Geometric Structures Lab / 기하구조론 연구실

Geometric Structures Lab / 기하구조론 연구실

Department of Mathematics in KAIST

We are interested in studying low-dimensional manifolds and geometric structures on the manifolds and associated representations of the fundamental groups into Lie groups. The basic questions here are on the existences and deformation spaces of geometric structures on manifolds. This study helps us in studying the representations of discrete groups in Lie groups. The study of representations of the fundamental groups of surfaces into Lie groups are of great importance. We have worked on PGL(3,R)-representations using elementary geometric methods. Classically SL(2,R)-representation spaces correspond to the study of Teichmüller spaces. Goldman generalized this study into many more Lie groups and Hitchin worked out many topological properties of semi-simple Lie group representations. Currently, our work has been significantly generalized into PGL(n,R)-representations for n > 3 and into other reductive groups by Labourie and Berger-Wienhard, and so on. In particular, we know that there are components of representations spaces which consist of discrete representations only. Currently, we are interested in 2-dimensional orbifold fundamental group representations into Lie groups.

Mainly, we are studying (G, X)-structures on manifolds or orbifolds where G is a Lie group acting on a manifold X. Orbifolds are generalizations of manifolds allowing some finite order singularities. The focus is on real projective, affine, and flat Lorentz structures. They form deformation spaces which classify these by isomorphisms. They admit a local homeomorphism to the character variety Hom(π(M), G)/G.

오비폴드나 다양체 위의 기하구조를 연구하는 것이 목적으로 주로 실사영구조, 아핀구조와 평탄한 로렌츠 구조를 연구하고 있습니다.

Some of our past results [과거 대표 연구 성과]:

1. Classified real projective structures on closed surfaces

2. Proved the Hitchin conjecture identifying the deformation spaces of convex real projective structures with a component of the character variety. (with Goldman)

3. Classified the deformation spaces of convex real projective structures on 2-orbifolds and proved the corresponding Hitchin conjecture. (with Goldman)

4. Computed the dimensions of deformation spaces of convex real projective structures on large classes of Coxeter orbifolds with or without boundary (with Gye-Seon Lee). Ludovic Marquis generalized our results towards complete solution for truncation types. Also, we classified Dehn surgery on convex real projective structures on Coxeter orbifolds (with Gye-Seon Lee, Ludovic Marquis)

5. Proved the tameness of flat Lorentz 3-manifolds with or without parabolic elements (with Goldman, Drumm)

6. Found global Darboux coordinates on the deformation spaces of convex real projective sturctures on surfaces and 2-orbifolds (with H.T. Jung, HC. Kim)

We plan to further research into these areas.

The topics you will be learning in this lab

Connections and principal bundles, geometric structures on manifolds
Orbifold theory
Group theory (presentations, hyperbolic groups)
Computational algebra and other computational methods using maple, mathematica and graphics
Discrete subgroups of Lie groups
Deformation spaces of geometric structures
Representations of the fundamental groups of surfaces and 2-orbifolds into Lie groups and associated structures
Hyperbolic structures and other geometric structures on 3-manifolds

Members

	Name	E-mail	Homepage	Current position
Professor	Choi, Suhyoung	schoi [at] math.kaist.ac.kr	http://math.kaist.ac.kr/~schoi/
PhD	Park, Seungyeol	sypark14 [at] kaist.ac.kr	https://sites.google.com/view/seungyeolpark/home
	Lee, Juseop	dlwntjq77 [at] kaist.ac.kr
	Seo, Yongho	seojm8 [at] kaist.ac.kr
	Kim, Kwanho	kkh0201 [at] kaist.ac.kr
MSc	Bae, Jaesung	dxp_bae [at] kaist.ac.kr	https://sites.google.com/view/jaesungbae/home/

Theses

Jae-Soon Ha, A study on trace functions of closed curves on projective orbifolds

Gye-Seon Lee, Projective deformations of hyperbolic Coxeter 3-orbifolds

Hongtaek Jung, Symplectic coordinates on PSL_3(R)-Hitchin components

Junho Seo, Classification of knots and links with mosaic number 6

Jean Paul Filpo Molina, On the Realization Spaces of Convex Polyhedra

Papers

Suhyoung Choi, Craig D. Hodgson and Gye-Seon Lee, Projective deformations of hyperbolic Coxeter 3-orbifolds , Geom. Dedicata 159 (2012), 125-167 (with Computational results)

Preprints

Talks

Gye-Seon Lee, Projective deformations of 3-dimensional hyperbolic Coxeter orbifolds, December 17, 2009, Joint Meeting of KMS and AMS

Gye-Seon Lee, Deformations of hyperbolic Coxeter orbifolds, June 9, 2009, The University of Melbourne

Gye-Seon Lee, Deformation spaces of projective structures on 3-dimensional Coxeter orbifolds, December 12, 2008, Tokyo Institute of Technology

Jae-Soon Ha, A study on trace functions of closed curves on projective orbifolds & Elimination, November 28, 2008, Tokyo Institute of Technology

Seminars

2009

Arithmetic of hyperbolic 3-manifolds
Reference: Arithmetic of hyperbolic 3-manifolds by Maclachlan and Reid

Algebraic geometry
Reference: Using algebraic geometry by D. Cox, J. Little and D. O'Shea

2008

Arithmetic geometry
Reference: An invitation to arithmetic geometry by D. Lorenzini

Hyperbolic manifolds (The space of hyperbolic manifolds and the volume function, The rigidity theorem: compact case)
Reference: Lectures on hyperbolic geometry by R. Benedetti and C. Petronio

Differential geometry (Lie groups and Lie algebras, structure of semisimple Lie algebras, symmetric spaces, decomposition of symmetric spaces)
Reference: Differential geometry, Lie groups, and symmetric spaces by S. Helgason

Hyperbolic reflection groups
Reference: E.B. Vinberg, Hyperbolic reflection groups, Uspekhi Mat. Nauk 40 (1985) 29-66

Real projective reflection groups
Reference: E.B. Vinberg, Discrete linear groups that are generated by reflections, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971) 1072-1112

Real projective structures on Coxeter 3-orbifolds
Reference: S. Choi, The deformation spaces of projective structures on 3-dimensional Coxeter orbifolds, Geom. Dedicata 119 (2006) 69-90

2006-2007

Hyperbolic manifolds
Reference: Foundations of hyperbolic manifolds by J.G. Ratcliffe

Presentations of groups
Reference: Combinatorial group theory by W. Magnus, A. Karrass and D. Solitar

Conferences

KAIST Geometric Topology Workshop 2024, August 19 - 23, 2024

2023 JNU/KAIST/SNU Geometry & Topology Workshop (4th), August 16 - 18, 2023

The 3rd JNU-KAIST Geometric Topology Fair: Introductory lectures, October 18 - 22, 2021

The 10th KAIST Geometric Topology Fair, August 13 - 17, 2012

The 9th KAIST Geometric Topology Fair, August 8 - 12, 2011

Intensive Lectures on Real Projective Structures, October 25 - 27, 2010

Hyperbolic geometry: algorithmic, number theoretic, and numerical aspects, March 15 - 19, 2010

The 8th KAIST Geometric Topology Fair, January 10 - 15, 2010