Search Results

Let $h \in \mathbb{R}[x, y]$ be an irreducible cubic polynomial. Consider the affine variety$$\{(x, y) \in \mathbb{R}^2: h(x, y) = 0\}.$$What are the qualitatively different possibilities for what thi …

In Wise1 Wise shows that hyperbolic graphs of free groups with cyclic edge groups are subgroup separable. In Hsu-Wise these are shown to be cubulated, and by Agol they're virtually special, so quasico …

Let $X$ be a Banach space. My question is: $X$ contains no copy of $l_{1}$ if and only if any operator from $X$ to $l_{1}$ is compact? I guess that the necessary part may be true. But is the sufficien …

Take for example the measure $\mu(n)=n^2$ on $\{1, \ldots, N\}$ and a random variable $X$ distributed according to the probability obtained by normalizing $\mu$. Does there exists a constant $K>0$ s …

Let $X \subseteq \mathbb{A}^n$ be algebraic, and let $x$, $y \in \mathbb{A}^n - X$. How do I see that there exists $f \in I(X)$ with $f(x) \neq 0$ and $f(y) \neq 0$.

Throughout this question we assume ZFC. If CH holds, then the following is obvious: (S) Every definable infinite subset of $\mathbb R$ has size either $\aleph_0$ or $2^{\aleph_0}$. (It's true b …

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$. What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$? Here we denote by $d(G)$ the smallest cardina …

The moduli space of K3 surfaces forms a 20-dimensional family with countably many 19-dimensional components $M_d$ corresponding to the polarized K3s $(X,L)$ with $L^2=d$. The moduli space $M_d$ has a …

I am trying to solve the following integral $$ \int_{-1}^{1}\;db\;||[t_{b}(A),J]||_{F}^{2} $$ where $t_{b}$ is the entrywise threshold of the matrix A ($0$ if $a_{ij}<b$, $a_{ij}$ if $a_{ij}>b$, with …

It is known that for smooth projective varieties $X,Y$ over $k=\bar k,$ $$CH^d(X_{k(Y)})=\varinjlim_{U\subset Y\ open}CH^d(X\times_k U)$$ I was wondering whether there was such an equality with algebr …

There are several questions about transverse complete intersection arising from L. Guth's paper: http://www.ams.org/journals/jams/0000-000-00/S0894-0347-2015-00827-X/home.html We say a polynomial $P …

Given a $\mathbb{C}$-scheme $S$, two $S$-schemes $X$ and $Y$ that are flat over $S$ and a coherent sheaf of $O_Y$-modules $F$. Assume we have a (faithfully) flat $S$-morphism $\pi: X \rightarrow Y$ a …

In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and commutat …

Let $A$ and $B$ be two square matrices with complex entries. Let $\lambda_1, \ldots, ,\lambda_n$ be the Eigenvalues of $A$ and $\mu_1, \ldots, ,\mu_m$ be the Eigenvalues of $B$. Then the Eigenvalues …

Suppose just for example $A\bigotimes{B}=2C+D^++E^-$ with irreps $A...E$. You have a dimension sum rule ($2*d_C+d_D+d_E=...$) and a Dynkin index sum rule ($2*i_C+i_D+i_E=...$). If $A=B$, you get anoth …