77 problems

If countable ICC property-$(T)$ groups $G$ and $H$ have isomorphic group factors $L(G)\cong L(H)$, must $G\cong H$?

Is Lukic’s weighted entropy condition for finitely many critical points equivalent to decomposing the Verblunsky coefficients into components localized at those points?

For a positive-measure set $E\subset(0,\infty)$, let $E'=\bigcup_{r\geq1}rE$. Is it true that for almost every $x$ there is $M(x)$ such that $nx\in E'$ for every integer $n>M(x)$?

Let $S\subset\mathbb R$ have infinite measure and suppose $x/y$ is never an integer for distinct $x,y\in S$. How fast can $|S\cap(0,x)|$ tend to infinity?

For every real $\alpha$, is the sequence of fractional parts $\{\alpha p_n\}$ along the primes necessarily not well-distributed in the strong sliding-window sense?

Can the angular discrepancy of the roots of a sparse complex polynomial be bounded by a constant times $\sqrt{n\log M}$, where $n$ is its number of nonzero coefficients and $M$ its…

For an infinite sequence $x_j\in(0,1)$, must the limsup exponential-sum amplitudes $A_k$ be unbounded as $k\to\infty$? Can one at least have $A_k=o(k)$?

For arbitrary interpolation nodes in $[-1,1]$, must the Lebesgue function on every fixed subinterval attain at least $(2/\pi-o(1))\log n$?

Among all nonconstant monic polynomials $f$ whose roots lie in $[-1,1]$, determine …

For Chebyshev interpolation nodes and a fixed evaluation point $x_0$, which closed subsets of $[-1,1]$ can occur as the finite cluster set of $L_nf(x_0)$?

Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below $(1+\varepsilon)n$ to have arbitrarily large uniform…

For a transcendental entire function, how fast can $|f(z)|$ be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus $M(r,f)$?

Is the Glöckner--Neeb multiplication-growth condition automatic for every Mackey-complete continuous inverse algebra?

How large can the universal annular threshold in Korenblum’s maximum principle for the Bergman space $A^2(\mathbb D)$ be?

Is there a universal Fourier summation process that recovers the topological degree for every $C^{0,\alpha}$ circle map when $\alpha\leq1/3$?

Do the remaining strict two-point inequalities in Weissler’s complex hypercontractivity parameter region hold?

Does the Kannan–Lovász–Simonovits variance inequality hold with a universal constant, and in particular for every quadratic form under an isotropic log-concave measure?

For unit-circle zeros $z_i$, put $M_n=\max_{|z|=1}|\prod_{i\leq n}(z-z_i)|$. Must $\sum_{k\leq n}M_k>n^{1+c}$ eventually for some $c>0$?

Let $X,Y$ be commuting positive contractions and $P$ a projection. If $A=PXP|_{\operatorname{ran}P}$ and $B=PYP|_{\operatorname{ran}P}$ commute and equality holds in the compressed…

Does there exist one Whittaker-model vector $W$ for $\mathrm{GL}_{n+1}(F)$ that yields a finite, nonzero local Rankin–Selberg integral for every generic representation $\pi$ of…

Given discrete sets $S_n\subset\mathbb C$, can one transcendental entire function have some derivative vanish on every point of each $S_n$?

Does a nonlinear entire function exist such that $x$ is rational exactly when $f(x)$ is rational?

If every root of a monic polynomial lies in $|z|\leq r<2$, must some component of its unit lemniscate have diameter greater than $2-r$?

Let $f\in\mathbb C[x]$ be monic with $m$ distinct roots, and let $c>0$ be small enough that $\{z:|f(z)|\leq c\}$ has $m$ connected components. Must all those components be convex?

Must every monic non-constant polynomial have a straight line onto which its unit lemniscate projects with length at most two?