# Papers
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21. Nearly Fuchsian surface subgroups of finite covolume Kleinian groups (abstract)

with Jeremy Kahn
(almost complete 44 page draft, available upon request)

Let Γ < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove that for all K > 1, there is a subgroup H < Γ that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a nearly Fuchsian surface subgroup.

20. Marked points on translation surfaces (abstract)

with Paul Apisa

We show that all GL(2,R) equivariant point markings over orbit closures of translation surfaces arise from branched covering constructions and periodic points, completely classify such point markings over strata of quadratic differentials, and give applications to the finite blocking problem.

19. Totally geodesic submanifolds of Teichmuller space (abstract)

We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space.

18. The algebraic hull of the Kontsevich-Zorich cocycle (abstract)

with Alex Eskin and Simion Filip

We compute the algebraic hull of the Kontsevich-Zorich cocycle over any GL(2,R) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.

17. Full rank affine invariant submanifolds (abstract)

with Maryam Mirzakhani

Duke Math. J. to appear

We show that every GL(2, R) orbit closure of translation surfaces is either a
connected component of a stratum, the hyperelliptic locus, or consists entirely
of surfaces whose Jacobians have extra endomorphisms. We use this result to
give applications related to polygonal billiards. For example, we exhibit
infinitely many rational triangles whose unfoldings have dense GL(2,R) orbit.

16. Cubic curves and totally geodesic subvarieties of moduli space (abstract)

with Curtis McMullen and Ronen Mukamel

Ann. of Math. to appear

In this paper we present the first example of a primitive, totally geodesic subvariety F of M_{g,n} with dim(F) > 1. The variety we consider is a surface F in M_{1,3}, defined using the projective geometry of plane cubic curves. We also obtain a new series of Teichmuller curves in M_4, and new SL_2(R)–invariant varieties in the moduli spaces of quadratic differentials and holomorphic 1-forms.

15. The boundary of an affine invariant submanifold (abstract)

with Maryam Mirzakhani

Invent. Math. to appear

We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann surfaces, modulo zero area components. The main result is a formula for the tangent space to the boundary.

We also prove finiteness results concerning cylinders, a partial converse to the Cylinder Deformation Theorem, and a result generalizing part of the Veech dichotomy.

14. From rational billiards to dynamics on moduli spaces (abstract)

Bull. Amer. Math. Soc. 2016

This short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces, and in particular the recent breakthrough result of Eskin, Mirzakhani, and Mohammadi.
We also discuss the context and applications of this result, and connections to other areas of mathematics such as algebraic geometry, Teichmuller theory, and ergodic theory on homogeneous spaces.

13. Finiteness of Teichmuller curves in non-arithmetic rank 1 orbit closures (abstract)

with Erwan Lanneau and Duc-Manh Nguyen

Amer. J. Math. to appear

We show that in any non-arithmetic rank 1 orbit closure of translation
surfaces, there are only finitely many Teichmuller curves. We also show that
in any non-arithmetic rank 1 orbit closure, any completely parabolic surface is Veech.

12. A smooth mixing flow on a surface with non-degenerate fixed points (abstract)

with Jon Chaika

J. Amer. Math. Soc. accepted pending revisions

We construct a smooth, area preserving, mixing flow with finitely many
non-degenerate fixed points and no saddle connections on a closed surface of
genus 5. This resolves a problem that has been open for four decades.

11. Translation surfaces and their orbit closures: An introduction for a broad audience (abstract)

EMS Surv. Math. Sci. 2015

Translation surfaces can be defined in an elementary way via polygons, and arise naturally in the study of various basic dynamical systems. They can also be defined as differentials on Riemann surfaces, and have moduli spaces called strata that are related to the moduli space of Riemann surfaces. There is a GL(2,R) action on each stratum, and to solve most problems about a translation surface one must first know the closure of its orbit under this action. Furthermore, these orbit closures are of fundamental interest in their own right, and are now known to be algebraic varieties that parameterize translation surfaces with extraordinary algebro-geometric and flat properties. The study of orbit closures has greatly accelerated in recent years, with an influx of new tools and ideas coming from diverse areas of mathematics. Many areas of mathematics, from algebraic geometry and number theory, to dynamics and topology, can be brought to bear on this topic, and known examples of orbit closures are interesting from all these points of view.

This survey is an invitation for mathematicians from different backgrounds to become familiar with the subject. Little background knowledge, beyond the definition of a Riemann surface and its cotangent bundle, is assumed, and top priority is given to presenting a view of the subject that is at once accessible and connected to many areas of mathematics.

10. Classification of higher rank orbit closures in H^odd(4) (abstract)

with David Aulicino and Duc-Manh Nguyen

J. Eur. Math. Soc. 2016

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H^{odd}(4) the only GL^+(2,R) orbit closures are closed orbits, the Prym locus Q(3,-1^3), and H^{odd}(4).
Together with work of Matheus-Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmüller curves) in H^{odd}(4) outside of the Prym locus.

9. Hodge-Teichmüller planes and finiteness results for Teichmüller curves (abstract)

with Carlos Matheus

Duke Math. J. 2015

We prove that there are only finitely many algebraically primitive Teichmüller curves in the minimal stratum in each prime genus at least 3. The proof is based on the study of certain special planes in the first cohomology of a translation surface which we call Hodge-Teichmüller planes.

We also show that algebraically primitive Teichmüller curves are not dense in any connected component of any stratum in genus at least 3; the closure of the union of all such curves (in a fixed stratum) is equal to a finite union of affine invariant submanifolds with unlikely properties. Results of this type hold even without the assumption of algebraic primitivity.

Combined with work of Nguyen and the second author, a corollary of our results is that there are at most finitely many non-arithmetic Teichmüller curves in H(4)^hyp.

8. Non-Veech surfaces in H^hyp(4) are generic (abstract)

with Duc-Manh Nguyen

Geom. Funct. Anal. 2014

We show that every surface in H^hyp(4) is either a Veech surface or a generic surface, i.e. its GL^+(2,R)-orbit is either a closed or a dense subset of H^hyp(4). The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Recent results of Eskin-Mirzakhani-Mohammadi, Avila-Eskin-Möller, and the second author are used.

Combined with work of Matheus and the second author, a corollary is that there are at most finitely many non-arithmetic Teichmüller curves (closed orbits of surfaces not covering the torus) in H^hyp(4).

7. Cylinder deformations in orbit closures of translation surfaces (abstract)

Geom. Topol. 2015

Let M be a translation surface. We show that certain deformations of M supported on the set of all cylinders in a given direction remain in the GL(2,R)-orbit closure of M. Applications are given concerning complete periodicity, field of definition, and the number of of parallel cylinders which may be found on a translation surface in a given orbit closure.

The proof uses Eskin-Mirzakhani-Mohammadi's recent theorem on orbit closures of translation surfaces, as well as results of Minsky-Weiss and Smillie-Weiss on the dynamics of horocycle flow.

6. The field of definition of affine invariant submanifolds of the moduli space of abelian differentials (abstract)

Geom. Topol. 2014

The field of definition of an affine invariant submanifold M is the smallest subfield of the reals such that M can be defined in local period coordinates by linear equations with coefficients in this field. We show that the field of definition is equal to the intersection of the holonomy fields of translation surfaces in M, and is a real number field of degree at most the genus. We show that the projection of the tangent bundle of M to absolute cohomology H^1 is simple, and give a direct sum decomposition of H^1. Applications include explicit full measure sets of translation surfaces whose orbit closures are as large as possible, and evidence for finiteness of algebraically primitive Teichmüller curves. The proofs use recent results of Artur Avila, Alex Eskin, Maryam Mirzakhani, Amir Mohammadi, and Martin Möller.

5. Schwarz triangle mappings and Teichmüller curves: the Veech-Ward-Bouw-Möller curves (abstract)

Geom. Funct. Anal. 2013

We study a family of Teichmüller curves T(n,m) constructed by Bouw and Möller, and previously by Veech and Ward in the cases n=2,3. We simplify the proof that T(n,m) is a Teichmüller curve, avoiding the use Möller's characterization of Teichmüller curves in terms of maximally Higgs bundles. Our key tool is a description of the period mapping of T(n,m) in terms of Schwarz triangle mappings.

We prove that T(n,m) is always generated by Hooper's lattice surface with semiregular polygon decomposition. We compute Lyapunov exponents, and determine algebraic primitivity in all cases. We show that frequently, every point (Riemann surface) on T(n,m) covers some point on some distinct T(n',m') .

The T(n,m) arise as fiberwise quotients of families of abelian covers of CP^1 branched over four points. These covers of CP^1 can be considered as abelian parallelogram-tiled surfaces, and this viewpoint facilitates much of our study.

4. Schwarz triangle mappings and Teichmüller curves: abelian square-tiled surfaces (abstract)

J. Mod. Dyn. 2012

We consider normal covers of CP^1 with abelian deck group, branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces.

We compute all individual Lyapunov exponents for abelian square-tiled surfaces, and demonstrate a direct and transparent dependence on the geometry of the period mapping. For this we develop a result of independent interest, which, for certain rank two bundles, expresses Lyapunov exponents in terms of the period mapping. In the case of abelian square-tiled surfaces, the Lyapunov exponents are ratios of areas of hyperbolic triangles.

3. Sums of Adjoint Orbits and L^2--Singular Dichotomy for SU(m) (abstract)

Adv. in Math. 2011

Let G be a real compact connected simple Lie group, and g its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in g, or a product of conjugacy classes in G, contains an open set.

Our general methods allow us to determine exactly which sums of adjoint orbits in su(m) and products of conjugacy classes in SU(m) contain an open set, in terms of the highest multiplicities of eigenvalues.

For su(m) and SU(m) we show L^2--singular dichotomy: The convolution of invariant measures on adjoint orbits, or conjugacy classes, is either singular to Haar measure or in L^2.

2. Operator Algebras with Unique Preduals (abstract)

with Ken Davidson

Canad. Math. Bull. 2011

This paper won the G. de B. Robinson Award

We show that every free semigroup algebras has a (strongly) unique Banach space predual. We also provide a new simpler proof that a weak*-closed unital operator operator algebra containing a weak* dense subalgebra of compact operators has a unique Banach space predual.

1. Regular Orbital Measures on Lie Algebras (abstract)

Colloq. Math. 2008

Let H be a regular element of an irreducible Lie Algebra g, and let mu be the orbital measure supported on the Adjoint orbit of H. We show that the k-th power of the Fourier transform of mu is in L^2(g) if and only if k > dim g/(dim g-rank g).