Mathematics Problem Archive

[S. V\ u Ng{\d o}c] Compute the Taylor series invariant of semitoric systems directly from the spectrum ``at'' the focus-focus critical value....

[S. V\ u Ng{\d o}c] Is the singular Bohr-Sommerfeld formal power series in $\hbar$ a spectral invariant?...

[\'{A}. Pelayo] What information about the principal symbols $f_1,\ldots, f_n$ of a quantum integrable system $T_{1,\hbar},\ldots, T_{n,\hbar}$ can be...

[\'{A}. Pelayo] Can one detect from the joint spectrum of a quantum integrable system $T_{1,\hbar},\ldots, T_{n,\hbar}$ that a singularity is degenera...

Quantise the Mischenko-Fomenko algebra $\mathcal F_a \subset \mathcal P(\goth g)$ for an arbitrary finite-dimensional Lie algebra $\mathfrak g$ {\/\rm...

[seems bad, but might be ugly] For a given sign pattern $\sigma,$ which admissible pairs $(pos,neg)$ are realizable by polynomials whose signs of coef...

(Paley's conjecture) For any entire function $f(z)$ of finite order $\rho$ in the plane, we have \[1\leq\liminf_{r\to\infty}\frac{\log M(r,f)}{T(r,f)}...

The analogue of Problem 1.7 may be asked for meromorphic functions. The proposers conjecture that in this case \[\sum\delta(a,f)\leq\max\{\Lambda_1(\r...

Let $f$ be meromorphic in $\mathbb{C}$, and let $f^{-1}$ denote any element of the inverse function that is analytic in a neighbourhood of a point $w$...

Determine the upper and lower estimates for the growth of entire and meromorphic solutions of algebraic ordinary differential equations (AODE). (This ...

Let $F$ be a polynomial in two variables, and let $y$ be a meromorphic solution of the algebraic ordinary differential equation $F(y^{(n)},y)=0$. Is i...

Let $f$ be meromorphic in $\mathbb{C}$, and suppose that the function \[F(z)=f^{(k)}(z)+\sum^{k-2}_{j=0}a_j(z)f^{(j)}(z)\] is non-constant, where $k\g...

Let $f(z)$ be an entire function. Then Boas (unpublished) proved that there exists a path $\Gamma_\infty$ such that, for every $n$, $$ \left|\frac{f(z...

If the entire function $f(z)$ has finite order $\rho$, and the maximal density of non-zero coefficients is $\Delta$, is it true that if $\rho\Delta<\f...

If $f(z)$ is an entire function, the iterates $f_n(z), n=1,2,\ldots$ are defined inductively by \[f_{n+1}(z)=f(f_n(z)),\hspace{1cm}f_1(z)=f(z).\] A po...

We can also compare $m_0(r,f)$ with the characteristic $T(r)$. We have \[\limsup_{r\to\infty}\frac{\log m_0(r,f)}{T(r)}\geq D(\lambda)\] and ask for t...

(A width conjecture) Given a power series $\sum^\infty_{k=0}a_kz^k$, suppose that there is a non-negative $\rho$ such that all of the partial sums $S_...

Let $\Gamma$ be a rectifiable curve. Suppose $f$ is a continuous function on the plane satisfying \[\int_{\sigma(\Gamma)}f(z)dz=0\hspace{1cm}\text{ fo...

Let $f$ be a rational function and $C$ be as in Problem 2.62. We say that $g$ is a limit function for $f$ if $g$ is defined in some component $G$ of $...

Let $f$ be an entire function satisfying the condition \[\log M(r,f)\leq(1+o(1))r^\rho,\hspace{1cm}\text{ as }r\to\infty.\] Suppose that there exists ...

(Fatou's conjecture) Show that the subset $U$ of functions $g$ in $R_d$, such that all the critical points of $g$ are in the basins of attraction of p...

[(a)] ; Show that, if a function $g$ in $R_d$ has the property that its Julia set $J(g) \neq \hat{\mathbb{C}}$, then $g$ does not leave invariant a no...

Let a function $f$ in $R_d$ have the property that \[f(z)=\lambda_\alpha z+O(z^2)\hspace{1cm}\text{ as }z\to0,\] where $\lambda_\alpha=e^{2\pi i\alpha...

Let $B$ denote the boundary of the Mandelbrot set (or, equivalently, the topological bifurcation set of the family $z\mapsto z^2+c$, $c\in\mathbb{C}$....

Let $P(z)$ be a polynomial whose zeros $z_1, z_2, \ldots, z_n$ lie in $|z|\leq1$. Is it true that $P'(z)$ always has a zero in $|z-z_1|\leq1$? (Bl. Se...

Let $f(z)=z^n+a_1z^{n-1}+\ldots+a_n$ be a polynomial of degree $n$. Cartan proved that the set $|f(z)|\leq1$, which we call $E^{(n)}_f$ can always be ...

Suppose that $P$ is a non-linear polynomial with real coefficients. Show that $P^2(z)+P'(z)$ has non-real zeros. We conjecture that the lower bound fo...

Let $\mathcal{P}$ denote the set of all polynomials of the form \[p(z)=\prod^n_{\nu=1}(z-\zeta_\nu),\] where $n\geq2$ and $|\zeta_\nu|\leq1$, $\nu=1, ...

(The corona conjecture) Let $D$ be an arbitrary domain in the plane that supports non-constant bounded analytic functions. Suppose that $f_1(z),\ldots...

Suppose that $f(z)=\sum^\infty_{n=0}a_nz^n$ is circumferentially mean $p$-valent and $f(z)\neq0$ in $\mathbb{D}$. (This latter condition is a conseque...

If $f$ in $S$, Baernstein has shown that \[\int^{2\pi}_0|f(re^{i\theta})|^p\,d\theta\leq\int^{2\pi}_0|k(re^{i\theta})|^p\,d\theta,\hspace{1cm}0<r<1,\h...

Let $\Omega$ be a simply-connected domain in the finite plane whose complement contains $n$ disjoint closed balls with centres on the interval $[0,1]$...

Show that each quasiconformal mapping of $\mathbb{R}^n$ onto $\mathbb{R}^n$ has a quasiconformal extension to $\mathbb{R}^{n+1}$ . This has been estab...

Suppose that $A$ and $B$ are disjoint linked Jordan curves in $\mathbb{R}^3$ which lie at a distance $1$ from each other. Show that the length of $A$ ...

Let $\mathcal{W}$ be a hyperbolic Riemann surface, $G_\omega$ the Green's function with pole $\omega\in\mathcal{W}$, and $\Gamma=\{[\gamma_n]\}$ the f...

Let $V$ be the zero set of some analytic function in a strictly pseudo-convex domain $\Omega$ in $\mathbb{C}^2$. If $V$ has finite area inside $\Omega...

Prove or disprove the following statements about analytic capacity $\gamma$. [(a)] ; If $E\subseteq\mathbb{C}$ is compact and $\phi$ is a $C^1$-diffeo...

Do there exist inner functions in any strictly pseudo-convex domain of $\mathbb{C}^n$, $n\geq2$? Alexandrov (no citation) and independently L{\o}w (no...

Let $\psi:S^1\to S^1$ be a homeomorphism. When is it true that $\text{Re }A = \text{Re }A \circ \psi$? That is, when is each function $f$ in the real ...

For nonconvex subsets of $\mathbb{R}$ and $\mathbb{R}^2$, is a set spectral if and only if it tiles by translations?...

Characterize the domains for which there exists a nonzero function whose integral vanishes over every congruent copy of the domain....

Establish global regularity, or exhibit breakdown, for solutions of the Vlasov–Maxwell equations from appropriate smooth initial data....

For viscous incompressible flow in $\mathbb{R}^3\setminus B$ past a fixed obstacle $B$, with velocity approaching a nonzero constant vector at infinit...

Develop a global-in-time theory of the Cauchy problem for the one-dimensional Euler–Fourier system for initial data constrained only by finite energy ...

Develop a compensated-compactness calculus for symmetric matrices when compensated compactness yields only inequalities. As a first application, prove...

For compressible Navier–Stokes equations with constant viscosities near vacuum, does the unphysical one-dimensional consequence identified by Hoff and...

Does the compressible Euler system in odd spatial dimension admit a nontrivial smooth eternal solution having finite nonzero mass and energy?...

Prove existence of a regular reflection for compressible flow against a wedge without assuming that the flow is irrotational....

Let $X$ be a centered Gaussian random vector in $\mathbb{R}^n$ with known covariance matrix, and for an orthonormal basis $B=(b_1,\ldots,b_n)$ let $N_...

Does there exist $c>0$ such that it is NP-hard, given a graph $G$ and $\phi>0$, to distinguish $h_G\le\phi$ from $h_G\ge c\sqrt{\phi}$?...