Chatha, Prayag George Singh (Prayag George Singh Chatha) (author), (Felipe Ramirez; Adam Fieldsteel; David Constantine) (Thesis advisor)
Diophantine approximation is a branch of number theory that concerns the metric relationship of the rationals and irrationals. Much of the present-day research in the subject approaches problems of approximation via dynamics. We introduce the Littlewood Conjecture, a longstanding problem that was almost completely proven in 2006 by a theorem of Einsiedler, Katok, and Lindenstrauss. After giving an exposition of classical Diophantine approxi- mation and fundamental ergodic theory, we set out a survey of the aforemen- tioned theorem's context, particularly the motivation of the authors' dynam- ical approach, rich connections to linear algebra and hyperbolic geometry, an exploration of measure rigidity, and an interpretation of the paper's main conclusions., Old URL: https://wesscholar.wesleyan.edu/etd_mas_theses/149, In Copyright – Non-Commercial Use Permitted (InC-NC)
The marked length spectrum of a metric on a compact Riemannian manifold records the length of the shortest closed curve in each free homotopy class.
It is known that a negatively curved Riemannian metric on a compact surface is uniquely determined by its marked length spectrum up to isometry. My results show that under certain conditions on the excluded homotopy classes, a partial marked length spectrum also uniquely determines such a metric.
It is also known that an inequality between the marked length spectra of two negatively curved Riemannian metrics on a compact surface implies a corresponding inequality between the area with respect to the metrics. I will show that an inequality between the partial marked length spectra is enough to imply the same conclusion., In Copyright – Non-Commercial Use Permitted (InC-NC)