Most of my research is in **Ergodic Theory and Dynamical Systems**,
and **Number Theory**.

My focus in dynamics is on group actions on homogeneous spaces. For example, I am interested in rigidity phenomena for actions by "large" groups, like higher-rank abelian groups or semisimple Lie groups.

Homogeneous dynamics interacts strongly with number theory, especially Diophantine approximation. Some of my current projects are in this area of interaction, as well as other related areas, like fractal geometry.

Khintchine's Theorem with random fractions,

Preprint.

A characterization
of bad approximability,

Submitted.

Rational approximation of affine
coordinate subspaces of Euclidean space,

with D. S. Simmons
and F. Sueess,

*Acta Arithmetica* **177** (2017), no. 1,
91-100.

Different speeds suffice for
rendezvous of two agents in arbitrary graphs,

with
E. Kranakis, D. Krizanc, E. Markou, and A. Pagourtzis,

*SOFSEM* 2017.

Counterexamples,
covering systems, and zero-one laws for inhomogeneous
approximation,

*International Journal of Number
Theory* **13** (2017), no. 3, 633-654.

Metric Diophantine
Approximation: aspects of recent work,

with
V. Beresnevich and S. Velani,

*Dynamics and Analytic
Number Theory*, London Mathematical Society Lecture Note
Series **437**, eds. D. Badziahin, A. Gorodnik,
N. Peyerimhoff, Cambridge University Press, 2016. (pp. 1-95)

Khintchine types of
translated coordinate hyperplanes,

*Acta Arithmetica* **170** (2015), no. 3, 243-273.

Recurrence to
shrinking targets on typical self-affine fractals,

with H.
Koivusalo,

*Proceedings of the Edinburgh Mathematical Society*, to appear.

Higher cohomology
of parabolic actions on certain homogeneous spaces,

*Journal d'Analyse Mathématique* **131** (2017), no. 1, 255-275.

Invariant
distributions and cohomology for geodesic flows and higher cohomology
of higher-rank Anosov actions,

*Journal of Functional
Analysis* **265** (2013), no. 6, 1002-1063.

Limit theorems for rank-one
Lie
groups,

with A.
Gorodnik,

*Proceedings of the AMS* **142** (2014), no. 4, 1359-1369.

Higher cohomology for Anosov
actions on certain homogeneous spaces,

*Journal d'Analyse
Mathématique* **121** (2013), no. 1, 177-222.

Smooth cocycles over homogeneous
dynamical
systems,

Ph. D. Thesis (2010).

Cocycles over
higher-rank abelian actions on quotients of semisimple Lie
groups,

*Journal of Modern Dynamics* **3** (2009), no. 3, 335-357.