Open
problems raised at the 2007 Workshop in Analysis and Probability
The problems here were either submitted specifically for the
purpose of inclusion in this page, or were taken from talks given
during the 2007 Workshop in Linear Analysis and Probability.
Problem #1 (Submitted by Hari Bercovici)
In Classical probablity theory it is easy to show that:
If X,Y are real independent random variables such that $Y=X$ in distribution,
$P(X\geq0) \geq 1/2$, and $P(X\leq0) \geq 1/2$ then
for all $t>0$, $1/2 P(|X|>t) \leq P(X+Y>t) \leq 2P(|X|>t/2)$
Is there any analagous result for free variables?
Problem #2 (Submitted by Leonid Kovalev)
If $f:S_X \rightarrow S_{X^*}$ is Lipschitz and $f(x)(x)\geq\delta>0$ then is $X$ 2-smoothable?
(has an equivalent norm with quadratic modulus of smoothness)
Problem #3 (Submitted by Rafal Latala)
In estimating the expected value of the operator norm of random matrix $(a_{ij}g_{ij})$, where
g_{ij} are i.i.d. $N(0,1)$, is it true that:
$E||(a_{ij}g_{ij})||_{(1)(2)} ~ E\sup_i(\sum_{j}a_{ij}^2 g_{ij}^2)^{1/2}+E\sup_j(\sum_{j}a_{ij}^2 g_{ij}^2)^{1/2} ?
Problems #4 and #5 (Submitted by Rafal Latala)
These are posed in this pdf file.
Problem #6 (Submitted by Aryeh Kontorovich)
This is posed in this pdf file.
Problem #7 (Submitted by Stanislaw Szarek)
Does there exists strictly increasing to infinity $(k_n)$ and a constant $C>0$ such that for all normed spaces X with dimension n, there
exists projection P on X whose rank is at least $k_n$ and $||P||< C$?
In particular, does $k_n=\sqrt{n}$ work?
Problem #8 (Submitted by Julio Bernues)
Conjecture: $|B^~^n_p \cap E^{\bot}|_{n-k} \geq1$ for all k-dim subspaces $E\subset R^n$.
This is known to be true for all cases other than the case: $1< p<2$ and $1< k<{n=1}/2$
Problem #9 (Submitted by Ted Odell)
Let $2< p< \infty$. Characterize when $X\subset L_p$ embeds into $(\sum\ell_2)_p$.
In particular, if every weakly null tree in $S_X$ has a branch equivalent to a block basis of the
standard basis for $(\sum\ell_2)_p$, must $X$ embed into $(\sum\ell_2)_p$?
Also, if $X$ is a quotient of a subspace of $(\sum\ell_2)_p$, must $X$ embed into $(\sum\ell_2)_p$?
Problem #10 (Submitted by Ted Odell)
Let $1< p\neq q< \infty. Characterize when a reflexive $X$ embeds into $(\sum E_n)_p\oplus(\sum F_n)_q$,
for some sequences of finite dimensional spaces $(E_n), (F_n)$.
In particular, is it enough to assume
that every weakly null tree in $S_X$ has a branch equivalent to a block basis of the standard basis for $\ell_p\oplus \ell_q$?
Problem #11 (Submitted by Mark Ptak)
$W(T_{z_1},T_{z_2})$ on $H^2(\Pi^2)$ is reflexive. Is it hyper-reflexive?
Problem #12 (Submitted by Joe Rosenblatt)
Does there exist $f\in L_2$ and invertible ergodic transformation $T$ such that
$(f\circ T^k : k\in\Z)$ is an o.n.b. for $L_2(X)$? The conjecture is no.
Problem #13 (Submitted by Joe Rosenblatt)
Let $S_N (x)=\sum_{k=1}^N exp(im_k x)$
for $m_1 \leq m_2 \leq ...$, is $S_N(x)$ almost everywhere recurrent?
(Anderson/Pitt) Yes, if $m_{k+1}/m_{k} \geq\gamma>1$.
Problem #14 (Submitted by Joe Rosenblatt)
$T:\Pi\rightarrow \Pi$ is Ergodic. Must $S_N (\gamma)=\sum_{k=1}^N T^{P_k}\gamma$ be recurrent a.e $\gamma$?
If not, for which $P_k$ must it be true?
Problem #15 (Submitted by Joe Rosenblatt)
If $(\psi_n)$ is an $L_1$-approximate identity, when does $\lim \psi_n * f=f$ a.e for all $f\in L_1$?
Problem #16 (Submitted by Joe Rosenblatt)
For $A=[cos(2^{n(i-1)+j}x)]_{1\leq i, j\leq n}$, must $||A||||A||^{-1}=O(n)$ with high probability?
Problem #17 (Submitted by Yehoram Gordon)
The famous Dvoretsky theorem states that for all $0<\epsilon<1$, if $n\leq c(\epsilon)log(N)$, then
any N-dim Banach space $X$ contains a random subspace $Y\subset X$, $dim(Y)=n$ such that $d(Y,\ell^n_2)\leq 1+\epsilon$,
with very high probability. V. Milman proved in 1971, using the Levy isoperimetric inequality that
$c(\epsilon)~\epsilon^2/log(2/\epsilon)$. Y. Gordon proved in 1985, using Gaussian embedding operators
that $c(\epsilon)~\epsilon^2$.
Is $\epsilon^2$ the best possible for random subspaces of high probability?
Problem #18 (Submitted by Dmitri Panchenko)
For $g=(g_1,...,g_N)$ i.i.d. $N(0,1)$, must the following hold:
$sup_v \E\frac{\int_{z^1*z^2\leq\epsilon} exp(v(g*z^1-g*z^2))dv(z^1)dv(z^2)}{(\int exp(v(g*z))dv(z))^2}
Problem #19 (Submitted by Haskell Rosenthal)
Does there exist a sequence $(c_n)$, $c_n\rightarrow\infty$, such that given an $nxn$ matrix $(a_{i,j})$
with $a_{i,j}=\pm 1$ for all $i,j$, then
$\sum_{i=1}^n(\sum_{j=1}^n a_{\lamda(i),\sigma(j))^2 \geq c_n n$, where the sup is over all permutations
$\lambda, sigma$ of $(1,...,n)$.
Problem #20 (Submitted by Staszek Szarek)
Is $\sup_d ((vol PPT states)/(vol all states))^{1/(d^2-1)} < 1 on \C^d for d=d_1*d_2?
Problem #21 (Submitted by Staszek Szarek)
Find a direct proof of: $M$ block positive implies $tr(M^2)\leq (tr(M))^2
Problem #22 (Submitted by Staszek Szarek)
Is $M^{sep}$ of constant height?
If you have a problem that you want posted, or want changes to be made, send an email to freeman@math.tamu.edu