# Papers
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8. Exceptional directions for the Teichmuller geodesic flow and Hausdorff dimension (abstract)

with Hamid al-Saqban , Alena Erchenko, Osama Khalil, Shahriar Mirzadeh , and Caglar Uyanik .

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 strenghthens results of Masur and Chaika-Eskin.

7. Rank One Orbit Closures in H^{hyp}(g-1,g-1) (abstract)

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.

6. Periodic Points in Genus Two: Holomorphic Sections over Hilbert Modular Varieties, Teichmuller Dynamics, and Billiards (abstract)

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.

5. Marked Points on Translation Surfaces (abstract)

with Alex Wright .

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.

4. GL(2,R)-invariant measures in marked strata: generic marked points, Earle-Kra for strata, and illumination (abstract)

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.

3. GL(2,R) orbit closures in hyperelliptic components of strata (abstract)

accepted to Duke Math J.

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.

2. A generalization of the Burnside basis theorem. (abstract)

with Benjamin Klopsch. J. Algebra 2014

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.

1. Groups with the Universal Mapping Property (abstract)

Bachelor's Honors Thesis

We study properties of groups with the universal mapping property - finite groups that are uniquely expressible as a quotient of a free group.