Papers $\def\Z{\mathbb Z} \def\Q{\mathbb Q} \def\Torelli{\mathcal I}$

The moduli space of twisted holomorphic 1-forms on Riemann surfaces, equivalently dilation surfaces with scaling, admits a stratification and GL(2, R)-action as in the case of moduli spaces of translation surfaces. We produce an analogue of Masur- Veech measure, i.e. an SL(2, R)-invariant Lebesgue class measure on strata or explicit covers thereof. This relies on a novel computation of cohomology with coefficients for the mapping class group. The computation produces a framed mapping class group invariant measure on representation varieties that naturally appear as the codomains of the periods maps that coordinatize strata.

We identify the moduli space of complex affine surfaces with the moduli space of regular meromorphic connections on Riemann surfaces and show that it satisfies a corresponding universal property. As a consequence, we identify the tangent space of the moduli space of affine surfaces, at an affine surface X, with the first hypercohomology of a two-term sequences of sheaves on X. In terms of this identification, we calculate the derivative and coderivative of the holonomy map, sending an affine surface to its holonomy character. Using these formulas, we show that the holonomy map is a submersion at every affine surface that is not a finite-area translation surface, extending work of Veech. Finally, we introduce a holomorphic foliation of some strata of meromorphic affine surfaces, which we call the isoresidual foliation, along whose leave holonomy characters and certain residues are constant. We equip this foliation with a leafwise indefinite Hermitian metric, again extending work of Veech.

We give a new short proof of the classification of rank at least two invariant subvarieties in genus three, which is due to Aulicino, Nguyen, and Wright. The proof uses techniques developed in recent work of Apisa and Wright.

We show that the only algebraically primitive invariant subvarieties of strata of translation surfaces with quadratic field of definition are the decagon, Weierstrass curves, and eigenform loci in genus two and the rank two example in the minimal stratum of genus four translation surfaces discovered by Eskin-McMullen-Mukamel-Wright

We introduce a construction of affine invariant subvarieties in strata of translation surfaces whose input is purely combinatorial. We then show that this construction can be used to construct the Bouw-Moeller Teichmueller curves and the seven Eskin-McMullen-Mukamel-Wright rank two orbit closures. The construction is based on the theory of Hurwitz spaces and is inspired by work of Delecroix, Rueth, and Wright.

The orbit closure of the unfolding of every rational right and isosceles triangle is computed and the asymptotic number of periodic billiard trajectories in these triangles is deduced. This follows by classifying all orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials. Additionally, given a fixed set of angles, the orbit closure of the unfolding of all unit area rational parallelograms, isosceles trapezoids, and right trapezoids outside of a discrete set is determined.

We classify the GL(2,R)-invariant subvarieties M in strata of Abelian differentials for which any two M-parallel cylinders have homologous core curves. This answers a question of Mirzakhani and Wright. As a corollary we show that outside of an explicit list of exceptions, if M is a GL(2,R)-invariant subvariety, then the Kontsevich-Zorich cocycle has nonzero Lyapunov exponents in the symplectic orthogonal of the projection of the tangent bundle of M to absolute cohomology.

We classify GL(2,R) orbit closures of translation surfaces of rank at least half the genus plus 1.

We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech, we show that the the moduli space of affine surfaces with fixed genus and with cone points of fixed complex order is a holomorphic affine bundle over the moduli space of Riemann surfaces. Similarly, the moduli space of dilation surfaces is a covering space of the moduli space of Riemann surfaces. We classify the connected components of the moduli space of dilation surfaces and show that any component is an orbifold K(G,1) where G is the framed mapping class group of Calderon-Salter.

We classify point markings over genus two Abelian differentials and show that exotic examples of orbit closures discovered by Kumar-Mukamel and Eskin-McMullen-Mukamel-Wright are unique. Applications to determining holomoprhic sections over Hilbert modular curves are given.

We classify a natural collection of GL(2,R)-invariant subvarieties which includes loci of double covers as well as the orbits of the Eierlegende-Wollmilchsau, Ornithorynque, and Matheus-Yoccoz surfaces. This is motivated in part by a forthcoming application to another classification result, the classification of "high rank" invariant subvarieties. We also give new examples, which negatively resolve two questions of Mirzakhani and Wright, clarify the complex geometry of Teichmuller space, and illustrate new behavior relevant to the finite blocking problem.

We introduce and study diamonds of GL(2,R)-invariant subvarieties of Abelian and quadratic differentials, which allow us to recover information on an invariant subvariety by simultaneously considering two degenerations, and which provide a new tool for the classification of invariant subvarieties. We investigate the surprisingly rich range of situations where the two degenerations are contained in "trivial" invariant subvarieties. Our main results will be applied in forthcoming work to classify large collections of invariant subvarieties; the statement of those results will not involve diamonds, but their proofs will use them as a crucial tool.

The set of directions from a quadratic differential that diverge on average under Teichmuller geodesic flow has Hausdorff dimension exactly equal to one-half.

We classify the periodic points on the regular n-gon and double n-gon translation surfaces and deduce consequences for the finite blocking problem on rational triangles that unfold to these surfaces.

We prove strong finiteness results on the orbits of marked translation surfaces under geodesic flow and apply the work to the finite blocking problem in rational billiards.

The Hausdorff dimension, starting at any translation surface, of the set of directions in which Birkhoff or Oseledets averages deviate a definite amount from the expected limit is strictly less than one. The dimension of the set of directions that diverge on average is bounded above by 1/2 and this bound is sharp. This strengthens results of Masur and Chaika-Eskin.

We classify orbit closures of marked translation surfaces when the unmarked translation surface is generic with respect to the GL(2,R)-action. The illumination and finite blocking problems on generic translation surfaces are resolved as corollaries.

Every GL(2,R)-orbit in hyperelliptic components of strata of abelian differentials in genus greater than two is either closed, dense, or contained in a locus of branched covers.

We show that all affine invariant submanifolds of complex dimension greater than three in hyperelliptic components of strata of abelian differentials are branched covering constructions, i.e. every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of abelian differentials. This result implies finiteness of algebraically primitive Teichmuller curves in all hyperelliptic components for genus greater than two. A classification of all GL(2,R) orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided conditional on the sparsity conjecture.

A finite group is called a B-group if all its irredundant generating sets are the same size. We classify B-groups with trivial Frattini subgroup and deduce a structure theorem for B-groups as a corollary.