows. An unramified Langlands parameter <span class="math-container">$\varphi:W_F\rightarrow \hat T$</span> is determined by its image of <span class="math-container">$\textrm{Fr}$</span>. Suppose that <span class="math-container">$\varphi(\textrm{Fr})=\textrm{diag}(a_1,\dots,a_r)\in \hat T$</span> with <span class="math-container">$a_i\in \mathbb{C}^\times$</span>. Then the corresponding <span class="math-container">$\mu:T\rightarrow \mathbb{C}^\times$</span> is determined by <span class="math-container">$\mu(1,\dots,1,\varpi,1,\dots,1)=a_i$</span>, where <span class="math-container">$\varpi$</span> is a uniformizer of <span class="math-container">$F$</span> in the <span class="math-container">$i$</span>-th position.</p>
<p>Here is one example I am interested in. Define <span class="math-container">$J_n\in \textrm{GL}_n$</span> inductively by
<span class="math-container">$$J_n=\begin{pmatrix} &1\\ J_{n-1}&\end{pmatrix}, J_1=(1).$$</span>
Let <span class="math-container">$G$</span> be the split <span class="math-container">$\textrm{SO}_5$</span>, which is defined by the matrix <span class="math-container">$J_5$</span>. Then <span class="math-container">$\hat G=\textrm{Sp}_4(\mathbb{C})$</span>, which is realized by the matrix
<span class="math-container">$$\begin{pmatrix} &J_2\\ -J_2&\end{pmatrix}.$$</span>
Let <span class="math-container">$\varphi:W_F\rightarrow \hat T$</span> by <span class="math-container">$\varphi(w)=\textrm{diag}(1,|w|,|w|^{-1},1)$</span>. Then what is the corresponding <span class="math-container">$\mu$</span>?</p>
<p>Any comments and references about the general case or the above special example are appreciated. </p>Thu, 15 Nov 2018 13:53:19 -0700Math Overflow Recent Questions: Easy cases of Herbrand's theorem
https://mathoverflow.net/questions/315393/easy-cases-of-herbrands-theorem
<p><span class="math-container">$\def\QQ{\mathbb{Q}}\def\ZZ{\mathbb{Z}}$</span> I recall Herbrand's theorem about class groups of cyclotomic fields: Let <span class="math-container">$p$</span> be an odd prime, let <span class="math-container">$\zeta$</span> be a primitive <span class="math-container">$p$</span>-th root of <span class="math-container">$1$</span> and let <span class="math-container">$K = \QQ(\zeta)$</span>, so <span class="math-container">$\mathrm{Gal}(K/\QQ)$</span> is canonically isomorphic to <span class="math-container">$(\ZZ/p \ZZ)^{\times}$</span>. Let <span class="math-container">$A$</span> be the class group of <span class="math-container">$K$</span> and let <span class="math-container">$V = A/p A$</span>, an <span class="math-container">$\mathbb{F}_p$</span> vector space. Then <span class="math-container">$\mathrm{Gal}(K/\QQ)$</span> acts on <span class="math-container">$V$</span> and <span class="math-container">$V$</span> splits accordingly into characters of <span class="math-container">$(\ZZ/p \ZZ)^{\times}$</span>; let <span class="math-container">$V = \bigoplus V_r$</span> where <span class="math-container">$a \in (\ZZ/p \ZZ)^{\times}$</span> acts by <span class="math-container">$a^r$</span> on <span class="math-container">$V_r$</span>.</p>
<p>For <span class="math-container">$1 \leq r \leq p-2$</span> odd, Herbrand's theorem tells us that, if <span class="math-container">$V_r \neq 0$</span>, then <span class="math-container">$p$</span> divides the numerator of the Bernoulli number <span class="math-container">$B_{p-r}$</span>. For example, since <span class="math-container">$B_2 = \frac{1}{6}$</span>, we always have <span class="math-container">$V_{p-2}=0$</span>. First question:</p>
<blockquote>
<p>Is there a way to see that <span class="math-container">$V_{p-2}=0$</span> without understanding Herbrand's proof?</p>
</blockquote>
<p>As an example of what I'm hoping for, it is straightforward to see that <span class="math-container">$V_{p-1}=V_0=0$</span>. If <span class="math-container">$V_{p-1}$</span> were nonzero, class field theory would give an unramified extension <span class="math-container">$K/\mathbb{Q}(\zeta_p)$</span> so that <span class="math-container">$K/\QQ$</span> is Galois with Galois group <span class="math-container">$(\ZZ/p) \times (\ZZ/(p-1))$</span>. But then the fixed field of <span class="math-container">$\ZZ/(p-1)$</span> is an unramified degree <span class="math-container">$p$</span> extension of <span class="math-container">$\QQ$</span>, violating Minkowski's theorem.</p>
<p>I ask because I am still thinking about this <a href="https://mathoverflow.net/questions/313505/the-roots-of-unity-in-a-tensor-product-of-commutative-rings">very challenging question</a>. If <span class="math-container">$V_{p-2}=V_{-1}$</span> were nonzero, I believe I could show that the ring of integers in the corresponding <span class="math-container">$(\ZZ/p) \rtimes (\ZZ/(p-1))$</span> extension of <span class="math-container">$\QQ$</span> would give a counter-example to this question. With similar motivation, I ask:</p>
<blockquote>
<p>Is there a straightforward way to see that the eigenspace <span class="math-container">$V_1$</span> is zero?</p>
</blockquote>
<p>This last occurs as Proposition 6.16 in Washington's <a href="http://www.math.hawaii.edu/~pavel/cmi/References/Washington_Introduction_to_Cyclotomic_Fields.pdf" rel="nofollow noreferrer">Introduction to Cyclotomic Fields</a>, but I can't figure out whether it is straightforward or whether it needs the 6 chapters that precede it.</p>Thu, 15 Nov 2018 13:10:34 -0700Math Overflow Recent Questions: A new generalisation of associativity?
https://mathoverflow.net/questions/315389/a-new-generalisation-of-associativity
<p>During my research I come accross, on this result :</p>
<blockquote>
<p><strong>Proposition :</strong> Let <span class="math-container">$E$</span> be a finite set. If <span class="math-container">$f,g$</span> binary laws on <span class="math-container">$E$</span>, with :</p>
<p><span class="math-container">$$\forall a,b,c \in E,\; f(a,f(b,c))=f(g(a,b),c)$$</span></p>
<p>then <span class="math-container">$$\forall a,b,c,x \in E, \; f(g(a,g(b,c)),x)=f(g(g(a,b),c),x)$$</span></p>
</blockquote>
<p>></p>
<blockquote>
<p><strong>Definition :</strong> In this case, we say <span class="math-container">$g$</span> is pseudo associative.</p>
</blockquote>
<p><strong>Remark :</strong> This question is very important for calculus <span class="math-container">$h^{N}(x)$</span> with <span class="math-container">$N>2^{10000}$</span></p>
<p>where we choose a good fonction <span class="math-container">$h$</span> and it's very important in cryptanalyse <strong>(1)</strong>.</p>
<blockquote>
<p><strong>Question :</strong> Have there been studies of the notion of pseudo associativity in the literature?</p>
</blockquote>
<p>></p>
<p>></p>
<p><strong>(1)</strong> : for example and
roughly, if <span class="math-container">$h([x,y])=[x+1,x\times y]$</span> then <span class="math-container">$h^N([1,1])=[N+1,N!]$</span>
This function can help to calculus the factoriel on set <span class="math-container">$\mathbb Z/ (p\times q)\mathbb Z$</span> and factorise <span class="math-container">$p \times q$</span></p>
<p>the question is choose the good function <span class="math-container">$h$</span> for break the security.</p>
<p>It's possible to do that for the discret log.</p>Thu, 15 Nov 2018 12:08:51 -0700