## Organizers

Hyungryul Baik (KAIST),

Sang-hyun Kim, Sanghoon Kwak, Javier de la Nuez-González, Carl-Fredrik Nyberg-Brodda (KIAS)

## How to join

Zoom https://kimsh.kr/vz

Meeting ID: 822 3235 0014

Passcode: 7998

Time Generally, Tuesdays or Thursdays 11 am KST

Length is typically for one-hour unless noted otherwise, although it's often extended by questions etc.

Mailing list Please contact one of the organizers to subscribe.

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## 2024

September 19th 2024, 11am - 12:00pm

Zoom https://kimsh.kr/vz (online only)

Miyachi Hideki (Kanazawa University)

Title: Regularity of the infinitesimal Teichmuller homeomorphism and Duality of the Teichmuller metric

Abstract: In 1940’s, Teichmuller invented the deformation space of marked Riemann surfaces of (topologically, analytically) finite type, which is recently called the Teichmuller space, to study Riemann’s moduli problem. Now a day, the Teichmuller space is widely known and applied in several aspects in Mathematics. Before Teichmuller, Grotzsch reseached in 1928 the extremal quasiconformal mappings, and he observed that the extremal quasiconformal mapping between rectangles in the plane (preserving the vertices) is affine. The extremality is discussed with the maximal dilatation of quasiconformal mappings. Grotzsch’s observation is dealt in Teichmuller’s ivention to showed that the Teichmuller space is naturally homeomorphic to the unit ball in the integrable holomorphic quadratic differentials (of a fixed reference Riemann surface), is recently developed to a deep theory for the extremality of quasiconformal mappings. In the theory of extremal quasiconformal mappings, a natural Finsler metric and distance on the Teichmuller space, which are recently called the Teichmuller metric and the Teichmuller distance, naturally appears.

The infinitesimal Teichmuller homeomorphism in the title of this talk is the infinitesimal version of Teichmuller’s homeomorphism above mentioned. This is a map from the holomorphic cotangent bundle (the bundle of the integrable holomorphic quadratic differentials) to the holomorphic tangent bundle of the Teichmuller space. The infinitesimal Teichmuller homeomophism gives a relation between the Teichmuller metric and the L^1-norm of the integralble holomorphic quadratic differentials,

In this talk, I will first discuss the regulity of the infinitesimal Teichmuller homeomorphism. I will show that the infinitesimal Teichmuller homeomorphism is real-analytic on each stratum of a natural statification of the cotangent bundle. Next, I will give a new duality property, called the infinitesimal duality, between the Teichmuller metric and the L^1-norm of the integrable holomorphic quadratic differenitals.

September 20th 2024, 11am - 12:30pm

KIAS 1424

& Zoom https://kimsh.kr/vz

Chris Leininger (Rice University)

Title: Billiards, symbolic coding, and geometry

Abstract: A billiard trajectory in a polygon P in the Euclidean plane is the path of a particle inside P, following straight lines until it encounters a side, and then bouncing off so that the angle of reflection equals the angle of incidence. A generic trajectory never encounters the corners in forward or backward time, and so produces a biinfinite symbolic coding: the itinerary of sides encountered by the trajectory. A natural question studied by Bobok and Troubetzkoy about 15 years ago is whether one can recover the shape of the polygon from the set of all itineraries: the symbolic coding. They observed that two polygons with vertical and horizontal sides that differ by an affine transformation have the same codings, and proved that among rational polygons (those whose interior angles are rational multiple of pi), this essentially accounts for the only ambiguity. I'll describe work with Duchin, Erlandsson, and Sadanand where we prove an analogous result with no restrictions on interior angles. Time permitting, I will also describe a companion result for billiards in hyperbolic polygons with Erlandsson and Sadanand, where the exceptional cases are much more robust.

September 24 - 27, 2024

Workshop Dynamical Group Theory IV - KIAS-Rice Workshop on Geometric Topology

Mini-course: Chris Leininger (Rice)

Invited talks: Hyungryul Baik (KAIST), Jeffrey Brock (Yale), Inhyeok Choi (Cornell / KIAS), David Fisher (Rice), Koji Fujiwara (Kyoto), Kazuo Habiro (U Tokyo), Inkang Kim (KIAS), Sang-hyun Kim (KIAS), Sanghoon Kwak (KIAS), Khánh Lê (Rice), Carl-Fredrik Nyberg-Brodda (KIAS), Alan Reid (Rice / KIAS), Yandi Wu (Rice)

### Organizers

Hyungryul Baik (KAIST)

Sang-hyun Kim, Sanghoon Kwak, Javier de la Nuez-González, Carl-Fredrik Nyberg-Brodda (KIAS)