Mathematics Problem Archive

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....

[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...

[ Maxwell, seems bad, no tools] For any system of $N$ isolated fixed point charges in $\mathbb{R}^3$, the number of points of equilibrium (assumed fin...

[folklore, very irritating] For any set of charges of the same sign in $\mathbb{R}^n$, the set of its points of equilibrium is finite....

[A. Gabrielov, D. Novikov, B. Sh., seems good, but no progress] Let $(x_1,y_1),(x_2,y_2),\dots, (x_N,y_N)$ be a collection of points in $\mathbb{R}^2$...

[B. Sh., looks bad, but very important] Does there exist an upper bound for the number of real roots valid for all non-trivial solutions of all equati...

[D. Khavinson, I. Itenberg, B. Sh., apparently bad] Find the maximal possible number $\#(2k, l)$ of isolated zeros for real non-negative polynomials o...

[G. Ottaviani, B. Sh., seems good] Find the maximal possible number $\widetilde\#(2k, l)$ of isolated zeros for real non-negative polynomials of degre...

[G. Ottaviani, B. Sh., seems good] For any number of variables, $\widetilde\#(2k,l)=k^l$....

[S. Fisk, seems bad, see , p. 575] Given a pair of real polynomials $(p,q),$ give restrictions on the location of the roots of $p+iq$ in terms of the ...

[P. Br\"anden, I. Krasikov, B. Sh., hopefully good, see ] A difference operator $T(p(x))=a_0p(x)+a_1p(x-1)+\cdots+a_kp(x-k)$ with constant coefficient...

If $p$ and $q$ are real-rooted polynomials of degree at most $d$ and of mesh $\geq 1$, then so is $p \bullet q$....

[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...

[J. Forsg\aa rd, V. Kostov, B. Sh, hopefully good, see ] For an arbitrary sign pattern $\sigma$, the only type of pairs $(pos,neg)$ which can be non-r...

[J. Forsg\aa rd, B. Sh., seems good, see ] Let $f(z) = \sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients, and consider the related (wei...

[J. Forsg\aa rd, B. Sh., seems good, see ] Let $f(z) = \sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients. Consider the differences \[ \...

[J. Forsg\aa rd, B. Sh., seems good, see ]] Let $f(z) = \sum_{k=0}^n a_k z^k$ be a polynomial with positive coefficients. Consider the differences \[ ...

[V. Kostov, B. Sh., looks ugly, see ] What additional restrictions besides [source label: eq:1] exist on configurations $\mathcal A_{f}=\{x^{(i)}_{l}\...

[looks ugly] What symbolic sequences can occur for strictly real-rooted polynomials of degree $n$?...

[B. Sh., seems good] For any real polynomial $p(x)$ of degree $k$ with simple real zeros, $$ \#_{r}\left[(k-1)(p'(x))^2-kp(x)p''(x)\right] \le \#_{nr}...

[B. Sh] For any real polynomial $p(x)$ of even degree, $$ \#_{r}\left[(k-1)(p'(x))^2-kp(x)p''(x)\right] + \#_{r}p(x)>0, $$...

[B. Sh] For any degree $k$ polynomial $p(x)$ with real coefficients, $$ \#_{r}P_{i}(x) \le \min\{{\deg{P_i(x)},k}\}. $$...

How big can the set of Valiron deficiencies be for functions in the plane? It is known that $$ N(r,a)=T(r,f)+O\big(T(r,f)^{\frac{1}{2}+\varepsilon}\bi...

If $f(z)$ is meromorphic of finite order $\rho$ and $\sum\delta(a,f)=2$, it is conjectured that $\rho=n/2$, where $n$ is an integer and $n\geq 2$, and...

Let $f(z)$ be an entire function of finite order $\rho$, and let $n_1(r,a)$ denote the number of simple zeros of the equation $f(z)=a$. If \[n_1(r,a)=...

Under what conditions can $\sum\delta(a,f)$ be nearly $2$ for an entire function of finite order $\rho$? Pfluger proved that if $\sum\delta(a,f)=2$, t...

Arakelyan has proved that, given $\rho>\frac{1}{2}$ and a countable set $E$, there exists an entire function $f(z)$ of order $\rho$, for which all the...

If $f(z)$ is an entire function of finite order $\rho$ which is not an integer, it is known that (see Pfluger and Hayman ), \[\sum \delta(a,f)\leq 2-K...

Following the notation in Problem 1.7, if $f(z)$ is meromorphic in the plane of order $\rho$, it is conjectured by Pfluger , that for $a\neq b$ \[\lim...