Mathematics Problem Archive

Calculate the degree of $\widetilde{W}_{k,d}$ for perfect pairs $(k,d)$....

[Fr\"oberg's Conjecture, 1985] Let $f_1, \dots, f_r$ be generic forms of degrees $d_1, \dots, d_r$, respectively. Then the Hilbert series of the quoti...

[Fr\"oberg-Iarrobino Conjecture] Given generic linear forms $\ell_1, \ldots, \ell_r$ and a positive integer $d$, let $I$ be the power ideal generated ...

[] For generic forms $g_1,\ldots,g_r$ of degree $d>1$, the ideal $(g_1^k,\ldots,g_r^k)$ has the same Hilbert series as the one generated by $r$ generi...

When are the WLP and the SLP true for $T_{n,d,k}$?...

For $R= S/(f_1,\ldots,f_r)$, where $f_1,\ldots,f_r$ are generic forms, does $R$ satisfy the $\mu$-Lefschetz property for all partitions $\mu$?...

Given a scheme of generic fat points $X \subset \mathbb{P}^{n_1-1}\times\ldots\times \mathbb{P}^{n_t-1}$, what is the multi-graded Hilbert function $\...

$\mathcal{S}$ is cyclotomic if and only if $k[\mathcal{S}]$ is a complete intersection....

Are $B_{n,m}$ and $B'_{n,m}$ finite for any pair $(n,m)$ with even $n$?...

For $n=1$ and given $p$, what are the (lengths of the) possible periods of $\phi$?...

For $n = 1$ and given $p$, find the minimal positive integer $i$ such that $\psi^i$ is the identity map on the space of polynomials of degree at most ...

Let $f$ be a form of odd degree $d$ in $E$. Is it true that $({\rm Ann}(f))_i = (f)_i$, for $i < (n-d)/2$?...

Let $f$ and $g$ be generic quadratic forms in $E$ and let $\ell_1$ and $\ell_2$ be two generic linear forms in $S$. Then the Hilbert series of $E/(f,g...

For a non-hyperelliptic algebraic curve, is its Clifford index determined by the extent to which its canonical embedding has linear syzygies?...

Determine for which orders Williamson matrices exist; such matrices give a construction of Hadamard matrices....

For each order $n$, determine the largest possible absolute determinant of an $n\times n$ matrix whose entries are all $1$ or $-1$....

Let $m,p\ge 2$, put $d=m+p-1$, and let $F(x)=(1,x,\ldots,x^d)$ be the rational normal curve. For $j=1,\ldots,mp$, let $X_j$ be the $p$-plane spanned b...

Classify all empty simplices of dimension $n$ up to lattice congruence....

Find a natural generalization of the Farey tessellation to higher-dimensional hyperbolic geometry....

{\bf (Jacobi's last theorem.)} Let $K$ be a totally real cubic number field. Consider arbitrary elements $y$ and $z$ of $K$ such that $0<y,z<1$ (here ...

Generalize continued fractions to describe 3-bridge knots....

Prove the existence of energy minimizers for elastostatics for quasiconvex stored-energy functions $W$ satisfying $W(A)\to\infty$ as $\det A\to0^+$ ....

Can the Lavrentiev phenomenon occur for elastostatics under growth conditions ensuring that all finite-energy deformations are continuous?...

Prove or disprove that, under reasonable growth conditions on $W$, energy minimizers satisfy the weak Euler-Lagrange equations....

Justify the Ciarlet-Nečas minimization problem, or an appropriate modification, in situations involving smooth self-contact....

Prove or disprove uniqueness of sufficiently smooth equilibrium solutions for pure-displacement problems in homogeneous bodies homeomorphic to a ball ...

Develop local and global bifurcation theories for nonlinear elastostatics with mixed displacement-traction boundary conditions....

Clarify the status of models based on the fracture energy functional (2.31) in the source relative to classical fracture and nonlinear elastostatics....

Prove global existence and uniqueness for suitable initial-boundary-value problems in dynamic nonlinear elasticity....

Develop criteria for dynamic stability and instability of equilibria in nonlinear elasticity....

For the set of energy-minimizing gradients $K(\theta)$ defined in the source, determine its quasiconvex hull $K(\theta)^{qc}$ for $\theta\le\theta_c$....

Give a rigorous derivation of models of rods, plates, and shells from three-dimensional elasticity as thickness tends to zero....

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....