11 problems

Let $M$ be a complex manifold and let $\nabla$ be a flat meromorphic connection on $M$. A formal decomposition of $\nabla$ is called good when its formal exponential factors satisf…

Sabbah's conjecture. There is a finite sequence of point blow-ups

Let $\mathbf{S}=(S_i)_{i=1,\ldots,d}$ be a collection of abstract spectral types. A collection is irreducibly realizable if it is the spectral type of an irreducible meromorphic…

Jimbo–Miwa–Ueno differential conjecture. One has

Jimbo–Miwa–Ueno differential conjecture. The observation that the Jimbo–Miwa–Ueno differential is obtained from the Hamiltonian differential by formal evaluation at $\hbar=0$, up t…

Higher-dimensional local decomposition conjecture. Meromorphic connections in higher dimensions should admit such local decompositions, at least after suitable blow-ups of the sing…

Abate and Tovena's conjecture. There is a neighborhood $U$ of $p_0$ such that every geodesic ray which enters into $U$ tends to $p_0$. The two rays of any geodesic which stays in…

Let $S^o=\mathbb{P}^1(\mathbb{C})\setminus\{p_0,p_1,\dots,p_r\}$, where $p_0,p_1,\dots,p_r$ are the poles of a meromorphic connection $\nabla$ on $\mathbb{P}^1(\mathbb{C})$. Let…

Let $(M,t^0)$ be an irreducible germ of a generically semisimple F-manifold. A generically semisimple F-manifold structure conjecture. The existence and parametrization assertions…

Let $R=\mathbb{C}[t]/(t^m)$ be a truncated polynomial ring with $m\in\mathbb{N}$, let $G_R$ be the Weil restriction to $\mathbb{C}$ of $G\otimes_{\mathbb{C}}R$, and consider the mo…

Biquard's conjecture. There exists a sequence of point blowing-ups such that the pull-back connection by this proper modification is good along the divisor of its poles.