The 7th KAIST Geometric Topology Fair

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

July 9 - July 11, 2007

Gyeongju TEMF Hotel, Gyeongju-city, Gyeongsangbuk-do, 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.

Organizers

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

Confirmed Participants

An, Byung Hee KAIST
Cha, Jae Choon ICU
Choi, Suhyoung KAIST
Dhrubajit Choudhury KAIST
Ha, Jae-soon KAIST
Hwag, Taek Kyu KAIST
Hong, Sungbok Korea University
Huh, Young Sik Hanyang University
Jeon, Choon Bae Daeduk College
Jo, Kyeonghee Mokpo National Maritime University
Kim, Hong Chan Korea University
Kim, Inkang Seoul National University
Kim, Jee Hyoun KAIST
Kim, Se-goo Kyonghee University
Kim, Tae Hee Konkuk University
Ko, Ki Hyoung KAIST
Lee, Eon-Kyung Sejong University
Lee, Gye-seon KAIST
Lee, Hwa Jeong KAIST
Lee, Jaejeong University of California
Lee, Jung Hun Konkuk University
Lee, Sang Jin Konkuk University
Lee, Sangyop Seoul National University
Park, Hyo Won KAIST
Park, Seo Jung KAIST

Overseas Visitors

Elisha Peterson United States Military Academy
Sean Lawton Kansas State University
William M. Goldman University of Maryland

Programs

July 9
08:50-09:10 Announcements
09:10-10:00 William M. Goldman Deformations of geometric structures and representations of fundamental groups
10:10-11:00 William M. Goldman Deformations of geometric structures and representations of fundamental groups
11:00-12:00 Sean Lawton Generators of SL(2,C)-Character Varieties of Arbitrary Rank Free Groups
12:00-01:30

--- Lunch ---

 
01:30-02:10 Inkang Kim Deformation of hyperbolic 3-manifolds
02:20-03:00

Jae Choon Cha

New Hirzebruch-type invariants from iterated p-covers
03:10-03:50

Sangyop Lee

Exceptional Dehn fillings
03:50-04:20

--- Coffee Break ---

 
04:20-05:00

Kyeonghee Jo

A characterization of cones in the projective space
05:10-05:50 Jaejeong Lee Fundamental domains of properly convex real projective structures
July 10
09:00-09:50 Elisha Peterson Trace Diagrams, Spin Networks, and Spaces of Graphs
10:00-10:50 Sean Lawton Central Functions and SL(2,C)-Character Varieties
11:00-11:50 Elisha Peterson Trace Diagrams and Character Varieties
11:50-01:30

--- Lunch ---

01:30-

--- Excursion ---

Presentation

You can use :
  • Electronic File via Notebook + Beam Projector
  • Manuscript on A4 or letter paper via Document Camera + Beam Projector
  • Registration

    Benefits for registeration include:
  • Participation in the academic programs
  • Rooms in the workshop venue (Gyeongju Temf Hotel) for three nights, July 8, 9, and 10
  • Dinner on July 8, breakfasts, lunches, and dinners on July 9 and 10, and breakfast on July 11 (accompanying people are also welcome)
  • Registration Fee

    Participants accompanied with family 200,000 (Korean Won)
    Participants not accompanied with family 100,000 (Korean Won)

    Local information

  • Lodging : Gyeongju Temf Hotel
  • Transportation from Incheon International airport
      1. Purchase Limousine tickets for Gimpo Airport. (It will take about 40 minutes to Gimpo by bus.)
      or Purchase AREX tickets for Gimpo Airport. (It will take about 40 minutes to Gimpo by train.)
      2. At Gimpo Airport, buy a ticket for Ulsan Airport. (It will take about 60 minutes to Ulsan by airplane.)
      3. At Ulsan Airport, take a taxi. (It will take about 40 minutes to Gyeongju TEMF Hotel by taxi.)
      The taxi driver will understand your destination if you show him the message.
      (It means that 'destination: Gyeongju TEMF Hotel'.)
  • Excursion : Tour Information
  • Contact

    For any inquiries on the workshop, please contact:

    Gye-Seon Lee
    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: smileabacus at kaist dot ac dot kr

    Abstracts

    William M. Goldman : Deformations of geometric structures and representations of fundamental groups

    In 1936, Charles Ehresmann initiated the study of locally homogeneous geometric structures on manifolds. Such structures are modelled on homogeneous spaces and include many familiar objects in differential geometry with strong local symmetry. The deformation theory of such structures is closely modelled on the space of representations of the fundamental group, and leads to an algebraic study of such representations.

    This theory is particularly rich for surfaces, for which the deformation spaces enjoy a rich symplectic/Poisson geometry as well as an action of the mapping class group. The first lecture will describe the general theory of Ehresmann structures on manifolds and examples of their deformation theory, and the second talk will describe in more detail dynamical systems arising from geometric structures on surfaces.

    Sean Lawton : Generators of SL(2,C)-Character Varieties of Arbitrary Rank Free Groups

    Sikora has shown that character varieties may be understood as spaces of graphs. These graphs deform to provide knot invariants related to Skein modules. Working directly with the SL(2,C) character varieties of rank n free groups, we give explicit derivations of the n/6(n^2+5) minimal generators of the coordinate ring of the character variety. We then show there are no relations among 3n-3 of these generators; maximally chosen with this property. Lastly, we briefly discuss a recent result of Florentino that the natural map from the character variety to C^(3n-3) is only surjective in the cases n=2 or 3; but almost surjective in general.

    Sean Lawton : Central Functions and SL(2,C)-Character Varieties

    We give a decomposition of the coordinate ring of SL(2,C)-character varieties of free groups and use representation theory to define a set of special functions we refer to as central functions. These functions correspond directly to "Peterson graphs," strongly related to "Sikora graphs," embedded in a surface with non-empty boundary.

    Elisha Peterson : Trace Diagrams, Spin Networks, and Spaces of Graphs

    Trace diagrams are marked graphs which may be identified functions between tensor powers of vector spaces. They generalize both spin networks and the Temperley-Lieb algebra by introducing a diagrammatic notation for group actions. We describe the functor between the categories of trace diagrams and that of functions, and discuss how trace diagrams fit into the more general context of planar algebras. We conclude by examining in more detail the special case of SL(2,C) group actions. In this case, the functions corresponding to diagrams are very simple to compute due to the fundamental binor identity.

    Elisha Peterson : Trace Diagrams and Character Varieties

    The language of trace diagrams gives rise to a simple expression of the coordinate ring of character varieties; in fact, the functions in this ring have very natural representations as trace diagrams. We demonstrate how to depict a basis of the coordinate ring using trace diagrams, and then compute several examples for the case of SL(2,C) character varieties.

    Inkang Kim : Deformation of hyperbolic 3-manifolds

    We resolve a Thurston's conjecture on limit of Kleinian groups and generalize his original conjecture.

    Jae Choon Cha : New Hirzebruch-type invariants from iterated p-covers

    We define new Hirzebruch-type invariants which are essentially intersection form analogues of well-known signature defects. Applications include various torsion problems on homology cobordism of 3-manifolds and concordance of links.

    Sangyop Lee : Exceptional Dehn fillings

    We estimate the number of exceptional slopes.

    Kyeonghee Jo : A characterization of cones in the projective space

    Any open cone has an accumulation point in the base by the action of its automorphism group. We prove the converse of this statement, more precisely, a projective domain D whose boundary has a locally flat point where an Aut(D)-orbit accumulates is a cone. We don't need any other assumption when D is convex. In the case of non-convex domains, the following conditions about the flat boundary piece P containing the accumulation point are necessary : (i) P is a component of < P > Ç­ cl(D)­­, (ii) P has no complete line. We also prove that a quasi-homogeneous affine domain with a flat boundary piece P satisfying (i) and (ii) is affinely equivalent to R+ ´ int(P).

    Jaejeong Lee : Fundamental domains of properly convex real projective structures

    Endowed with the Hilbert metric, every properly convex domain D in RPn becomes a complete metric space on which any discrete subgroup G of Aut(D) acts properly by isometries. However, bisectors (i.e. equidistant hypersurfaces) of two points of D with respect to the Hilbert metric are not necessarily totally geodesic and it is not clear whether the action of G on D has a convex fundamental domain. We show that the action of G indeed admits a convex fundamental polyhedral domain. Conversely, we present local conditions under which one can obtain a (properly) convex domain D in RPn by gluing together convex polytopes in RPn via projective facet-pairing transformations.


    Last Updated: July 24th, 2007