| Tue, Oct 22, 2024
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Analysis Seminar
3:00 PM
MSCS 509 | | Diophantine properties of affine diffeomorphisms of a lattice surface
Joshua Southerland, Indiana University
| | | Abstract: A translation surface can be thought of as a polygon in the plane with an even number of sides, where we identify sides by translation. Some translation surfaces have large groups of affine diffeomorphisms, diffeomorphisms mapping the surface to itself which, in a particular set of coordinates, looks like the action of a matrix. Among these translation surfaces are lattice surfaces, surfaces whose collection of derivatives of these affine maps comprise a lattice in $\mathrm{SL}_2(\mathbb R)$. In this talk, we will discuss ongoing work that shows that on a lattice surface, these affine diffeomorphisms exhibit Diophantine-like properties. Fix a lattice surface, an affine diffeomorphism $\phi$, and a point on a surface $y$ and ask what the optimal values are of $\alpha$ so that $| \phi(x) - y | < \| d\phi \|^{- \alpha}$ has a solution for a.e. $x$, for infinitely many $\phi$. $\| d\phi \|$ indicates the norm of the matrix $d\phi$. This is joint work with Chris Judge. |
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