23 problems

Let $(M,g)$) be a closed Riemannian manifold. Define … and let $0_M$ denote the image of the zero section of $T^*M$. For a compactly supported Hamiltonian diffeomorphism…

Let $(M_k,\ker\alpha_k)$ be an orderable contact manifold, and let $c_+^{\alpha_k}$ be the associated contact spectral invariant. A map is $\alpha_k$-spectral when it satisfies the…

Let $X$ be a derived critical locus, and let $f$ range over potentials whose derived critical locus is $X$. Derived spectrum equivalence conjecture. The derived spectrum is the sam…

A derived spectrum is the spectrum associated with the derived critical-locus construction of a potential for the moduli space of $\mathop{SL}(2,\mathbb{C})$ representations of a 3…

Let $c_{-}^{\alpha}(\varphi_t)$ and $c_{+}^{\alpha}(\varphi_t)$ be the order measurements of a contact isotopy, and let $c_{\alpha}(1;\varphi_t)$ be the spectral invariant of the u…

Let $(M,\xi)$ be a closed orderable contact manifold, meaning that it admits no positive contractible loop of contactomorphisms. Let $\alpha$ be any contact form supporting $\xi$,…

Let $(M,\omega)$ be a monotone symplectic manifold admitting a pseudo-rotation. Here $\gamma_\mathbb{Z}$ denotes the spectral norm over the integers, and $\phi^k$ denotes the $k$-t…

Existence and characterization conjecture. The following assertions are conjectured:

Let $d\geq 2$, and let $f,g$ be non-Lattès rational maps of degree $d$. Let $\tau_d$ denote the multiplier spectrum map, and let $\overline{g}$ be the coefficientwise complex conju…

For every $d\geq 2$, let $\mathcal{M}_d$ be the moduli space of rational maps of degree $d$, defined over $\mathbb{R}$. For $f\in \operatorname{Rat}_d(\mathbb{C})$, let its length…

Let $\alpha$ be a supporting contact form and let $c_\pm^\alpha$ be the associated maps on the contactomorphism spaces $\mathcal{G}$ and $\widetilde{\mathcal{G}}$. Spectral-selecto…

Let $(M,\xi)$ be a contact manifold, let $\alpha$ be a supporting contact form, and let $c_\pm^\alpha$ denote the selectors defined on the relevant contactomorphism or Legendrian s…

Let $(Y_1,lambda_1)$ and $(Y_2,lambda_2)$ be closed connected contact three-manifolds with nonzero contact invariants, and let $k$ be a nonnegative integer. Write…

A derived spectrum is a spectrum proposed for the non-commutative, non-Abelian Hodge-theoretic construction associated with a 3-manifold and its Heegaard splitting. Derived spectru…

Let $(M,\omega)$ be a symplectically aspherical manifold, meaning that $\omega$ and the first Chern class vanish on every spherical homology class. A Hamiltonian diffeomorphism is…

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 $N$ be a closed Riemannian manifold and let $DT^*N=\{(q,p)\in T^*N\mid |p|_g\leq 1\}$. For an element $L\in\widehat{\mathfrak L}(T^*N)$, its $\gamma$-support is denoted by…

Let $N$ be a closed Riemannian manifold, and let … An exact Lagrangian $L$ is geometrically bounded when it is contained in $DT^*N$. Geometrically bounded implies spectrally bounde…

Let $(M,\omega)$ be a symplectic manifold and let $L$ be an exact Lagrangian. For any Hamiltonianly isotopic exact Lagrangian $L_1$ such that $\gamma(L,L_1)$ is sufficiently small,…

HYMNs conjecture. The HYMNs of an adinkra carry all information about its shape isomorphisms: two adinkras are isomorphic if and only if their HYMNs are the same.

Vanishing action-gap conjecture. There is a sequence $kitoto atext{infinity}$ such that

Let $M$ be a symplectic manifold with symplectic form $\omega$, and let $\widetilde{\phi}\in\widetilde{\operatorname{Ham}}(M,\omega)$ be Floer non-degenerate. A generalized path fr…