8 problems

Let $k$ be the field of constants, let $Y$ be a smooth algebraic variety over $k$, let $Z$ be a divisor in $Y$, and let $({\mathcal M},\nabla) \in {\bf MIC}((Y \setminus Z)/k)$. Fo…

Associated-connection conjecture. The connection $\eta$ can be seen as an associated connection to a principal connection on $(\hat{P}, M, \pi_{\hat{P}, M}, \operatorname{Aut}(G))$…

Let $(E, abla)$ be a connection in suitably generality, and suppose that its $p$-curvature vanishes for almost all primes $p$. Grothendieck–Katz conjecture. The monodromy of the co…

Let $X$ be a smooth quasi-projective variety over $\mathbb C$, and let $(E,\nabla)$ be an integrable algebraic connection. Choose a finite-type scheme $S$ over $\mathbb Z$ over whi…

Over the complex numbers, replacing Artin–Schreier sheaves by rank-one connections of exponential type produces a family of connections on $\bA^1$. Airy-connection identification c…

Let $G$ be a simple complex group with Lie algebra $\operatorname{\mathfrak{g}}$. Fix a Coxeter $G$-formal type $\mathscr{A}$ of slope $r/h$ with $\gcd(r,h)=1$. A nilpotent orbit t…

Spin-connection convergence conjecture. One might conjecture that

Convex-invariance conjecture. Finsler geometry is characterized by the following data: