Mathematics Problem Archive

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

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...

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

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...

[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 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...

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?...

In a symmetric space, is every Killing tensor a sum of symmetric products of Killing vectors? Equivalently, is it true that the algebra of all polynom...

{\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...

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...

Assume that the exterior of an outer billiard table is foliated by invariant curves. Prove that the table is an ellipse....

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...

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...

{\rm (A. Glutsyuk, S. Tabachnikov )} Given two nested closed convex hypersurfaces, assume that the billiard transformations on the set of oriented lin...

{\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?...

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

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...

Construct real analytic examples of Nijenhuis operators on closed two-dimensional surfaces whose eigenvalues are real and generically distinct....

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...

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....

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] 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] 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] Write Bohr-Sommerfeld rules for focus-focus singularities in Berezin-Toeplitz quantisation....