# Flat Surfaces and Dynamics of Moduli Space (16w5010)

Arriving in Oaxaca, Mexico Sunday, May 8 and departing Friday May 13, 2016

## Organizers

Ferran Valdez (Universidad Nacional Autonoma de Mexico)

Jayadev Athreya (University of Washington)

Martin Möller (Goethe Universität Frankfurt Am Main)

## Objectives

We want to bring together international experts on moduli spaces of curves and flat surfaces. For many of the above topics there is a dichotomy of type of results: results that hold almost everywhere and results for especially rich examples. These complementary approaches make the topic challenging both for mathematicians with analytic (or measure-theoretic) backgrounds and for those with rather algebraic backgrounds. \par We remark that in Mexico there are well stablished research groups on moduli spaces of curves, foliations and dynamical systems, however flat surfaces are not yet widely known. By including researches working in Mexico and by locating the conference in Oaxaca, we expect to also expand the local community in flat surfaces and moduli spaces. For these reasons, our first preference for the venue of the proposed meeting is BIRS station in Oaxaca, Mexico.

In recent years, workshops and conferences on flat surfaces and their relation to the moduli space of curves (and Teichm\"uller space) have taken place at various institutions around the world.

%REFERENCES

*However, we can also carry this meeting in BIRS station in Banff, Canada.*\\ \\ Talks on flat surfaces and dynamics on moduli space subject have been given at the last five international congresses of mathematicians: by H.~Masur in Z\"urich, 1994; by A.~Eskin in Berlin, 1998; by G.~Forni in Beijing, 2002; by Y.~Minsky and A.~Zorich in Madrid, 2006; by A.~Avila, U.~Hamenst\"adt, and M.~Mirzakhani in Hyderabad, 2010. {\it{Moreover, in Seoul 2014, two Fields medals have been given to A. Avila and M. Mirzakhani for their work on the proposed topic.}}In recent years, workshops and conferences on flat surfaces and their relation to the moduli space of curves (and Teichm\"uller space) have taken place at various institutions around the world.

- May--August 2010 (Trimester at the Hausdorff-Institute, Bonn): ``Geometry and dynamics in Teichm\"uller space'', organized by U. Hamenstaedt, M. M\"oller, A. Zorich.
- May 2011 (Oberwolfach) ``Billiards, flat surfaces and dynamics on moduli space'', organized by H. Masur, M. M\"oller, A. Zorich.
- July 2011 (IAS Summer school, Park City): ``Moduli Spaces of Riemann Surfaces'', organized by B. Farb, E. Looijenga, D. Hain.
- April 2012 (CIRM, Luminy): ``The horocyclic flow in different situations'', organized by F. Dal'bo, P.~Hubert, A.~Zorich.
- September 2012 (Roscoff, Bretagne): ``Algebraic geometry for the flats'', organized by P.~Hubert, E.~Lanneau, A.~Zorich.
- August 2012 (University of Illinois at Urbana-Champaign): ``Geometric structures And Representation varieties'', organized by J.~Athreya, C.~Leininger and S.~Bradlow,
- August 2013 (CCM, Morelia): ``International Conference and Workshop on Surfaces of Infinite type'', organized by J.~Bowman, P.~Hooper,R.~Trevino, F.~Valdez, G.~Weitze-Schmith\"usen.
- November 2013 (ICERM, Providence) ``Low-dimensional Topology, Geometry, and Dynamics'', organized by M.~Duchin, P.~Hubert, H.~Masur, R.~Schwartz, A.~Zorich.
- March 2014 (Oberwolfach): ``Flat surfaces and dynamics on moduli space'', organized by H. Masur, M. M\"oller, A. Zorich.

%REFERENCES

#### Bibliography

- [AEZ] AtEsZo} J.~Athreya, A.~Eskin,A.~Zorich: \textit{Right-angled billiards and volumes of moduli spaces of quadratic differentials on $\mathbf{CP}^{1}$ preprint, arXiv:math/1212.1660
- [AtCh] AtCh} J.~Athreya, J.~Chaika: \textit{The distribution of gaps for saddle connection directions. Geometric and Functional Analysis, Volume 22, Issue 6 (2012), 1491-1516. \bibitem[ANW]{ANW}D.~ Aulicino, M.~Nguyen, A. Wright: \textit{Classification of higher rank orbit closures in $H^{odd}(4)$}. J. Eur. Math. Soc. to appear
- [AH] AvilaHub}A.~Avila, P.~Hubert: \textit{Recurrence for the wind-tree model Annales de l'Institut Henri Poincar\'e - Analyse non lin\'eaire, to appear.
- [AvVi] AvVi} A.~Avila, M.~ Viana: \textit{Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture. Acta Mathematica 198 (2007), 1--56.
- [BaMo] BaMo} M.~Bainbridge, M.~M\"oller: \textit{Deligne-Mumford-compactification of the real multiplication locus and Teichmueller curves in genus three. Acta Math.~208 (2012), 1--92.
- [BoVa] BowVa} J.~Bowman, F.~Valdez: \textit{Wild singularities of flat surfaces. Israel J. Math. 197 (2013), no. 1, 69--97.
- [ChHe] ChaikaHensel}J.~Chaika, S.~Hensel: \textit{The set of uniquely ergodic IETs is path-connected. preprint, arXiv:1405.0767
- [Ch] Ch} J.~Chaika: \textit{Every transformation is disjoint from almost every IET. preprint, arXiv:0905.2370, to appear in Ann.\ Math.\
- [Che] Cheung} Y.~Cheung: \textit{Hausdorff dimension of the set of nonergodic directions. Ann.\ of Math.~(2) 158 (2003), no.~ 2, 661--678. \bibitem [CHM]{CHM} Y.~Cheung, P.~Hubert, H.~Masur: \textit{Dichotomy for the Hausdorff dimension of the set of nonergodic directions.} Inventiones Math. (183) (2011) 337-383
- [DHL] DHL} V.~Delecroix, P.~Hubert, S.~Leli{\`e}vre: \textit{Diffusion for the periodic wind-tree model. Ann. ENS, Volume 47, fascicule 3 (2014)
- [EKZ] EKZ} A.~Eskin, M.~Kontsevich, A.~Zorich: \textit{Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. preprint, arXiv:1112.5872, Publications Math\'ematiques de l'IHES, to appear
- [EMZ] EMZ} A.~Eskin, H.~Masur, A.~Zorich: \textit{Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants. Publications Math\'ematiques de l'IHES No. 97 (2003), 61--179.
- [EsMi] EsMi} A.~Eskin, M.~Mirzakhani: \textit{Invariant and stationary measures for the $SL(2,{\mathbb{R}m \mathbb{R}})$ action on moduli space. preprint, available on http://www.math.uchicago.edu/~eskin/
- [EsMiMo] EsMiMo} A.~Eskin, M.~Mirzakhani,A.~Mohammadi: \textit{Isolation, Equidistribution, and Orbit Closures for the $SL(2,{\mathbb{R}m \mathbb{R}})$ action on moduli space.) preprint, available on http://www.math.uchicago.edu/~eskin/
- [EsOk] Eskin:Okounkov} A.~Eskin, A.~Okounkov: \textit{Asymptotics of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent.\ Math.~145 (2001), 59--104.
- [Fi] Filip13 S.~Filip: \textit{Splitting mixed Hodge structures over affine invariant manifolds}, preprint, arXiv:1311.2350
- [FrUl] FrUl K.~Fraczek, C.~Ulcigrai: \textit{Non-ergodic Z-periodic billiards and infinite translation surfaces}, Inventiones Mathematicae, to appear.
- [Ha] Ha U.~Hamenst\"adt: \textit{Symbolic dynamics for the Teichmueller flow}, preprint, arXiv:1112.6107
- [HW] HooWei P.~Hooper, B.~Weiss: \textit{Generalized staircases: recurrence and symmetry}, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 4, 1581-1600.
- [Hoo] Hoo P.~Hooper, \textit{Immersions and the space of all translation structures}, preprint, available on http://wphooper.com/docs/
- [KMS] KMS} S.~Kerckhoff, H.~Masur, J.~Smillie: \textit{Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2) 124 (1986), no. 2, 293--311.
- [Ko] Kontsevich} M.~Kontsevich: \textit{Lyapunov exponents and Hodge theory. ``The mathematical beauty of physics'' (Saclay, 1996) 318--332, Adv. Ser. Math. Phys., 24, World Sci. Publ.\ (1997).
- [LaNg] LaNg} E.~Lanneau, D.-M.~Nguyen: \textit{Teichmueller curves generated by Weierstrass Prym eigenforms in genus three and genus four. Journal of Topology (2013), (2014) 7 (2): 475-522 .
- [MaSm] MaSm} H.~Masur, J.~Smillie: \textit{Hausdorff dimension of sets of nonergodic measured foliations. Ann. of Math. (2) 134 (1991), no. 3, 455--543.
- [Mc] McMDyn} C.~McMullen: \textit{Dynamics of $\textrm{SL}_2(\mathbb{R})$ over moduli space in genus two. Ann. of Math. (2) 165 (2007), no. 2, 397--456.
- [Mi] Mi3} M.~Mirzakhani: \textit{Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann.\ of Math.~168 (2008), no.~1, 97--125.
- [Tre] Tre} R.~Trevi\~{n}o,\textit{On the ergodicity of flat surfaces of finite area. Geom. Funct. Anal. 24 (2014), no. 1, 360-386.
- [Ve] Ve} W.~Veech: \textit{Teichm\"uller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent.~Math.\ 97 (1989), 533--583
- [Zo] Zo} A.~Zorich: \textit{How do the leaves of a closed 1-form wind around a surface. "Pseudoperiodic Topology'', Translations of the AMS, Ser.~2, vol.~197, Providence, RI (1999), 135--178.