Speaker Ross Street (Macquarie University)

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Title Basic structures for higher-dimensional algebra

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Lecture 1: Substitudes and convolution.

Lecture 2: Higher dimensional categories.

Abstract

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Emmy Noether emphasized that Euler characteristic, genus, and
other numerical invariants of spaces should be extracted after passing
to invariant algebraic structures such as homology and homotopy groups.
Categories provided a language for such functorial passages from
geometry to algebra. Sometimes, as when we pass from a compact
Hausdorff space to its commutative C*-algebra, the passage is perfect
and we have a duality, thereby diminishing the division between geometry
and algebra. The distinction between the language and the mathematics
must also be abandoned as we see an increase in the use of categorical
structures themselves as invariants. Examples are the fundamental
groupoid of a space and the monoidal category of appropriate
representations of a topological group. When we work in a traditional
algebra we may view it as 0-dimensional: the elements are points that
may or may not be equal. When we work in a category we consider paths of
morphisms that may be equal or not: this is 1-dimensional algebra. In a
2-category or bicategory, we work with "pasting diagrams" of
2-dimensional regions containing 2-cells. In this way we are led to
higher-dimensional algebra. In the first lecture I will describe some
low dimensional categorical structures and show how we work in them to
understand processes lying behind such areas as Tannaka duality and
vertex algebras. In the second lecture I will define the basic
structures for doing n-dimensional algebra for all n.

Speaker Alexei Davydov (Macquarrie University)

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Title K-theory of braided monoidal categories

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Abstract

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The commutativity constraint of braided category gives rise to an
algebraic structure on its K-theory known as Gerstenhaber algebra. If,
in addition, the braiding has a compatible balanced structure the
Gerstenhaber bracket on K-theory is generated by a Batalin-Vilkovisky
differential. We use these algebraic structures to prove a
generalization of Anderson-Moore-Vafa theorem which says that the order
of the twist in a semi-simple ribbon (tortile) category with finitely
many simple objects is finite.

Speaker Valerio Toledano (University of Paris, Jussieu, Paris)

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Title Lectures on Operator Algebras and Conformal Field Theory

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Abstract

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I will describe an operator algebraic solution to the problem
of fusion for the Wess-Zumino-Witten model of Conformal Field
Theory. This approach, initiated by A. Wassermann, relies on
the use of Connes' tensor product of bimodules over a von
Neumann algebra to define a multiplicative operation (Connes
fusion) on the positive energy representations of a loop group
LG = C^{}(S^{1},G) at a given level. The notion of bimodules
arises by restricting these representations to loops with support
contained in an interval I of
S^{1} or of its complement.

This method has also been used by T. Loke to define and compute
the fusion ring corresponding to the discrete series representations
of Diff(S^{1}) and is also applicable to the case where the target
group G isn't simply connected. One of the interesting problems
it poses is to relate it to Kazhdan and Lustzig's definition of
fusion.

I will show that the multiplicities of tensor product decompositions
(fusion rules) are given by the Verlinde rules.
The computation is done in the spirit of invariant theory à l
H. Weyl and rests ultimately on the detailed study of the primary
fields of the theory, which allows to prove that they define
operator-valued distributions and on the explicit computation
of the monodromy of a family of Knizhnik-Zamolodchikov equations
corresponding to a suitable 'vector' representation of the loop
group. An interesting, and perhaps novel feature here is that
the exceptional groups
E_{6},E_{7},F_{4},G_{2} turn out to
possess such vector representations.

Speaker Paolo Piazza (La Sapienza, Rome)

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Title
Dirac index classes and the non commutative spectral flow

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Abstract

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I shall present some joint work with Eric Leichtnam.
Let M--->B be a fibration of compact manifolds, for example
a fibration of spin manifolds.
Let t be in [0,1] and assume that for each t we are given a vertical
family
of generalized Dirac operators D(t) = (D(t)_{b})_{b in B}.
D(t) can be given, for example, by a smooth variation of the vertical
metrics defining the Dirac operators on the fibres. We assume that
D(t) depends continuously on t and that Ind(D(0)) = 0 in K^{i} (B),
i = 0,1 depending of the dimension of the fibres.
Of course we then have Ind(D(t)) = 0 for each t.
Under these assumptions Dai and Zhang have defined the notion
of higher spectral flow of {D(t)}_{t in [0,1]}; this is an
element in K^{i+1}(B).
I shall present a generalization of this notion to a noncommutative
context,
with D(t) a 1-parameter family of operators acting on the sections
of a A-bundle, with A a C^{*}-algebra. We shall be mainly
interested in the reduced C^{*}-algebra of a discrete group
and to the 1-parameter family of operators induced by a family
of G-invariant Dirac operators on a Galois G-covering.
In this case the noncommutative spectral flow is, when defined , an
element
in K_{*}(C^{*}_{r} (G)).

I shall give geometric applications of this concept to index
theoretic and geometric situations, including the
problem of establishing when the Novikov's higher signatures of a
closed
manifolds are cut-and-paste invariants.

Speaker Danny Stevenson (Adelaide University)

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Title
The K-theory of Bundle Gerbes

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Abstract

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In the presence of a non-trivial B-field, D-brane charges in certain
string theories are classified by twisted K-theory. The appearance of
twisted K-theory is due to the fact that D-branes have `twisted' bundles
on their world volumes. This twisting arises from a class in
H^{3}(M; ** Z **)
coming from the field strength of the B-field. We shall describe some
recent joint work with Bouwknegt, Carey, Mathai and Murray which allows
for a description of these twisted bundles as honest bundles on the total
space of a projective unitary bundle together with an action of a certain
gerbe. We shall give an example of this in the case where the twisting
class in H^{3}(M; ** Z **) is torsion.
We shall also discuss the twisted Chern
character in this setting.

Speaker Peter Bouwknegt (Adelaide University)

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Title
Branes on Group Manifolds and Fusion Rules

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Abstract

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We discuss the classification of D-brane
charges for branes on group manifolds
in the context of boundary conformal
field theory. We compare the results
to the K-theoretic classification of
D-brane charges.

Speaker Krzysztof Wysocki (Melbourne University)

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Title
Pseudoholmorphic curves and Hamiltonian dynamics

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Abstract

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We shall describe new useful tools in the study
of three-dimensional flows. The tools are based on a first order partial
differential equations of Cauchy-Riemann type.
Families of solutions of the partial differential equations can be
used to construct a natural foliation
of a star-shaped energy surface. This foliation gives rise to a geometric
decomposition of the star-shaped energy surface
into a few very special periodic orbits *P* of the
Hamiltonian vector field and a 2-dimensional foliation in the
complement consisting of embedded surfaces transversal to the flow and
bounded by periodic orbits from *P*. As a consequence for
dynamical systems one concludes, for example, that every non-degenerate
Hamiltonian vector field on the star-shaped energy surface
possesses either 2 or infinitely many periodic orbits. Joint work with
H. Hofer and E.Zehnder.