I am organizing a zoom workshop for graduates students in the fall 2021 academic semester. All students are welcome, and there will be an opportunity to present if desired. We tentatively plan to meet on Thursdays via Zoom. Our first meeting will be on Thursday, August 26 at 9:45-10:45am. If you want to attend, send me an email at Dragomir.Saric”@” and I will send you an invite.

Fall 2021: Student Workshop on Quasiconformal Maps, Hyperbolic Geometry, and Low-dimensional Topology

In this student workshop, we will study Teichmuller spaces of finite and infinite Riemann surfaces, and the universal Teichmuller space via hyperbolic and quasiconformal invariants. Our focus will be on topological,  analytic, and combinatorial properties of Teichmuller spaces through the lens of tessellations and shear coordinates. We will start with a discussion of the Farey tessellation of the unit disk which is arguably one of the most beautiful (and visually appealing) objects in mathematics. The Farey tessellation has deep connections to the Teichmuller theory, mathematical physics, number theory, to name a few (for example, see “Decorated Teichmuller Theory” by R. Penner).

In the first meeting we will have a short overview of some elements of the Teichmuller theory. Our discussion will be at a very elementary level with no assumption on the background of the students. The goal is to build a solid understanding of the shear coordinates in terms of the hyperbolic geometry and the relationship to the deformations of the hyperbolic structures-i.e. the Teichmuller space (for example, see “Circle Homeomorphisms and Shears” D. Saric, G&T). Following this basic understanding, we will develop some of the analytic properties of the shear coordinates on the Teichmuller space and pose a number of open problems. After that, we will discuss some topological/combinatorial applications of the tessellations and some number theoretical relationships of the Farey tessellation to the shear coordinates. 

Workshop schedule (to be updated)

  • August 26, 2021: Teichmuller space (Speaker: Dragomir Saric)
  • September 2, 2021: Elementary hyperbolic geometry (Speaker: Dragomir Saric)
  • September 9, 2021: Collar Lemma (Speaker: Dragomir Saric)
  • September 16, 2021: Classification of isometries of the hyperbolic plane (Speaker: Dragomir Saric)
  • September 23, 2021: Lengths of closed geodesics under qc deformations; Shear coordinates for finite surfaces (Speakers: Michael Pandazis and Dragomir Saric)
  • September 30, 2021: Shear coordinates for finite surfaces (Speaker: Dragomir Saric)
  • October 7, 2021: Shears for the quasisymmetric maps (Speaker: Dragomir Saric)
  • October 14, 2021: Shears for the quasisymmetric maps (Speaker: Dragomir Saric)
  • October 21, 2021: Triangulations of infinite surfaces (Speaker: Casey Whitney)
  • October 28, 2021: Triangulations of infinite surfaces (Speaker: Casey Whitney)
  • November 4, 2021: Minkowski ? function, (Speaker: Ajith Nair)
  • November 11, 2021: Hausdorff dimension of simple geodesics, (Speaker: Yuan (Jessica) Liu)
  • November 18, 2021: Fenchel-Nielsen coordinates, (Speaker: Michael Pandazis)
  • December 2, 2021: Intro to modulus of curves and some estimates, (Speaker: Michael Pandazis)