Program
Schedule
- On Friday the talks will be in JFB B103.
- On Saturday and Sunday the talks will be in JWB 335.
- Poster session and dinner on Saturday will be in the LCB Loft.
Please refer to the interactive campus map to locate the buildings (JFB, JWB and LCB)
Abstracts
Dawei Chen
Title: Moduli of Differentials and Teichmüller Dynamics
Abstract: A holomorphic differential defines a flat metric with conical singularities such that the underlying Riemann surface can be realized as a polygon with edges pairwise identified via translations. Varying the shape of such polygons by affine transformations induces an action on the moduli space of differentials, called Teichmüller dynamics, whose study has provided fascinating results in many fields. In this lecture series I will give an elementary introduction to this beautiful subject, with a focus on a combination of algebraic, analytic, combinatorial and dynamical viewpoints as well as some recent developments.
References:
Anton Zorich, Flat Surfaces, https://arxiv.org/abs/math/0609392
Maxim Kontsevich and Anton Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, https://arxiv.org/abs/math/0201292
Alex Wright, Translation surfaces and their orbit closures: An introduction for a broad audience, https://arxiv.org/abs/1411.1827
Dawei Chen, Teichmüller dynamics in the eyes of an algebraic geometer, https://www2.bc.edu/dawei-chen/bootcamp.pdf
Diana Davis
Title: Tiling billiards and interval exchange transformations
Abstract: Tiling billiards is a dynamical system where a beam of light refracts through a planar tiling. It turns out that, for a regular tiling of the plane by congruent triangles, the light trajectories can be described by interval exchange transformations. For a very special choice of triangle, the trajectory approaches a space-filling curve, whose finite approximations approach the Rauzy fractal. I will explain this surprising behavior, and the behavior of the system for other interesting tilings, including one where we use an infinite translation surface to understand the system. There may also be a magic trick.
Christopher Leininger
Title: Teichmüller space, mapping class groups, and moduli space
Abstract: In these lectures, I will define hyperbolic and complex structures on a surface of finite type, and explain how to pass back and forth between them. Next we will discuss the Teichmüller space of equivalence classes of such structures, then describe its topology and an important metric on it called the Teichmüller metric. Finally, we will define the mapping class group and describe properties of its action on Teichmüller space and its quotient space, the moduli space.
Chaya Norton
Title: Moduli space of vector bundles and the monodromy map
Abstract: We develop the notion of a non-abelian Cauchy kernel for a framed vector bundle of rank n and degree ng on a fixed Riemann surface of genus g. This Cauchy kernel can be used to define an affine connection holomorphically varying in the moduli space of vector bundles. Thus we pose and answer the question regarding how the complex symplectic structure on the moduli space of Higgs bundles (or the cotangent bundle to the moduli space of vector bundles) relates to the Goldman symplectic structure. Based on work in progress with Marco Bertola and Giulio Ruzza.