The 8th KAIST Geometric Topology Fair

A workshop on geometric topology
including knot theory, manifold theory, and related topics

This year's concentration area: Arithmetic group theory

January 11^th to January 13^th , 2010

#1501, Natural Science Building(E6), KAIST, Daejeon, KOREA

The aim of this workshop is to encourage academic activities in the area of geometric topology and promote friendly relations between researchers in this area. Our main focus is knot theory and its relationship to manifold theory. We hope this workshop contributes to acceleration of research in the field.

In this year, we will concentrate on arithmetic group theory and rigidity. The main aim is to introduce the graduate students to this area with applications to geometric topology.

Organizers

Ki Hyoung Ko (KAIST), Gyo Taek Jin (KAIST) and Suhyoung Choi (KAIST)

Confirmed Participants

Overseas Visitors

Programs

Presentation

You can use :

• Electronic File via Notebook + Beam Projector

• Manuscript on A4 or letter paper via Document Camera + Beam Projector

Venue

The department of Mathematical Sciences, KAIST, Daejeon, South Korea

Registration

Accomodations: We will reserve a room for you in KAIST if possible. Please let us know the type of room desired and

the dates in between January 10-14. Send us e-mails with names and the other informations.

Tour

There will be a. tour of Seoul for foreign speakers. Anyone interested can join us at Kyongbokgung Palace 10: 30 in the morning of Jan 14^th. One has to pay fees for yourselves

Registration Fee

10,000 Won

One has to pay for your rooms and meals yourselves in KAIST.

There will be some lecture fees.

Local information: See http://www.kaist.ac.kr/ , http://tour.daejeon.go.kr/

how to arrive at KAIST

Contact

For any inquiries on the workshop, please contact:

Kang-Hyun Choi
Korea Advanced Institute of Science and Technology
373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea
Phone: +82-42-869-2772
Email: kchoi1982 at kaist dot ac dot kr

Title, Abstracts and notes

Dave Morris: Introduction to arithmetic groups (Note)

Arithmetic groups are fundamental groups of locally symmetric spaces. We will see how they are constructed, and discuss some of their important properties. For example, although the Q-rank of an arithmetic group is usually defined in purely algebraic terms, we will see that it provides important information about the geometry and topology of the corresponding locally symmetric space. Algebraic technicalities will be pushed to the background as much as possible

Jeong-Rae Kim: Topology and Dynamics of Biological Networks (Note)

Complex cellular behaviors can be seen as a result of interactions between numerous intracellular or extracellular biomolecules such as DNAs, RNAs, proteins and metabolites. To figure out cellular behaviors, it is therefore important to investigate the topology of cellular circuits and corresponding dynamical characteristics. As a means of conducting such investigations, network motifs have been proposed and studied in various cellular circuits. In this talk, I will address recent results on the relation between topology and dynamics of biological networks in terms of network motifs such as feedback loops and feed-forward loops.

Dong Hwa Shin: y-coordinates of elliptic curves (Note)

We explain the uniformization theorem for elliptic curves over C. By an elliptic curve we usually mean a smooth curve of genus 1 as a Lie group which can be viewed as the locus of a Weierstrass equation of two affine variables x and y. By a change of variables we obtain new y-coordinates of elliptic curves. Utilizing these y-coordinates as modular functions together with the elliptic modular function we generate the modular function? fields of level N(>=3). Furthermore, by means of the singular values of the y-coordinates we construct the ray class fields modulo N over imaginary quadratic fields as well as normal bases of these ray class fields.

Youngju Kim: Quasiconformal nonstability for isometry groups in hyperbolic 4-space(Note)

It is known that a geometrically finite Kleinian group is quasiconformally stable. We prove that this quasiconformal stability cannot be generalized in 4-dimensional hyperbolic space. This is due to the presence of screw parabolic isometries in dimension 4. We also prove that a Fuchsian thrice-punctured sphere group has a large deformation space in hyperbolic 4-space which is in contrast to lower dimensions where the Fuchsian thrice-punctured sphere group has a trivial deformation space. However, the thrice-punctured sphere group is still quasiconformally rigid.

Alexander Stoimenow: Around Bennequin's inequality (Note)

I will try to explain some relation between Bennequin's inequality, some description of particular types of unknot diagrams, the positivity problem of the signature, and the conjecture that only finitely many positive links are concordant.

Shintaro Kuroki: Equivariant cohomological and cohomological rigidity of toric hyperKahler manifolds(Note)

In Toric Topology, the cohomological rigidity problem is one of the central problem. The cohomological rigidity problem asks whether cohomology rings determine diffeomorphism types (or homeomorphism types) of class of certain manifolds or not.

Of course, in general, the answer of this problem is negative (e.g. homotopy complex projective spaces). However, for the important classes in Toric Topology, this problem is still open (e.g. (quasi)toric manifolds).

The cohomological rigidity problem can be also asked to the equivariant cohomology. In this case, Masuda proved that the equivariant cohomology could distinguish toric manifold as algebraic variety. In this talk, we will prove two rigidity theorems of toric hyperKahler manifolds, where the toric hyperKahler manifold is the hyperKahler analogue of toric manifolds.

Roughly, we will prove that the equivariant cohomology distinguishes toric hyperKahler manifolds up to hyperhamiltonian T-isometry, and the (ordinary!) cohomology distinguishes them up to diffeomorphism.

Seon Hee Lim: Entropy rigidity and measures on the boundary of negatively curved metric spaces (Note)

We will start with a short survey on entropy rigidity motivated from Mostow's strong rigidity. We will then describe several measures such as Patterson-Sullivan measure and Liouville measure on the boundary of hyperbolic spaces (e.g. hyperbolic manifolds, hyperbolic buildings, etc.) which are related to entropy rigidity, and mention some applications.

Taejung Kim: A moduli space of Higgs bundles and a character variety (Note, Note2, Note3)

In this lecture series, we will survey the theory of the equivalence of three deformation theories, the Betti groupoid, de Rham groupoid, and Dolbeault groupoid, and we will characterize theses spaces in terms of a dynamical system point of view.

In the first lecture, we will explain a toy model of this equivalence, namely, the equivalence for rank 1 case. In the simplest setting, it will show formal aspects of more complicated cases for a higher rank.

In the second lecture, we will generalize this equivalence to a higher rank. The principal tool for showing this equivalence is the Hitchin's self-duality equation. Moreover, we will also explain the applications of this approach.

In the third lecture, we will study the Dolbeault groupoid, i.e., the moduli space of Higgs bundles, as a dynamical system, i.e., the Hitchin system. We will prove a cohomological criterion for the linearity of Hitchin flows.