Mathematics Problem Archive

Does every Reeb vector field on a closed contact manifold have at least one periodic orbit?...

In the $n$-body problem with $n \geq 4$, can non-collision singularities occur in finite time?...

Under which additional assumptions does this principle become a rigorous theorem?...

Describe the symplectic invariants of stable rank-one singularities described by V. V. Kalashnikov. For such singularities, one of the action variable...

[N. T. Zung] Study the topology and geometry of these singular fibers and their small neighbourhoods....

[N. T. Zung] Give a clear description of these special singular fibers....

Complete the above table: {\rm (1)} construct new examples of natural Hamiltonian systems on closed two-dimensional surfaces admitting polynomial inte...

Construct stationary axially symmetric 4-dimensional Einstein metrics admitting Killing tensors of higher order....

[Gilkey ] In the Riemannian case, is every $1$-homogeneous manifold locally homogeneous?...

{\it Construct a natural Hamiltonian system on the 2-sphere with a nonconstant potential which is superintegrable by integrals of degree $\ge 3$ and a...

Prove the algebraic version of Birkhoff conjecture for outer billiards in a non-Euclidean surface of constant curvature....

Prove Conjecture [source label: Descon] it the case that the foliation admits (i) a rational, (ii) an algebraic first integral....

Is it possible to choose the domain so that the dynamics of the corresponding billiard map are locally (near the 2-periodic orbit) conjugated to the d...

Are there plane billiards, other than ellipses, that possess rational caustics with two different values of the rotation numbers? Same question for ou...

{\rm (V. Retakh) Establish complete integrability of the noncommutative version of the leapfrog map. Define noncommutative versions of the pentagram m...

Consider the action of $ Aut (V,J)$ on $V$. Describe the partition of $V$ into $ Aut (V,J)$-orbits. More generally, describe the action of $ Aut (V,J)...

Find necessary and sufficient conditions for the bi-Lagrangian Grassmannian $LG(V,J)$ to be a smooth algebraic variety. Describe the partition of $LG(...

Do bi-integrable systems exist for each algebraic type?...

Are there any restrictions on the algebraic type of the pencils $\mathcal{A}_{x+\lambda a}$? Which algebraic types can be realised by means of an appr...

Study examples of ``quadratic $+$ linear'' Poisson pencils. Compute their algebraic types and construct complete families of polynomials in bi-involut...

Is it true that for any quadratic Poisson bracket (defined on a vector space), there exists a polynomial integrable system?...

Describe closed manifolds $M$ which admit Nijenhuis operators $L(x)$ that are algebraically regular at each point $x\in M$....

Let us fix a certain algebraic type of a linear operator, i.e., its Segre characteristic (see above). Does there exist a Nijenhuis operator $L$ in $\m...

Describe all the functions $f(x,y)$ of two variables defined in a neighbourhood of $(0,0)\in\mathbb{R}^2$ such that ; $f_y(0,0)\not\equiv 0$; ; $f_y(0...

Consider a smooth map $\Phi=(\sigma_1,\dots,\sigma_n): U(0) \to \mathbb{R}^n$, where $U(0)$ is a neighbourhood of the origin $0\in \mathbb{R}^n$. We a...

Describe/classify the collections of algebraically independent homogeneous polynomials $\sigma_1, \dots, \sigma_n$, $\deg \sigma_k = k$, in $n$ variab...

Describe the structure of singularities of $\Phi=(\sigma_1,\dots,\sigma_n): M \to \mathbb{R}^n$ in terms of the singular points of the recursion opera...

Which foliations with affine leaves can be described as the image of the moment map?...

Consider a Poisson manifold $M$ of even dimension such that it is symplectic on a dense set $U\subset M$. Assume that $M$ is endowed with a toric acti...

[S. V\ u Ng{\d o}c] Given a set of semiclassical operators that verify conditions [source label: item:self-adjoint ] and [source label: item:commute] ...

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

[\'{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?...

[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...

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?...

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...

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...

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...

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...

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...

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

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...

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...

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...

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

Let $X=\{0,1\}^{\mathbb{Z}}$ and $Y=\{y\in\{0,1,2\}^{\mathbb{Z}}:y_i\neq y_{i+1}\}$. Both are mixing shifts of finite type with entropy $\log2$, but t...

Let $T:[0,1)\to[0,1)$ be the doubling map $x\mapsto2x\pmod1$, and let $\mu$ be an ergodic measure for $T$ with $0<h(\mu)<1$. Call $f:\mathbb{R}\to\mat...

Let $A=\{0,1,\ldots,d-1\}$ and let $\sigma$ be the shift on $A^{\mathbb{Z}}$. Define $(v_n(x))_i=\#\{0\leq j\leq n:x_j=i\}$ and $(w_n(x))_i=\#\{0\leq ...