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Interactions between Representation Theories, Knot Theory, Topology, Quantum Field Theory, Category Theory, and Mathematical Physics.

            ABSTRACTS

                Talks        Posters

 

                   Talks

John Robert Armstrong
The familiar notion of a representation of a group or of a Lie algebra can be extended to that of a representation of a category. This algebraic technique can be applied to categories of tangles, and thus to knot theory. In particular, Kauffman bracket evaluations can be seen as
restrictions of a certain family of representations of the category
{\bf FrTang} of framed, unoriented tangles.

Marta Asaeda
I will talk about TQFT constructed from subfactors--pairs of certain von Neumann algebras--.

Stephen J Bigelow
A classical problem is to understand the representation theory of the
symmetric group S_n. There is a beautiful solution to this problem over
characteristic zero. The behavior in characteristic p is similar to the
behavior of the Hecke algebra when the parameter q is a primitive pth root of unity. A conjecture by Dipper gives a precise correspondence in the case p^2>n. I will discuss a topological approach to this conjecture.

Vladimir Chernov
Let M be an oriented n-dimensional manifold. We study
the causal relations between the wave fronts W_1 and
W_2 that originated at some points of M. We introduce a
numerical topological invariant CR(W_1, W_2) (the so-
called causality relation invariant) that, in particular,
gives the algebraic number of times the wave front W_1
passed through the point that was the source of W_2
before the front W_2 originated. This invariant can be
easily calculated from the current picture of wave fronts
on M without the knowledge of the propagation law for the
wave fronts. Moreover, in fact we even do not need to know
the topology of M outside of a part P of M such
that W_1 and W_2 are null-homotopic in P.
We also construct the Affine winding number invariant
win which is the generalization of the winding number to
the case of nonzero-homologous shapes and manifolds other
than R^2. The win invariant gives the algebraic number
of times the wave front has passed through a given point
between two different time moments without the knowledge of
the wave front propagation law.
The invariants described above are particular cases of the
general affine linking invariant al of nonzero homologous
submanifolds N_1 and N_2 in M introduced by us. To
construct al we introduce a new pairing on the bordism
groups of space of mappings of N_1 and N_2 into M.
For the case N_1=N_2=S^1 this pairing can be regarded as
an analog of the string-homology pairing constructed by
Chas and Sullivan, and it is a generalization of the
Goldman Lie bracket.

Sam Lomonaco
A map phi:A->S from a group group A to a set S is said to have hidden subgroup
Structure if there exists a subgroup K (called a hidden subgroup) and an injection
iota: A/K -> S such that phi = iota composed with nu, where nu:A -> A/K is the
natural map.
We show how Shor’s quantum factoring algorithm can be viewed as a quantum hidden
subgroup (QHS) algorithm. We then show how to create QHS
algorithms on a variety of groups, including the additive group of reals,
and the circle (i.e., the additive group of reals mod 1).
Finally we discuss a QHS algorithm for functional integrals, which is highly speculative.
The algorithm, in the spirit of Feynman, is based on functional integrals whose existence
is difficult to determine, let alone approximate. However, in the light of Witten's
functional integral approach to the knot invariants, this algorithm has the advantage
of suggesting a possible approach toward the development of a QHS algorithm for
the Jones polynomial.
This work is in collabration with Lou Kauffman
This paper can be found at the URL:
http://xxx.lanl.gov/abs/quant-ph/0304084..
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BEGIN ABSTRACT2 .................................
We first define in mathematical terms what is meant by the physical phenomenon
of quantum entanglement (Q.E.). We then show how Lie group invariants can be used to
quantify and classify Q.E. We then explore the possible relation between Q.E. and
topological entanglement.
This talk is in collaboration with Lou Kauffman.
A paper on Lie group invariants of quantum entanglement can be found at the URL:
http://xxx.lanl.gov/abs/quant-ph/9811056

Louis Crane
Abstract 1: We explain the connections between categorical algebra and the foundations of quantum physics, and propose categorical state sums as fundamental constructions.Abstract 2: We give a self contained introduction to 2-categories, and
explain their physical applications.

Olaf Dreyer
I report on some advances in the quantum theory of black holes. One is a
hint that the degrees of freedom are spin one excitations. The second one
is the presence of a large symmetry group near the horizon.

Peter Freyd
abstract is not available yet.

Fred Goodman
Take a parallelogram with angles 60 and 120 degrees and with
integral side lengths a and b. Tile the parallelogram with equilateral
triangles. It is possible to define rules for folding the parallelogram
so that the number of foldings is equal to the dimension of the irreducible representation of sl_3 with highest weight (a+b, b).
This fact reflects a canonical version of Littelmann's path model in which
the objects which are folded are not paths at all but rather convex regions in the weight space determined by the origin and a dominant integral weight.
The fact that Littelmann operations on shapes are well defined is related to tableau algorithms (bumping and sliding) which were recently realized for types B, C, D by Lecouvey. A relatively simple argument works for all simply laced types.
This is joint work, in progress, with my student Holly Hauschild.

Louis H Kauffman
1. Virtual knot theory is the study of knots in thickened surfaces up to stabilization by addition
and subtraction of empty handles. There is a convenient planar (or spherical) diagrammactic expression for this theory, that enables one to see how to extend many classical invariants to invariants of virtual knots, and to compare properties of virtual knots with classical knots. Alternatively, the representation of virtual knots via surfaces can give important information.
This talk will survey recent results in this theory.
2. This talk will give a survey of work of David Radford and the speaker. The talk can be
taken as an introduction to the relationships between Hopf algebras and topological invariants
of knots and three manifolds.
3. This talk is an introduction to the subject of quantum
entanglement and to the relationships of linking (via unitary braiding
operators) and entanglement of quantum states. We will also discuss the problem of designing
a quantum computer that computes the Jones
polynomial.

Samuel J. Lomonaco
1.A map phi:A->S from a group group A to a set S is said to have hidden subgroup
Structure if there exists a subgroup K (called a hidden subgroup) and an injection
iota: A/K -> S such that phi = iota composed with nu, where nu:A -> A/K is the
natural map.
We show how Shor’s quantum factoring algorithm can be viewed as a quantum hidden
subgroup (QHS) algorithm. We then show how to create QHS
algorithms on a variety of groups, including the additive group of reals,
and the circle (i.e., the additive group of reals mod 1).
Finally we discuss a QHS algorithm for functional integrals, which is highly speculative.
The algorithm, in the spirit of Feynman, is based on functional integrals whose existence
is difficult to determine, let alone approximate. However, in the light of Witten's
functional integral approach to the knot invariants, this algorithm has the advantage
of suggesting a possible approach toward the development of a QHS algorithm for
the Jones polynomial.
This work is in collabration with Lou Kauffman
This paper can be found at the URL:
http://xxx.lanl.gov/abs/quant-ph/0304084
2.We first define in mathematical terms what is meant by the physical phenomenon
of quantum entanglement (Q.E.). We then show how Lie group invariants can be used to
quantify and classify Q.E. We then explore the possible relation between Q.E. and
topological entanglement.
This talk is in collaboration with Lou Kauffman.
A paper on Lie group invariants of quantum entanglement can be found at the URL:
http://xxx.lanl.gov/abs/quant-ph/9811056

Susan Montgomery
One of the difficulties in classifying Hopf algebras, and in
classifying semisimple Hopf algebras in particular, is the lack
of suitable invariants which can be computed for a Hopf algebra
or its representations. We give an introduction to semisimple
Hopf algebras and then discuss one such invariant, namely the
Frobenius-Schur indicator of an irreducible representation; it
generalizes the usual notion of indicator for representations of
finite groups.
We define this invariant and give examples of how it can be computed;
as one example, we consider the Drinfel'd double of the symmetric
group S_n. The indicator has already been used in the classification
theory, and in looking at possible dimensions of representations.
It is also of interest in conformal field theory, and can be extended
to the twisted Drinfel'd double of a finite group, which is a
quasi-Hopf algebra. The work we discuss comes mostly from work of the speaker, V. Linchenko, Y. Kashina, G. Mason, R. Ng, Y. Sommerhaeuser, and Y. Zhu.

Dror Bar Natan
The minor advantage of homology theory over the Euler characteristic is that it is a finer
invariant. The major advantage is that it is a functor:
Given a map between spaces there is a map between their homologies. Think of almost any
major theorem in algebraic topology and you'll find that the functoriallity of homology is
deeply involved. In my talk, I will explain in elementary terms what seems to be the
corresponding property of Khovanov's homology: that it is a functor from the category of
links and cobordisms to the category of vector spaces (see Jacobsson'sea arXiv:math.GT/0206303 and Khovanov's arXiv:math.QA/0207264). My proof of this
property is in the spirit of Khovanov's, but it is both simpler and more general. It involves
the extension of the theory to the canopoly of tangle cobordisms, with values in several
related canopolies. What's a canopoly? No, that would go in the talk; not here. It's an
object with a rather messy formal definition but a very simple visual image.

Amnon Neeman
Abstract: In his Grothendieck Festschrift paper, Thomason showed how to prove a
localisation theorem for the K-theory of singular schemes. For reasons coming from surgery,
it is interesting to generalise to non-commutative rings. We will explain work by Ranicki
and the speaker, which addresses the problem.

J'ozef H. Przytycki
Talk 1: In classical knot theory substantial effort was directed into
search for unknotting moves on links. Despite of this effort many problems remained
unsolved,including several classical problems concerning unknotting operations.
We solve several of them.Our solution uses a new concept -- Burnside groups of links
which establishes unexpected relation between knot theory and group theory.
Talk 2: Rational moves on links can be deformed to skein relations leading to invariants
of links in $S^3$ and skein modules in 3-manifolds.
We outline the theory of skein modules from the ``deformation"
perspective and discuss, in more detail, the case of the 3-move and
the cubic skein module. We speculate on the meaning of Burnside
obstructions to the Montesinos-Nakanishi 3-move conjecture
for the theory of cubic skein modules.

David Radford
1. We start with basic concepts in the theory of Hopf algebras
and proceed to discuss the Drinfeld double and related structures in
detail. Our emphasis will be on the aspects of Hopf algebra theory related
to certain invariants of knots and links.

2. The Drinfel'd duouble of a finite-dimensional Hopf algebra H
over a field has a natural oriented quantum algebra structure which
together with many "trace" functions give rise to invariants of oriented
knots and links. Representions of the double have an oriented
quantum algebra structure.
We will explore these structures in detail for certain finite-dimensional
Hopf algebras H. Our analysis is based on a new way of understandng
irreducible representations of finite-dimensional doubles.

Nicolai Reshetikhin
A construction of invariants of tangles with
flat connections in their complement will be given
in the first lecture. Connections are in the trivial
principle $G$-bundle over the complement. The construction
is based on solutions to the holonomy Yang-Baxter equation
which will also be introduced. In the second talk I will
explain how to obtain such solutions from quantized universal
algebras at roots of 1.

Fernando Jose Oliveira de Souza
Categories of diagrammatic morphisms have been used extensively through
various graphical calculi in category theory, algebra, topology, physics,
as well as classical and quantum information processing and logic. There
are various approaches to those categories, developed at different levels
of rigor and justification. In this talk, we review some of those
approaches, revising the relationship between them.We first recall a number of categorical structures and their corresponding
diagrammatic, structured categories freely generated by a given category
C. We show how those categories are used to provide C itself with a
graphical calculus when C happens to be structured, and to construct
algebraic objects (including some universal ones.) We use planar
immersions of rigid-vertex graphs in order to recognize Penrose's arrow
notation as the diagrammatic representation of (eventually traced)
symmetric categories. These were originally characterized combinatorially
via words, which correspond to ordered incidence relations encoded by
Penrose's tensor notation. Finally, we identify the arrow notation with
Kuperberg's contraction diagrams (which do not have distinguished
directions for composition and tensor product), and understand them
in operadic terms. We briefly mention Jones' planar algebras

David Yetter
Unpublished work of Barrett and Mackaay has shown that Kapranov and Voevodsky's
2-VECT is inadequate to the task of providing a setting for the representation theory of
2-groups with non-profinite groups of 1-arrows. In particular the Poincare 2-group of
Baez admits
only representations indexed by characters of the principal group,
and trivial reprsentations of the base group (using the crossed
module terminology).
We propose that a 2-category whose objects are categories
of measurable fields of Hilbert spaces and bounded measurable fields of
operators is the correct "infinite dimensional" analog of Kapranov
and Voevodsky's 2-VECT. We give representability
results for invertible additive functors between such categories, examine the functorial
properties of the classical direct integral construction, and introduce enough of the
representation theory of 2-groups to make connection with formal constructions of
4-dimensional TQFT's and conjectural constructions for Minkowskian quantum gravity.

 

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       Posters Abstracts

 

Ivelina Bobtcheva
Title: The mysteries in the center or how many HKR invariants of 3-manifolds
can be constructed from the same Hopf algebra?
Authors: Ivelina Bobtcheva, Maria Grazia Messia
Abstract: The input of the HKR (Hennings-Kauffman-Radford) invariants of
links is a finite-dimensional unimodular ribbon Hopf algebra $A$ and an
element in a quotient $\hat{Z}^S(A)$ of its center which determines a trace
function on $A$. Little is known about the special subset $T_s$ of trace
functions which bring to invariants under the band-connected sum of two link
components (a necessary condition to obtain invariants of 3-manifolds), except that $T_s$ contains the identity, the algebra integral and, when $A$ is a quantum group, an element $z_{RT}$ corresponding to the Reshetikhin-Turaev invariant of 3-manifolds.
We define a new product structure on $\hat{Z}^S(A)$, denoted with *, and a
subset $T_Z$ of $\hat{Z}^S(A)$, and we make those explicit for the quantum
sl(2) with the following conclusions:
(a) the nilpotent subalgebra of $\hat{Z}^S(A)$ with respect to
the *-structure is isomorphic to the fusion algebra of the semisimple
quotient of the category of representations of the quantum sl(2);
(b) $T_Z$ is a commutative monoid of $\hat{Z}^S(A)$ with respect to
the usual and the *- product structures. Moreover, $T_Z$ consists of
four elements, three of which are the identity, the algebra integral
and $z_{RT}$.
We conjecture that $T_Z=T_s$.

 

Heather Dye
Title: Using Surfaces to Detect Non-Trivial Virtual
Knots
Abstract: Virtual knot theory is a generalization of classical knot theory that was introduced
by Kauffman in 1996. Classical knot invariants were generalized to virtual knot diagrams,
but they often fail to detect non-trivial diagrams. We show how surfaces and the bracket polynomial may be used to detect non-trivial virtual knot diagrams.

Billie J. Rinaldi
Title: A Cellular Automaton Inverse Problem.
Abstract: This poster presents an introduction to a cellular automaton inverse
problem. Applications of this inverse problem are shown, and current work
on this inverse problem is discussed. Techniques are also presented for
filtering noise by means of this inverse problem.

 

Contact: Kazem Mahdavi, mahdavk@potsdam.edu

 

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