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Teichmüller dynamics

19–23 March 2018

Organisers: Corinna Ulcigrai, Anton Zorich, John Smillie

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Symposium Poster

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Schedule, Titles and Abstracts

The cotangent bundle of Teichmüller space can be identified with the space of quadratic differentials on the surface. Such quadratic differentials arise naturally in a range of areas, including notably the analysis of polygonal billiards. The moduli space of all quadratic differentials on a surface decomposes naturally into strata, consisting of quadratic differentials with specified types of zeroes and poles. These strata share many properties with moduli spaces of Riemann surfaces (i.e. are the quotient of Teichmüller space by the mapping class group) and with homogeneous spaces.

Strata have a natural $SL(2, \Bbb{R})$ action which connects the dynamical and geometric properties of individual quadratic differentials to dynamical properties of the action. Beautiful examples of quadratic differentials with small orbit closures were constructed by Veech. For these examples it is possible to analyse quite precisely the dynamic properties of the quadratic differentials.

Important work of Eskin-Mirzakhani-Mohammadi and Filip show that the $SL(2, \Bbb{R})$ orbit closures in each stratum are manifolds and even algebraic varieties, mirroring the situation for the $SL(n, \Bbb{R})$-action on the homogeneous space $SL(n, \Bbb{R})/SO(n)$ famously analysed by Ratner, Margulis and others. McMullen has determined the complete list of orbit closures in genus 2 but the corresponding list for strata in higher genus is not known and is a very active area of research. Computer investigations based on a program written by Eskin have played a fundamental role in the discovery of these and orbit closures and continue to play an important role.

Compactifications of strata should provide another tool for understanding the $SL(2, \Bbb{R})$-action, and such compactifications have recently attracted the interest of algebraic geometers. Saddle connections on quadratic differentials are studied by using the $SL(2, \Bbb{R})$-action but they also arise in connection with symplectic geometry and string theory. Strata are also natural examples of spaces of Bridgeland stability conditions and the study of their properties by symplectic geometers parallels their study by Teichmüller dynamicists.

The workshop will explore the interactions between Teichmüller dynamics and homogeneous dynamics, computation, algebraic geometry and symplectic geometry. It will investigate whether tools from the symplectic geometry community can be used in the flat surfaces community and also whether flat tools can answer symplectic questions. It will explore the use of computational methods in finding new orbit closures and look at the extent to which the algebraic geometers completions of strata can be used to answer questions related to the $SL(2, \Bbb{R})$ action.


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See also:
Mathematics Research Centre
Mathematical Interdisciplinary Research at Warwick (MIR@W)
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