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.
|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|
--- Lunch ---
|01:30-02:10||Inkang Kim||Deformation of hyperbolic 3-manifolds|
Jae Choon Cha
|New Hirzebruch-type invariants from iterated p-covers|
|Exceptional Dehn fillings|
--- Coffee Break ---
|A characterization of cones in the projective space|
|05:10-05:50||Jaejeong Lee||Fundamental domains of properly convex real projective structures|
|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|
--- Lunch ---
--- Excursion ---
|Participants accompanied with family||200,000 (Korean Won)|
|Participants not accompanied with family||100,000 (Korean Won)|
|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'.)|
For any inquiries on the workshop, please contact:
Korea Advanced Institute of Science and Technology
373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea
Email: smileabacus at kaist dot ac dot kr
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.
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.
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.
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.
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.
We resolve a Thurston's conjecture on limit of Kleinian groups and generalize his original conjecture.
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.
We estimate the number of exceptional slopes.
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).
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.