Mathematics Problem Archive

[S. V\ u Ng{\d o}c] Define (and detect) the quantum Chern class....

[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...

[\'{A}. Pelayo] Can one make progress in counting the number of fixed points by studying the spectrum of the quantisation of $\mu \colon M \to S^1$?...

What are necessary and/or sufficient conditions on a metric $g$ such that every polynomial integral of its geodesic flow is quantisable?...

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

[Miranda-Presas-Solha ] Modify this scheme to get finite dimensional representation spaces for focus-focus and hyperbolic singularities that still cap...

Let $f$ be a nonlinear entire function and let $D$ be a Fatou component on which all limit functions of the iterates $f^n$ are constant. Can the set o...

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...

Is the number of degenerate Herman rings of a rational function finite, and can it be bounded in terms of the degree of the rational function?...

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 real matrices $A\in\operatorname{Mat}_{n\times n}$, $B\in\operatorname{Mat}_{n\times p}$, and $C\in\operatorname{Mat}_{m\times n}$ with $n=mp$, de...

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...

What is the coarse Ricci curvature of discrete or continuous nilpotent groups? In particular, for the natural random walk generated by $a,b$ on the di...

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...

Suppose the global infimum of coarse Ricci curvature is zero, while its infimum on every finite-radius ball about an origin is positive. Is there a sy...

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 ...

Define curvature up to $\delta$ by $$T_1(m_x,m_y)\leq(1-\kappa(x,y))d(x,y)+\delta.$$ Which theorems for positive coarse Ricci curvature extend to this...

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...

Define a scalar-curvature candidate by $S(x)=\int\kappa(x,y)\,dm_x(y)$, possibly with a distance-dependent weight. Does this quantity have useful geom...

Under the strengthened transport estimate $$T_1(m_x^{*t},m_{x'}^{*t'})\leq e^{-\kappa\min(t,t')}d(x,x')+C\frac{(\sqrt t-\sqrt{t'})^2}{2d(x,x')},$$ the...

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

Does there exist a family of bounded-degree expander graphs, with spectral gap bounded away from zero and diameter tending to infinity, whose coarse R...

For permutation groups with the transposition random walk, coarse Ricci curvature is positive but gives concentration of the wrong order. Can this dis...

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...

Are there geodesic flows or Birkhoff billiards with fractal caustics? More specifically, for every $1\leq s<2$, is there a caustic of a convex billiar...

If two Birkhoff billiard maps $T_1$ and $T_2$ satisfy $T_1=ST_2S^{-1}$ for a homeomorphism $S$, must their tables be similar? Relatedly, can one hear ...

(a) Is the set of $n$-periodic orbits of a smooth strictly convex Birkhoff billiard nowhere dense for every $n$? (b) Does every polygonal Birkhoff bil...

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...

Does there exist a Hamiltonian system with a smooth invariant torus on which the induced dynamics is mixing? No such mixing can occur on a two-dimensi...

For a Hamiltonian system, call the good set the maximal invariant subset on which the invariant Liouville measure is almost periodic; call the bad set...

In the space of area-preserving $C^1$ diffeomorphisms of a compact manifold, is it generic that the dynamics is either hyperbolic or has zero Lyapunov...

For the infinite-dimensional Calogero–Moser system, in which particles on the real line interact through the inverse-square potential, does the dynami...

Are there quasiperiodic global minimals for metrics on the torus that are close to a flat three-dimensional torus? Here a geodesic is a global minimal...

Let $p$ be a prime for which $$f(u_1,u_2)=1+u_1u_2+u_1^2u_2+u_1^3u_2+u_1^4+u_2^2+u_1^4u_2^2$$ is irreducible, and consider the algebraic $\mathbb{Z}^2...

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...

Is there an analogue of Pesin theory for suitably defined smooth maps of the objects that arise naturally in algebraic dynamical systems—compact sets ...