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1. Nonvarying, affine and extremal geometry of strata of differentials

   https://doi.org/10.1017/S0305004123000567  

   CHEN, DAWEI (October 2023, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata ofk-differentials of infinite area are affine varieties for allk. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics. 

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2. TEICHMÜLLER CURVES IN HYPERELLIPTIC COMPONENTS OF MEROMORPHIC STRATA

   https://doi.org/10.1017/S1474748025000088  

   Möller, Martin; Mullane, Scott; Bakker, Benjamin; Mullane, Scott (July 2025, Journal of the Institute of Mathematics of Jussieu)

   Abstract We provide a complete classification of Teichmüller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichmüller curves intersect the boundary of the moduli space we derive a contradiction to the algebraicity of any candidate outside of Hurwitz covers of strata with projective dimension one, and Hurwitz covers of zero residue loci in strata with projective dimension two. 

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3. Billiards and Teichmüller curves

   https://doi.org/10.1090/bull/1782  

   McMullen, Curtis (April 2023, Bulletin of the American Mathematical Society)

   A Teichmüller curve V ⊂ M g V \subset \mathcal {M}_g is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems. 

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4. A tale of two moduli spaces: logarithmic and multi-scale differentials

   https://doi.org/10.46298/epiga.2025.11278  

   Chen, Dawei; Grushevsky, Samuel; Holmes, David; Möller, Martin; Schmitt, Johannes (October 2025, Épijournal de Géométrie Algébrique)

   Multi-scale differentials were constructed by M.~Bainbridge, D.~Chen, Q.~Gendron, S.~Grushevsky, and M.~Möller, from the viewpoint of flat and complex geometry, for the purpose of compactifying moduli spaces of curves together with a differential with prescribed orders of zeros and poles. Logarithmic differentials were constructed by S.~Marcus and J.~Wise, as a generalization of stable rubber maps from Gromov--Witten theory. Modulo the global residue condition that isolates the main components of the compactification, we show that these two kinds of differentials are equivalent, and establish an isomorphism of their (coarse) moduli stacks. Moreover, we describe the rubber and multi-scale spaces as an explicit blowup of the moduli space of stable pointed rational curves in the case of genus zero, and as a global blowup of the incidence variety compactification for arbitrary genera, which implies their projectivity. We also propose a refined double ramification cycle formula in the twisted Hodge bundle which interacts with the universal line bundle class. 51 pages 

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5. Stable Strata of Geodesics in Outer Space

   https://doi.org/10.1093/imrn/rnx269  

   Algom-Kfir, Yael; Kapovich, Ilya; Pfaff, Catherine (January 2018, International Mathematics Research Notices)

   Abstract In this article, we propose an Outer space analog for the principal stratum of the unit tangent bundle to the Teichmüller space $${\mathcal{T}}(S)$$ of a closed hyperbolic surface $$S$$. More specifically, we focus on properties of the geodesics in Teichmüller space determined by the principal stratum. We show that the analogous Outer space “principal” periodic geodesics share certain stability properties with the principal stratum geodesics of Teichmüller space. We also show that the stratification of periodic geodesics in Outer space exhibits some new pathological phenomena not present in the Teichmüller space context. 

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