The Optimal Paper Moebius Band by Richard Schwartz (Brown)

KIAS Mathematics Colloquium

March 12th (Wed) 2025, 4 pm

KIAS 1503 & Zoom https://kimsh.kr/vz

A paper Moebius band is made  by twisting a 1 x L rectangular strip of paper in space and gluing together the length-1 sides. If L is large, this is easy to do. If L is small this is impossible. What is the cutoff value?  This question goes back at least to Wunderlich in 1962 and is most likely much older.  In 1977 B. Halpern and C. Weaver conjectured that L>sqrt(3) is a necessary and sufficient condition. In this talk I will explain my proof of the Halpern-Weaver Conjecture and I will also prove that a minimizing sequence of examples converges to a unique limiting shape, the equilateral triangle.

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Le Retour de Pappus (The Return of Pappus) by Richard Schwartz (Brown)

Parts I & II : March 7th (Fri) 2025, 10 -  11:50 am, KIAS--Springer Lectures

Part III : March 14th (Fri) 2025, 11 -  11:50 am

KIAS 1503 & Zoom https://kimsh.kr/vz

Pappus's Theorem is a classic theorem in projective geometry, going back to antiquity. In this talk I will explain how Pappus's Theorem is related to Farey addition, representations of the modular group, and pleated surfaces contained in the rank 2 symmetric space associated to SL_3(R).    Some of these ideas go back to my 1993 paper, "Pappus's Theorem and the Modular Group", and some of the ideas are things I discovered when I returned to the topic this year.  I hope to explain  everything from scratch in the talk.

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March 20 (Th), 11 am, KIAS 1423 & zoom https://kimsh.kr/vz

Homin Lee (Northwestern, KIAS)

Random dynamics on surfaces

In this talk, we will discuss about smooth random dynamical systems and group actions on surfaces. Random dynamical systems, especially understanding stationary measures, can play an important role when we consider a group action. For instance, when a group action on torus is given by toral automorphisms, using random dynamics, Benoist-Quint classified all orbit closures. 

In this talk, we will study on non-linear actions on surfaces using random dynamics. We will discuss about absolutely continuity of stationary measures, classification of orbit closure, and exact dimensionality of stationary measures. This talk will be mostly about the joint work with Aaron Brown, Davi Obata, and Yuping Ruan.

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March 25 (Tu), 10:30 am, KIAS 1503 & zoom https://kimsh.kr/vz

Eugen Rogozinnikov (KIAS)

Parametrizing spaces of positive representations

Higher Techmüller theory deals with spaces of representations of the fundamental group of a surface into a reductive Lie group $G$, modulo the conjugation, especially with the connected components (called higher Teichmüller spaces) that consist entirely of injective representations with discrete image.

In the last two decades in works of Fock, Goncharov, Burger, Iozzi, Guichard, Wienhard, and others researchers, it was discovered that the most interesting higher Teichmüller spaces are emerging from the groups $G$ having a positive structure, i.e. certain submonoid $G_+$ with no invertible non-unit elements. Some of these submonoids have been known since 1930’s as totally positive matrices and then generalized by Lustzig for split real Lie groups. However it left out a large class of non-split reductive Lie groups such as $SO(p,q)$. O. Guichard and A. Wienhard filled this gap in 2018 by introducing the Theta-positivity, which also includes submonoids $SO(p,q)_+$ sitting in the unipotent group of $SO(p,q)$ and $Sp(2n,R)_+$ which is the set of upper uni-triangular block 2x2-matrices with a symmetric positive definite matrix in the upper right corner.

In my talk, I introduce the Theta-positivity for Lie groups and explain how the spaces of positive representations of the fundamental group of a punctured surface into a Lie group with a positive structure can be parametrized, and how we can describe the topology of these spaces using this parametrization. This is a joint work with O. Guichard and A. Wienhard.

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