
Organizers 
Topics from subfactor theory and TQFT
by
Marta M. Asaeda
University of Maryland, College Park
I will talk about the fusion algebras arising from orbifold systems of Uq(sl(n)) for q root of unity, especially about the fixed point resolution. These algebras are expected to play an important role in understanding quadratic (Hecke) skein modules of 3dimensional manifolds, S_{3, \infty}(M) = RL/(v^{1}L_{+}  vL_{} zL_{0}). The connection is very speculative but we can repeat after Joseph Conrad: "To follow the dream, and again to follow the dream and so always usque ad finem..."
Date received: May 1, 2003
Automorphisms of the Fricke characters of free groups
by
Richard Brown
American University
The set of all special linear characters of a Free group is an algebraic variety that can be realized as a subset of complex space cut out via a minimal set of polynomial manifestations of the Magnus Relation. Automorphisms of the free group induce automorphisms of this variety which preserve "volume" up to sign. We establish that these automorphisms extend to polynomial automorphisms of the ambient space which also preserve "volume" up to sign. When the free group is the fundamental group of a surface, this leads to a good algebriac model for the study of the dynamics of mapping class actions on surface character varieties.
Date received: April 30, 2003
Rational moves on links
by
Mieczyslaw K. Dabkowski
GWU
Coauthors: Jozef H. Przytycki (GWU)
The pth Burnside group of a link is an invariant of p moves which can be used to analyze p moves on links. This invariant, in particular, allows us to disprove MontesinosNakanishi 3move conjecture and HarikaeNakanishi (2, 2)move conjecture, and answer negatively Kawauchi's question concerning link homotopy and Nakanishi's 4move conjecture.
Date received: May 1, 2003
Flat Lorentz 3manifolds: an overview
by
Bill Goldman
University of Maryland
This talk will survey topological, geometric and dynamical aspects of an unusual class of geometric structures on noncompact 3manifolds: quotients of R^3 by proper actions of discrete groups of affine transformations. In 1977 Milnor asked whether a nonabelian free group admits such an action, and in 1983 Margulis proved such actions exist. In 1990 Drumm gave an explicit geometric construction. I will discuss the history of these examples, and present the current status of their classification.
Date received: April 28, 2003
Links in spatial embeddings of graphs
by
Brenda Johnson
Johns Hopkins University/ Union College
In 1983, Horst Sachs, John Conway, and Cameron Gordon proved that K_6, the complete graph on 6 vertices, is intrinsically linked, i.e., any spatial embedding of K_6 contains a nontrivial link. Sachs made a conjecture about the collection of all intrinsically linked graphs that was later proved by N. Robertson, P.D. Seymour, and R. Thomas. After describing the theorems of Sachs, ConwayGordon, and RobertsonSeymourThomas, I will give a (partial) survey of questions and results inspired by their work.
Date received: May 2, 2003
Quantum algebra and the Four Color Theorem
by
Paul Kainen
Georgetown University
It is shown that the Four Color Theorem (4CT) provides a natural connection between quantum algebra and physics. In particular, the natural operation which relates two edge 3colored rooted cubic plane trees by an elementary ``associational switch'' surgery is characterized by inducing a zeromapping between iterated tensor products of the usual qubit space (a complex vector space of complex dimension 2). But the 4CT is actually equivalent to a combinatorial rule for propagating twisting force, so a connection between physics and quantum algebra manifestly does exist. An application of the quaternions shows that in extending a coloring from one rooted cubic plane tree to another, the extension cannot fail on the last step.
Date received: May 5, 2003
4dimensional surgery: the A, Bslice problem
by
Slava Krushkal
University of Virginia
The A, Bslice problem is a reformulation of the 4dimensional surgery conjecture. I will give a survey of this problem, and will describe a new approach to it.
Date received: May 2, 2003
Khovanov Invariant
by
Eun Soo Lee
University of Illinois, Urbana, IL 61801
Khovanov invariant is an invariant of (relatively) oriented links, recently constructed by Mikhail Khovanov, which generalizes the Jones polynomial. In this talk, some results and open problems on Khovanov invariant will be discussed.
Date received: April 29, 2003
Representations of braid groups of Lie type
by
John J. Millson
University of Maryland
Coauthors: Valerio Toledano Laredo
I will discuss recent joint work with Toledano Laredo on the monodromy of the Casimir connection. The monodromy representation of this connection leads to a functor from representations V of a simple complex Lie algebra g to representations of the corresponding braid group of Lie type Bg ( on the zero weight space V[0] of V). I will discuss the problem of when this braid group representation is reducible answering a question of Procesi and Knutson. I will also give counterexamples using the same idea to a related conjecture of Kwon and Lusztig on representations of Bg coming from the quantum group associated to g.
Date received: April 28, 2003
Quantum subgroups, canonical basis, and boundary topological invariants
by
Adrian Ocneanu
PennState University
Topological quantum field theories with data given by Lie groups extend to the theories with boundary corresponding to subgroups of the original group. For quantum SU(N), N=2, 3,4, this quantum subgroups can be classified, using modular theory. This generalizes Coxeter ADE graphs. Quantum subgroups of SU(2) produce naturally canonical basis for the simple Lie groups. the other quantum subgroups produce new root and weight lattices.
Date received: May 1, 2003
"Topology" of protein folds
by
Teresa Przytycka
NIH
Proteins are fundamental building blocks of modern organisms whihc carry out important functions as enzymes, signal processors, carriers; to name a few. Still, the function of huge number of proteins is unknown and a significant effort is directed towards methods of predicting protein function. Protein function itself is a direct consequence of the protein 3dimensional structure, the so called fold. About 17 000 of such structures are known and much more are expected to be solved as the result of the outgoing projects. The efforts towards prediction the structure form the aminoacid sequence are also underway. However a visual inspection of a protein structure alone seldom elucidates its function. In contrast, the discovery that the structure of a given protein is similar to a structure of another protein for which the function is known can provide a valuable insight to the possible function of the first protein. Therefore fast and reliable methods for comparison of protein folds and their classification are of particular interest. This talk will serve as an introduction to the world of protein structure. Methods of structure classifications, including one based on speakers work and one based on Vassiliev invariant of knots will be presented.
Date received: April 30, 2003
KauffmanHarary Conjecture holds for Montesinos knots
by
Jozef H. Przytycki
George Washington University
Coauthors: Marta M. Asaeda (UMD), Adam S. Sikora (IAS)
We consider the conjecture by Kauffman and Harary on Fox colorings of alternating diagrams. We prove the conjecture for Montesinos knots. We propose a generalization of this conjecture to alternating links formulated in terms of homology of the double branched cover of S^{3} branched along a link. We prove this homology conjecture for Montesinos links. We speculate about the relation of the conjecture to a question on incompressible surfaces in the exterior of alternating links.
KauffmanHarary conjecture: Consider a knot K with the determinant, D(K), equal to a prime number p. Then for any alternating diagram of K with no nugatory crossings, every non trivial Fox pcoloring of the diagram colors different arcs with different colors.
Homology Conjecture: If K is an alternating diagram of a prime knot (more generally prime link) without a nugatory crossing then different arcs represent different elements of H_{1}(M_{K}^{(2)}, Z), where M_{K}^{(2)} is the double branched cover of S^{3} branched along K.
Theorem: The Homology Conjecture holds for all alternating Montesinos links.
Date received: April 29, 2003
Equivariant Euler Operators and Characteristics
by
Jonathan Rosenberg
University of Maryland
Coauthors: Wolfgang Lück (Münster)
The Euler characteristic of a compact manifold can be computed analytically in two different ways: by counting (with appropriate signs) the zeros of a "generic" vector field, and by taking the index of the "Euler characteristic operator" d + d* (acting on differential forms, graded by parity of the degree). We discuss the "correct" analogues of these calculations in the situation of a (possibly noncompact) manifold with a proper cocompact action of a discrete group. In particular we answer the question of what information is encoded in the equivariant Khomology class of the Euler characteristic operator.
Date received: April 30, 2003
modular circle quotients and PL limit sets
by
Richard Schwartz
University of Maryland, College Park
I will discuss a theoretical analogue of the question "What does your tennis racket look like if it is strung so tightly that the individual strings collapse into points". More precisely, I will consider topological quotients of the circle based on patterns of geodesics in the hyperbolic plane which have modulargroup symmetry. Given one of these patterns, one identifies two points of the circle if they are the endpoints of a geodesics in the pattern. I will show how to realize the resulting quotient spaces as limit sets of groups acting on the sphere  of possibly high dimension  by piecewise linear homeomorphisms.
Date received: May 2, 2003
Relations between the Sl(2, C)representations of knot groups and the Jones polynomial of a knot.
by
Adam S. Sikora
Institute for Advanced Study and SUNY Buffalo
We will describe the currently emerging connections between the Sl(2, C)representations of knot groups and the (colored) Jones polynomial of a knot.
Date received: April 27, 2003
C*algebras and (locally) Compact Quantum Groups  an overview
by
Piotr Stachura
The George Washington University (on leave from University of Warsaw)
I will present an outline of C*algebraic approach to Quantum Groups (Woronowicz's approach).
Date received: May 1, 2003
Dynamics of the modular group acting on GL(2, R)characters of a oncepunctured torus.
by
George Stantchev
University of Maryland at College Park
The SL(2, C) character variety of a free group of rank 2 contains certain components that correspond to representations in PGL(2, R), the full isometry group of hyperbolic 2space. I will discuss how to identify these components with R^{3} and how to parametrize the subsets that correspond to discrete embeddings. I will also discuss the action of the modular group on the set of characters of discrete embeddings and compare it with the action on the complement of this set in R^{3}.
Date received: April 30, 2003
Copyright © 2003 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas.