12 problems

Let $\mu$ be a complex number that is not a root of unity and $|\mu|=1$. A number $\mu$ is Brjuno when it satisfies the Brjuno condition referred to in the source. Let $Q_{\gamma}$…

Let $K_{\mathrm{CNF}}(I,J_2,\dots,J_n)$ be a truncated classical normal form near an index-1 saddle, where $I$ is the reaction integral and $J_k$ are the bath actions. On the norma…

Normal-form conjecture. In the second form, $u$ and $v$ satisfy restriction (a); in the third form, they satisfy restriction (b); and in the fourth form, they satisfy both restrict…

Consider the original Treschev problem, with coefficients $q_{2n}$ and a Diophantine rotation number. Here, $k>0$ is arbitrary and $C(k)$ denotes a constant depending on $k$. Tresc…

Let … denote the spherical water waves equation. Lifespan conjecture. If the Diophantine equation … , the lifespan goes beyond $\varepsilon^{-(1+\alpha)}$ as $\varepsilon\to0$. Fin…

Alternate normal form conjecture. Any such unfolding is orbitally conjugate to an unfolding of the form

Convergence conjecture. If the eigenvalues additionally satisfy good Diophantine properties, then the formal Taylor series defining the function $f$ is locally convergent. The form…

Let $u(t)=0$ be the steady state of a state-dependent delay differential equation, and suppose its expansion has terms through a given order involving only constant delays, with al…

In a smooth integrable dynamical system, let the system be defined near a nondegenerate singular point, meaning a singular point satisfying the paper's nondegeneracy condition for…

Zhang's conjecture. The diffeomorphism $\varphi$ is not embedded in a $C^{\infty}$ (respectively, analytic) flow.

Let $A_*$ be a $G$-algebra, let $A=\operatorname{Quot}(A_*)$, and let … where $R_*$ is a $G$-algebra. For a matrix $M\in R^{p\times p}$, there are square matrices $U,V,D$ with entr…

Let $R$ be a simple Euclidean domain and let $M$ be a square matrix in diagonal form … The quantity $\sum\deg(m_i)$ is an invariant of the module $R^{r\times r}/M$. Choose polynomi…