29 problems

Let $N$ be a Fibonacci number, let the Fibonacci lattice be the corresponding lattice point configuration on the torus, and let $f$ be a potential in the broad class of potentials…

Let $Q(x)=x^T A x$ be a positive definite quadratic form with $A$ symmetric, let $\Delta_Q=Q(\partial_1,\partial_2,\partial_3)$ on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$, and def…

Let $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$, let $\lambda>2$ and $\delta<1$, and define … Let … so that $P_{\lambda,\delta}f=\Phi_{\lambda,\delta}*f$. Kernel conjecture. If…

Discrepancy lower-bound conjecture. For all matrices, not only badly approximable ones, the factor $k^{n/2}$ in the denominator of the lower bound in Theorem lowerbound should be r…

Let an $n\times n$ torus be a square Domineering board of side length $n$, with Hepzibah and Vera as the two players; ${\rm 1st}$ and ${\rm 2nd}$ denote the first and second player…

For the quantization of an irrational skew translation on the two-torus, let $N$ denote the inverse Planck parameter and let the expectation values of observables in eigenstates co…

Let $T^2$ be the two-dimensional torus, and consider an integrable geodesic flow on its cotangent bundle. An integral quadratic in momenta is a first integral of the geodesic flow…

Let $g$ be a metric on $T^3$ that is close to the flat metric. A harmonic $1$-form on $T^3$ is a $1$-form $\theta$ satisfying $\Delta\theta=0$, and it is nowhere vanishing when…

Let $g$ be a metric on the two-dimensional torus $\mathbb{T}^2$. A metric is polynomially integrable when its second conserved quantity is a polynomial in the momentum variables, a…

Let $g_0$ be a Liouville metric on $\mathbb{T}^d$, let $r$ be the regularity exponent appearing in the $C^r$ norm, and let $g$ be another Riemannian metric. Say that $g$ is Laplace…

Folklore conjecture. Liouville metrics are the only integrable Riemannian metrics on $\mathbb{T}^2$. This is viewed as a closed-manifold analogue of the Birkhoff conjecture and is…

Let $f^t$ be a $C^{\infty}$-flow on the torus without singularities. A quasiminimal is the topological closure of a recurrent trajectory; a closed trajectory or a fixed point is ca…

Let $T^k=(S^1)^k$ be the $k$-dimensional torus, and consider its combinatorial triangulations by abstract simplicial complexes. Vertex-minimality conjecture for the k-torus. Every…

Let $T^d$ denote the $d$-dimensional torus, and let $\mathcal{T}_{(1,\ldots,1)}$ be the triangulation with $d+1$ ones in its subscript, having $2^{d+1}-1$ vertices. Lutz's minimal…

Let $G=\mathbb{T}^d$ be the $d$-dimensional torus with Haar probability measure $\mu$. For compact, non-empty sets $A,B\subseteq G$, write $A\cdot B=\{a\cdot b:a\in A,b\in B\}$. Bo…

For an even integer $p$, let $\mathcal{A}'_{\lambda,0}$ denote the lattice points on the circle of radius $\lambda$, and let $\mathbb{E}_3(\mathcal{A}'_{\lambda,0})$ be the number…

Let $g$ be a Liouville metric on ${\mathbb{T}}^2$, and consider an integrable deformation of $g$ that preserves all but finitely many rational invariant tori. Deformational rigidit…

Folklore conjecture. Liouville metrics are the only integrable metrics on ${\mathbb{T}}^2$.

Let $n\geq 3$, let $\lambda\geq 1$, and let $f$ be a function on the torus $\mathbb T^n$. For a spectral parameter $\lambda$ and unit-width spectral localization, write…

Let $X$ be an $n$-torus with universal covering $\tilde X$, and let $\Gamma=\mathbb Z^n$ act on $\tilde X$. A combinatorial class $({\cal P},\alpha_i)$ topologically exhaustively d…

A geodesic flow on the 2-torus is a Hamiltonian system for the metric … with Hamiltonian $H=\frac{1}{2}g^{ij}p_ip_j$. It is called integrable if there is a first integral…

Product decomposition conjecture. Given $\mu\in\mathrm{Dens}({\mathbb T}^n)$, there exist $\eta\in\mathrm{Diff}_{\mathrm{d}x}({\mathbb T}^n)$ and functions…

Let $\mathbb{T}$ be the axisymmetric torus with radii $0<r<R$, and let $h\in\mathbb{Z}\times\mathbb{Z}$ denote a winding number. A local minimizer is understood for the one-constan…

Let $\varphi=\varphi_E$ be an eigenfunction of the Laplacian on the $n$-dimensional torus $\Bbb T^n$, with $n\geq 2$, so that $-\Delta\varphi_E=E\varphi_E$. For $p<\frac{2n}{n-2}$,…

Let $T_0$ be the torus example considered in the source, let $\EuF(T_0)$ be its Fukaya category, and let $\EuA$ be the associated $A_\infty$-algebra. Assume there is an isomorphism…