28 problems

Relative Pu conjecture. There is a point $p\in X$ such that

Pu–Loewner inequality conjecture. Every such metric satisfies

Universal stable 2-systolic inequality conjecture. There is a numerical constant $C$, independent of $X$, such that for every metric $\mathbf{g}$ on $X$,

Pu's relative 1-systole conjecture. For any such $S$ and $\hat\phi$,

Fix a symmetric space $S$ of non-compact type without Euclidean de Rham factors, and let an arithmetic $S$-manifold be a quotient $M=\Gamma\backslash S$ by a torsion-free arithmeti…

Let $N\geq 2$, let $1\leq k\leq 2N-2$, set $d=k+1$, and let $\mu>0$. For the Berger metric $g_\mu$ on $\mathbb {RP}^{2N-1}$, define the mod-two homological systole by … For a real…

Let $n>2$, and let $(S^n,g)$ be an $n$-dimensional Riemannian sphere. A Besicovitch-type product inequality asserts that there is a constant $C(n)>0$, depending only on the dimensi…

Let $X$ be a translation surface. Write $\mathop{\mathrm{KVol}}(X)$ for its KVol and $\mathop{\mathrm{SysVol}}(X)$ for its systolic volume, defined as the volume divided by the squ…

Gromov's systolic conjecture. There exists an absolute constant $C$ such that, for every essential closed Riemannian manifold $M^n$,

Homotopical $2$-systole conjecture. Under these hypotheses,

Let $(P_n)$ be a sequence of quasi-optimal pseudomanifolds representing $n\,\mathbf a$, and let $\varphi_n:\pi_1(P_n)\to\pi_1(X)$ be the induced homomorphism. Write…

Let $(P_n)$ be a sequence of quasi-optimal pseudomanifolds representing multiples of a homology class, equipped with the piecewise-flat metric in which every $m$-simplex is isometr…

Let $\Sigma$ be the underlying manifold, let $C\in H^2_{\mathrm{dR}}(\Sigma)$, and let $\Omega_*$ be a Zoll odd-symplectic form with cohomology class $C$. For forms…

Maximal systolic ratio conjecture. The supremum of the systolic ratio over genus two surfaces $(X,d_{\omega})$ equals

Systolic capacity conjecture. There is a constant $D_n$ depending only on $n$ such that

Let $S_g$ be a closed hyperbolic surface of genus $g$, and let $D_{\operatorname{inj}(S_g)}$ denote an embedded metric disk of radius equal to the injectivity radius of $S_g$. Schm…

Let $K\subset {\mathbb{R}}^N$ be an embedded torus with one boundary component, and suppose that $\partial K$ is a unit circle. A closed curve in $K$ is non-null-homotopic if it is…

Let $K$ be a surface of genus $g$ embedded in ${\mathbb{R}}^N$ by a bilipschitz map, and let $\gamma_1,\dots,\gamma_{3g-3}$ be the boundary curves of a pants decomposition of $K$.…

Let $K$) be a surface embedded in ${\mathbb{R}}^N$, of genus $g$, and let $\gamma_1,\dots,\gamma_{3g-3}$ be the boundary curves of a pants decomposition of $K$. Here…

Let $\Sigma$ be a reversible optical hypersurface in the cotangent bundle $T^*\mathbb{R}P^n$ of real projective $n$-space, enclosing a volume $V$. A periodic characteristic on…

Loewner's bound. Every aspherical surface satisfies

Let $(B_2,g_0)$ be the singular Riemannian manifold considered in the source. A metric is extremal for the isosystolic inequality if it realizes the optimal systolic constant in it…

Let $(\mathbb{R} \mathbb{P}^3,sph)$ denote real projective 3-space endowed with its metric of constant curvature. A metric is extremal for the isosystolic inequality if it realizes…

Let $g$ be a Riemannian metric on the three-dimensional torus $T^3$, and let $f:T^3\rightarrow\mathbb{R}^2$. Fiber-length conjecture. There is a function $f$ such that, for every…

Fix a scale $r>0$. For a Riemannian metric on a closed manifold, let $Scal_r(p)$ denote the macroscopic scalar curvature at scale $r$ at $p$, defined by comparing the volume of the…