18 problems

Higher-dimensional torus conjecture. There exists a $T^2$-invariant contact structure $\xi$ on $T^{2n+1}$ such that

Let $K$ be a knot in $S^3$. Its augmentation variety is the locus in $(\mathbb{C}^*)^2$ of pairs $(\lambda_0,\mu_0)$ for which the framed knot DGA of $K$, specialized at…

Let $K$ be a codimension $2$ submanifold of a manifold $M$, let $LK$ be its unit conormal Legendrian submanifold in the contact manifold $ST^*M$ of unit covectors, and let the cord…

Let $K$ be a knot, let $\Sigma_2(K)$ be its double branched cover, and let $HC_0^{\operatorname{ab}}(K)$ denote the abelianized degree-zero contact homology, identified with the co…

Let $(Y,\xi)$ be a closed contact manifold with contact form $\alpha$. Write $\mathfrak{gap}_\sigma(Y,\alpha)$ for the spectral gap associated with a class $\sigma\in CH(Y,\xi)$. P…

Let $(Y,\xi)$ be a closed contact manifold with a periodic contact form $\alpha$ of period $T$. Let $CH(Y,\xi)$ denote contact homology, $\mathfrak{s}_\sigma(Y,\alpha)$ the spectra…

Let $(Y,\xi)$ be a closed contact manifold with a periodic contact form $\alpha$. The closed Reeb orbits form components $S$ of the orbit orbifolds, and to each component associate…

Let $E \subset \mathbb{C}^n$ be an ellipsoid and let $(\partial E,\lambda|_{\partial E})$ be its contact boundary, where $\lambda$ is the standard Liouville form. Irie's strong clo…

Eliashberg–Hofer conjecture. The differential satisfies $d^2=0$, and the homology $H_*(C_*,d)$ is independent of the contact form $\alpha$ for $\xi$ and of the complex structure…

Bourgeois–Ekholm–Eliashberg cyclic surgery conjecture. Under this isomorphism, the surgery exact triangle for cyclic Legendrian contact homology is identified with the exact triang…

Let $W$ be a Liouville cobordism corresponding to attaching critical handles along a collection $\Lambda$ of Legendrian spheres, and let $L{\mathbb H}^{\text{Ho}}(\Lambda)_*$ and…

Let the assumptions of Lemma(c) hold, and consider the inequality in Remark. Equality conjecture. Under these assumptions, the inequality is an equality. This proposed equality wou…

Legendrian knots are Legendrian submanifolds in the contactization of a one-dimensional base, and their Legendrian contact homology is encoded by a differential graded algebra gene…

Let $M$ be a closed $3$-manifold with a geometric structure, and let $\xi$ be a universally tight contact structure on $M$. Consider the growth rate of contact homology associated…

A nondestabilizable Legendrian knot is a Legendrian knot that cannot be obtained from another Legendrian knot by stabilization. Its Legendrian contact homology is the homology asso…

Left-handed stabilization conjecture. The left-handed stabilization of $(M,\alpha)$ along $L$ has vanishing contact homology:

Connected-sum conjecture. If $(N,\eta)$ is algebraically overtwisted, then

Nonvanishing conjecture.