Conjecture 2.2 — Suppose C∞(Hom(π,G )/G) D− →C∞(Hom(π,G )/G) is a differential operator which commutes with the ModΣ-action on Hom(π,G…

Definitions and mathematical statement

Set $X=\operatorname{Hom}(\pi,G)/G$, where $G$ acts on representations by conjugation. The intended $\pi$, group $G$, surface $\Sigma$ and quotient category are not specified in the extract; conventionally $\pi=\pi_1(\Sigma)$ and mapping classes act by precomposition with induced outer automorphisms.

Conjecture 2.2 states that if a differential operator \[ D:C^\infty(X)\longrightarrow C^\infty(X) \] commutes with the $\operatorname{Mod}_\Sigma$-action, then $D$ is a scalar multiple of the identity. Writing $U_f$ for the induced pullback action on functions, its hypothesis and conclusion are \[ DU_f=U_fD\quad(f\in\operatorname{Mod}_\Sigma) \qquad\Longrightarrow\qquad \exists c:\ D(h)=ch\quad(h\in C^\infty(X)). \] The conclusion uses one scalar for the whole function space.

In a usual linear finite-order interpretation on a smooth manifold, $D$ is locally a finite sum $\sum_{|\alpha|\le m}a_\alpha\partial^\alpha$ with smooth coefficients. The source does not explicitly state linearity, order convention, real versus complex functions, or the meaning of smooth differential operators on this potentially singular quotient. A nonlinear differential operation, or a different invariant-smooth-function model, would change the assertion. Connectedness and component permutations also matter: invariant locally constant multiplication operators could otherwise obstruct one global scalar conclusion. These missing hypotheses are recorded rather than supplying an arbitrary character-variety case in which to test the conjecture.

Short English statement

The conjecture says every differential operator on the specified representation quotient commuting with the mapping class action is one global scalar operator. The representation group, smooth quotient model and operator conventions are missing and are necessary to determine the intended assertion.

Sources

• Local UnsolvedMath registry, permanent ID $11000187$, exact source code catalogue entry, Conjecture 2.2, PDF page 216. No individual proposer or proposed year is supplied; the title is truncated.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The volume has 329 explicitly typeset numbered items across chapters, repeated local numbering distinguished by PDF page.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and calls for a current primary-literature audit. Registry status open and classification OPEN-TRIAGE; research summary says only "Literature status: Truncated." No separate primary definition or resolution reference is supplied.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.