Mathematics Problem Archive

Consider two Lagrangian fibrations $\phi: M^4 \to B$ and $\phi': {M'}^4 \to B'$. Assume that $B$ and $B'$ are affinely equivalent in the sense that th...

Do local symplectic invariants exist for diffeomorphic degenerate singularities? How many and of what kind are they? This question makes sense even in...

Assume that we know explicit formulas for the action variables $I_1,\dots, I_n$ so that we are able to analyse their asymptotic behaviour in a neighbo...

Describe all topological types of singularities that may appear in algebraically integrable systems with a small ($\leq 3$) number of degrees of freed...

[\'{A}. Pelayo] Extend the classification of semitoric systems $F=(J,H)$ in to allow for $F$ having non-degenerate singularities with hyperbolic block...

[\'{A}. Pelayo] Consider a compact connected $2n$-dimensional symplectic manifold $M$, endowed with a Hamiltonian $(S^1)^{n-1}$-action; these are call...

Construct a natural Hamiltonian system with a nonconstant potential on $S^2$ that admits a nontrivial polynomial integral of degree $5$ and does not a...

Construct new examples of natural Hamiltonian systems on higher dimensional manifolds which are integrable in the class of integrals polynomial in mom...

Does there exist a non-conformally flat metric on the sphere $S^n$, $n>2$, whose geodesic flow is maximally superintegrable (in the class of integrals...

Suppose that a billiard table has a sequence of convex caustics with rotation numbers converging to some number $\omega \in (0,1/2)$. Does it imply th...

{\rm (M. Bialy)} Are there multi-dimensional convex billiards, other than ellipsoids, having invariant hypersurfaces in the phase space? Same question...

Compute the Jordan--Kronecker invariants for the most interesting classes of Lie algebras, and particularly for the following {\/\rm(}a{\/\rm)} semidi...

Extend the action-angle theorem in a neighbourhood of a Liouville torus at singular points of the Poisson structure....

Extend the action-angle theorem in a neighbourhood of a Liouville torus at singular points of the Poisson structure for splittable integrable systems....

Determine obstructions to the global existence of action-angle coordinates on regular Poisson manifolds....

[S. V\ u Ng{\d o}c] Find natural examples of integrable systems in classical mechanics with non-trivial Duistermaat--Chern classes....

Assume that a classical integrable system $(p_1,\dots,p_n)$ is given on $M$. Does there exist a quantum integrable system $(P_1,\dots,P_n)$ such that ...

[S. V\ u Ng{\d o}c] Write Bohr-Sommerfeld rules for focus-focus singularities in Berezin-Toeplitz quantisation....

[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...

Let $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be rational with Julia set $J$, and suppose that a component $D$ of $\widehat{\mathbb C}\setminus J$...

How many completely invariant components can the Fatou set of a transcendental entire function have? In particular, can there be more than one?...

Does there exist a rational function having an analytic invariant Jordan curve on which it is topologically conjugate to an irrational rotation, where...

Let $C$ be an analytic invariant curve of a rational function $f$, suppose $f:C\to C$ is not a homeomorphism, and assume $C$ contains a repelling fixe...

For Ollivier's coarse Ricci curvature, smooth uniformly strictly log-concave measures on $\mathbb{R}^N$ have positive curvature. What can be said for ...

The space $\mathbb{R}^N$ equipped with an $L^p$ norm has zero coarse Ricci curvature. Does this observation yield useful results for Finsler manifolds...

For a continuous-time Markov semigroup $(m_x^t)$ define $$\kappa(x,y)=\liminf_{t\to0^+}\frac1t\frac{d(x,y)-T_1(m_x^t,m_y^t)}{d(x,y)}.$$ Under a natura...

Positive coarse Ricci curvature gives a spectral-gap bound for reversible random walks and on finite spaces. What spectral-radius, operator-norm, or P...

For the $\varepsilon$-step random walk on an $N$-dimensional Riemannian manifold, the coarse-curvature argument gives the lower spectral-gap estimate ...

Can estimates based on a uniform lower bound for coarse Ricci curvature be extended to spaces where curvature has only a controlled number of negative...

Can the bounded-local-variance hypothesis used for concentration under positive coarse Ricci curvature be relaxed while retaining estimates governed b...

Can concentration consequences of positive coarse Ricci curvature be formulated as transportation or other functional inequalities? In a coarse settin...

What is the relationship, if any, between Ollivier coarse Ricci curvature and the Sturm–Lott–Villani displacement-convexity notion, including its $CD(...

Is there an analogue, for positive coarse Ricci curvature, of the Bishop–Gromov theorem or the isoperimetric form of the Gromov–Lévy theorem? Identify...

Does positive coarse Ricci curvature imply a useful exponential entropy-decay statement analogous to that obtained from logarithmic Sobolev inequaliti...

Let the metric of a Markov space evolve by $$\frac{d}{dt}d(x,y)=-\kappa(x,y)d(x,y),$$ where $\kappa$ is computed from the current metric, with either ...

Replace the $T_1$ distance in the coarse-Ricci definition by $L^\infty$ transport, requiring a coupling that moves every point by at most $d(x,y)$. Do...

Do spaces with positive sectional curvature in the sense of Alexandrov have positive coarse Ricci curvature for a natural choice of Markov kernels? Ca...

What is the measure of the set of initial conditions of the Newtonian $N$-body problem that lead to global solutions? The complementary set of singula...

Does a free gas coupled to a convex rigid container by conservation of momentum converge weakly to equilibrium, with the container—which moves only by...

For $\mathbb{Z}^d$-actions by automorphisms of a connected group, mixing actions are mixing of all orders. Can this result be proved using simpler ide...

For each $1\le r\le\infty$, characterize the entropy structures that can occur for $C^r$ diffeomorphisms of compact Riemannian manifolds....

For a general $T$-periodic Riccati equation $$\frac{dx}{dt}=A_2(t)x^2+A_1(t)x+A_0(t),$$ give effective criteria determining whether it has a continuum...

Determine the maximum number $\mathcal{C}_n$ of centers for planar polynomial differential systems of degree $n\geq4$....

For the family of reversible quadratic centers $$\begin{cases}\dot x=-y+xy,\\ \dot y=x+Dx^2+Fy^2,\end{cases}$$ is $2$ the maximum number of critical p...

Is the period function associated with the period annulus of the origin for $$\dot z=iz+(z\bar z)^n z^{k+1},$$ where $n$ and $k$ are positive integers...

Determine the entries currently marked unknown in the following comparison between quadratic systems and planar piecewise-linear systems with a straig...