Expository/review papers

°  Bull. of AMS book review of Wiess' ``Twisted Teichm\"uller Curves";
   Bulletin of the AMS; Volume 53, Number 2, April 2016, Pages 313--324
              The directions to reviewers for the Bulletin are to give an extensive overview of the field, and then to say a few things about the book in question. Comments: In Section 3 of the review, ``primitive" is used in the sense of ``algebraically primitive" (genus and degree of trace field of translation surface are same). The more natural notion is ``geometrically primitive" (the given translation surface is not formed by the full-back of some abelian differential on a lower genus Riemann surface). Here it should be noted that McMullen's Prym construction (mentioned in Section 9) also gives strata with infinitely many geometrically primitive Teichm\"uller curves.
       

Continued Fractions: Old and New;
   in: Natural extension of arithmetic algorithms and S-adic system, 1--18, RIMS Kôkyûroku Bessatsu, B58, Res. Inst. Math. Sci. (RIMS), Kyoto, 2016.
              In this expository note, we attempt to point out some interesting recent developments in the theory and applications of continued fractions. The level of exposition is naturally uneven, with tendency towards informality a necessity. The notes are an abridged version of talks given at the Tambara Institute of Mathematical Sciences of The University of Tokyo, at a teaching workshop entitled ``Natural extensions of arithmetic algorithm and S-adic systems'' and organized by Professor Shigeki Akiyama. One aspect of the Tambara lecture that we kept in these short notes is the use of the natural extension (without measure theory) to derive diophantine results for continued fractions. As far as the author knows, this approach was originated by Jager and Kraaikamp. We also hint at the wide variety of settings for continued fractions by mentioning continued fractions over function fields, semi-regular continued fractions, and continued fractions related to various Fuchsian groups. We end with a brief overview of an approach of Arnoux and co-author giving a heuristic method for determining invariant measures, by way of natural extensions, for certain interval maps. Here we emphasize the connection of this to cross-sections for the geodesic flow on the unit tangent bundle of a corresponding hyperbolic surface.
       

°  with P. Hubert,  H. Masur, A. Zorich  Problems  on  billiards,  flat  surfaces  and  translation surfaces;
   in: Problems on mapping class groups and related topics,  B. Farb, ed. Proc. Symp. Pure Math., 74. AMS (2006)
              We pose a series of questions about the matters of the title.   Extremely brief motivation and background are given.