Masur-Veech Equidistribution of Individual Twist Tori

Definitions and mathematical statement

The supplied record attributes the following assertion to Conjecture 7 of the cited Oberwolfach report. Its complete mathematical setting is not supplied.

Let $T$ denote a twist torus in the family intended by that conjecture, let $\tau$ denote its intended torus measure, let $\mathcal O$ denote the map from the torus to the relevant stratum, and let $g_t$ denote the expanding flow appearing in the source. Write $X$ for the connected component of that stratum containing $\mathcal O(T)$, and write $m_X$ for the Masur--Veech--Smillie measure intended on $X$. These symbols are placeholders for the source objects: the supplied excerpt does not define the twist-torus family, its parameters, the map, the flow, the stratum, or the measure normalizations.

For a measurable map $F$ and a measure $\tau$, its pushforward is defined by \[ (F_*\tau)(A)=\tau(F^{-1}(A)) \] for measurable sets $A$ in the target. The intended conjectural assertion is that, for every admissible twist torus and every admissible choice of its source parameters, \[ \mu_t:=(g_t\circ\mathcal O)_*\tau \ \xrightarrow[t\to+\infty]{\mathrm{weak}\text{-}*}\ m_X. \] This is convergence along the full time parameter, with no averaging in time and no deletion of exceptional times.

If the intended setting uses Borel probability measures on a metrizable stratum component and weak convergence of probability measures, the assertion takes the test-function form \[ \lim_{t\to+\infty}\int_T f\left(g_t(\mathcal O(x))\right)\,{\,\mathrm{d}}\tau(x) =\int_X f\,{\,\mathrm{d}} m_X \qquad\text{for every }f\in C_b(X), \] where $C_b(X)$ denotes the bounded continuous real-valued functions on $X$. This formulation is conditional on that normalization and topology; neither is specified in the supplied excerpt.

The original passage writes the torus and its measure using corrupted notation involving $b,P,L$. Their definitions and admissible ranges are absent. Consequently, the quantifier over admissible tori cannot presently be made explicit, and the word "each" in the cleaned statement must be understood relative to the unspecified family of Conjecture 7. The excerpt does not justify extending the assertion to every object that might elsewhere be called a twist torus.

Short English statement

Conjecture 7 asks whether each twist torus in the source's intended family equidistributes under the expanding flow, along all times and without averaging, toward the Masur--Veech--Smillie measure on its stratum component. The supplied excerpt lacks the definitions and hypotheses needed to specify that family and its measures precisely. The supplied dated assessment labels the problem partially solved; that assessment is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006023$, title "Masur-Veech Equidistribution of Individual Twist Tori," exact source code catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2024.
• Supplied primary citation: Riemann Surfaces: Random, Flat, and Hyperbolic Geometry (2025), Oberwolfach Reports, pp. 2369--2424, DOI 10.4171/owr/2024/41. The excerpt identifies the target as Conjecture 7. https://doi.org/10.4171/owr/2024/41
• Supplied literature source labeled "OWR report": https://doi.org/10.4171/owr/2024/41.
• Supplied literature source labeled "Calderon--Farre": https://arxiv.org/abs/2405.12106. No title or theorem statement for this reference is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation reports convergence outside a zero-density set for broad twist-torus families and several identified limiting measures, while describing the full-time Masur--Veech--Smillie convergence conjecture as remaining open. This is supplied dated curation, not an independently verified literature assessment.
• The preceding source exposition describes recovering an equidistribution result in $M_g$ from Theorem 2, identifying a measure $\nu$ using work attributed to Apisa--Wright [1] concerning high-rank affine invariant submanifolds, and estimating support rank using cylinder deformations [8], with details in [4, Section 5]. The supplied record contains no bibliographic entries for [1], [4], or [8], and does not state Theorem 2 or its hypotheses.