14 problems

Let $G$ be the fundamental group of a complex hyperplane arrangement, let $\theta_n(G)$ denote its Chen ranks, and let $\mathcal{R}^1(G;\mathbb{C})\subseteq H^1(G;\mathbb{C})$ be i…

Let $\mathcal{A}$ be a hyperplane arrangement with complement $M$, let $\Phi_{\lambda}$ be its master function for weights $\lambda$, and let $\chi(M)$ denote the Euler characteris…

Universality conjecture. The predicted ordering of three-frequency resonances may be universal in a large class of dynamical systems with three interacting frequencies.

Consider the dispersive Hamiltonian normal form … where $k_{n+1,n}=d_{n+1,n}+ic_{n+1,n}$, and suppose the first relevant resonance occurs at a higher harmonic…

Typicality conjecture. The repeating resonance structure for large $\tau$ is a typical phenomenon of periodically forced delay differential equations.

Let $\mathfrak{g}$ be a finite-dimensional nilpotent Lie algebra over $\mathbb{k}$, and set $A=\operatorname{CE}(\mathfrak{g})$. Let $S$ be the symmetric algebra governing the Kosz…

Let $(A,d)$ be a connected, $q$-finite, and $q$-formal $\mathbb{k}$-CDGA. For $i\leq q$ and $s\geq 1$, let $\mathcal{R}^{i,s}(A)$ and $\mathcal{R}_{i,s}(A)$ denote the correspondin…

Higher-torsional-mode amplitude conjecture. The gap function reduces in amplitude when $\overline f_j$ is in resonance with higher torsional modes.

Purely imaginary parameter conjecture. No solutions of the fixed-point equation exist when $\varepsilon$ is purely imaginary, because the zero divisors can be regarded as resonance…

Unique-flex-point conjecture. All the resonant lines emanating from $(0,\ell)$, except $\omega\equiv 1$, are graphs of strictly increasing functions with a unique flex point.

Resonant-tori multiplicity conjecture. Under a convexity assumption on $H_0$, for any $r\leq n-1$ and for most families of unperturbed tori whose frequency vectors are resonant wit…

Let the resonant set in the model case and the resonant set in the real case be the sets considered above, with parameter $\lambda$. As $\lambda$ grows large, the term…

Let $H_\epsilon$ be a time-reversible Hamiltonian with stable and unstable manifolds parameterized by $\mathbf{\Gamma}^s$ and $\mathbf{\Gamma}^u$, and let $\gamma_h$ be a primary s…

A spatially inhomogeneous sine-Gordon equation with a time-periodic drive models a microwave-driven long Josephson junction with phase shifts. The junction may have a spatially non…