### KAIX Summer School 2018

* Lecture Notes *
KAIST undergraduate student, Haesong Seo, wrote a fairly complete lecture notes of all lectures given at this year's topology summer school.

KAIST Advanced Institute for Science-X (KAIX) hosts its first thematic program this summer. As a part of the program, there will be a summer school on mathematics in June. This year's theme is "Introduction to the recent developments in PDE and Topology, and their intersection." Topology session is organized by me, and PDE session is organized by Prof. Soonsik Kwon.

Topology session's title is "Topics in Geometric Group Theory".
PDE session's title is "Dynamics of Partial Differential Equations".

One can also take this officially as a course in summer semester of 2018;
MAS481(25.481): Topics in Mathematics I<Topics in Geometric Group Theory>.

All talks are at the building E6-1, Room 2413.
The schedule is as in the table below.

 June 25 June 26 June 27 June 28 June 29 July 2 9:00-10:30 Bestvina Bestvina Fujiwara Koberda Bestvina 10:30-12:00 Bestvina Bestvina Fujiwara Koberda Bestvina 14:00-15:30 Bestvina Koberda Ohshika Ohshika Fujiwara 15:30-17:00 Bestvina Koberda Ohshika Ohshika Fujiwara

Title: Introduction to Out(F_n)
Abstract: The following topics will be covered.
1. Stallings folds and applications.
2. Culler-Vogtmann's Outer space, contractibility, and consequences for Out(F_n)
3. Lipschitz metric on Outer space, train track maps and growth of automorphisms.
Vogtmann's brief introduction of Outer spaces [Link]

Koji Fujiwara (Kyoto University)
Title: Group actions on quasi-trees and application.
Abstract: A quasi-tree is a geodesic metric space that is quasi-isometric to a tree.
With Bestvina-Bromberg, I introduced an axiomatic way to construct a quasi-tree and group actions on it.
I explain the basic of it, then discuss some applications including some recent ones.

Kenichi Ohshika (Osaka University)
Title: Kleinian groups and their deformation spaces
Abstract: Historically, deformation spaces of Kleinian groups appeared as generalisations of Teichmuller spaces. Thurston’s work in the 1980s gave a quite novel viewpoint coming from his study of hyperbolic 3-manifolds. In this talk, I shall describe the theory of deformations of Kleinian groups starting from classical work of Bers, Maskit and Marden, and then spend most of time explaining Thurston’s framework. If time permits, I should also like to touch upon the continuity/discontinuity of several invariants defined on deformation spaces.

Thomas Koberda (University of Virginia)
Title: Regularity of groups acting on the circle
Abstract: There is a rich interplay between the degree of regularity of a group action on the circle and the allowable algebraic structure of the group. In this series of talks, I will outline some highlights of this theory, culminating in a construction due to Kim and myself of groups of every possible critical regularity $\alpha \in [1,\infty)$.There is a rich interplay between the degree of regularity of a group action on the circle and the allowable algebraic structure of the group. In this series of talks, I will outline some highlights of this theory, culminating in a construction due to Kim and myself of groups of every possible critical regularity $\alpha \in [1,\infty)$.