4/29/05

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Abstracts
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Teodor Banica, Universite Paul Sabatier (Toulouse III)

Random walks on discrete quantum groups

Abstract: We consider random walks on various quantum algebraic 
objects - quantum permutation groups, discrete quantum groups, subfactor 
graphs - with a few explicit calculations of associated probability 
measures.
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Florin Boca, University of Illinois at Urbana-Champaign

Some Statistical Problems Related to the Equation ad-bc=1

Abstract: We will present some results concerning the spacing
statistics of the Farey sequence and the limiting distribution
of the linear flow on the punctured 2-torus (joint work with
Zaharescu), and on the distribution of reduced quadratic irrational
numbers. The last topic is related to the continued fraction
algorithm and to certain number-theoretic spin chain models in
thermodynamics. In this framework we will introduce a certain
C*-algebra that is related to irrational rotation algebras.
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Dietmar Bisch, Vanderbilt University

A Notion of Free Product for Planar Algebras

Abstract: The Fuss-Catalan algebras, discovered by Jones and myself, are 
the fundamental symmetries associated to a subfactor whenever an intermediate 
subfactor is present. These algebras arise as certain ``free products'' of 
Temperley-Lieb algebras when viewed as planar algebras. In general, the 
dimension generating function of a free product planar algebra can be 
computed via Voiculescu's free multiplicative convolution.
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Ken Dykema, Texas A&M University

Multilinear function series and transforms in free probability theory

Abstract:  We consider the algebra of formal multilinear function series,
analogous to formal power series over an algebra, and we show that the R-
an S-transforms in free probability theory have unsymmetrized versions in
this setting.
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Alex Furman, University of Illinois at Chicago

Measure Equivalence and rigidity of higher rank lattices.

Abstract: Measure Equivalence of groups can be viewed as either measurable
analogue of quai-isometries or as a convenient framework for the study
of Orbit Equivalence of free, probability measure-preserving group
actions. In this context we shall focus on rigidity phenomena of higher rank
lattices.
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Shamindra Ghosh, University of New Hampshire

Representations of a Planar Algebra

Abstract: This talk will describe `representations of a planar algebra' 
in the sense of Vaughan Jones and classification of the irreducible ones 
according to their weights.

To every representation of a planar algebra, one can associate a power
series called `dimension'. We will show that the radius of convergence of
the dimension of any irreducible representation of a finite depth planar
algebra is at least as big as the inverse-square of the modulus of the
planar algebra.
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Pinhas Grossman, UC Berkeley

Intermediate Subfactors with No Extra Structure

Abstract: This talk will present joint work with Vaughan Jones on classifying
quadrilaterals of II_1 factors which have "no extra structure", in the sense
that the standard invariants of the elementary inclusions are Temperley-Lieb
algebras. It turns out that there are only two possibilities for the planar
algebra of a noncommuting quadrilateral with no extra structure.
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Adrian Ioana, UCLA

Amalgamated Free Products of w-rigid Factors and Calculation of
their Symmetry Groups, part II

Abstract:  We present several unique decomposition results (\'a la
Bass-Serre) for amalgamated free products of w-rigid factors.  We apply
this to prove rigidity results for isomorphisms between such factors and
to explicitly calculate their symmetry groups, i.e. their fundamental
groups and automorphism groups.  This is joint work with Jesse Peterson
and Sorin Popa.
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Kenley Jung, UCLA

Some free entropy dimension inequalities for subfactors of finite index

Abstract: Given an inclusion of $II_1$-factors $N \subset M$ and a finite 
generating set $F$ for $N$ we construct finite generating sets $X$ for $N$ 
and $Y$ for $M$ such that $\delta_0(X) \geq \delta_0(Y)$ where $\delta_0$ 
denotes Voiculescu's microstates free entropy dimension.  Applying this 
result to the basic construction yields a lower bound for $\delta_0(Y)$ 
in terms of $\delta_0(X)$ and $[M:N]$.  We apply these results to obtain 
certain nonisomorphism theorems.
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Yasuyuki Kawahigashi, University of Tokyo

Classification of extensions in algebraic quantum field theory
and vertex operator algebras

Abstract: We will explain how to study local and non-local extensions
of local nets of factors on the circle.  Relations of this problem
to classification of diffeomorphism covariant nets with c < 1, boundary
conformal field theory, the Goodman-de la Harpe-Jones subfactors, and
vertex operator algebras such as the moonshine VOA are presented.
This is joint work with Roberto Longo.
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Wolfgang Lueck, Universitaet Muenster

Middle algebraic K-theory of von Neumann algebras, group C^*-algebras 
and group rings

Abstract: We compute the lower and middle algebraic K-groups
of von Neumann algebras. Since $K$-groups commute with finite products,
it suffices to treat the various types separately. The middle $K$-groups
vanish if the type is different from I_f or II_1. The groups can be
computed in this cases via center valued traces and certain
determinants. We use this to study the
corresponding K-groups for group C^*-algebras and group rings.
In particular it is interesting to study the passage from the integral
group ring to the complex group ring, to the reduced group C^*-algebra
and finally to the group von Neumann algebra.
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Matilde Marcolli, Max-Planck Institute Bonn

KMS states and complex multiplication

Abstract: In this joint work with A.Connes and
N.Ramachandran, we construct a quantum statistical mechanical system that
generalizes the Bost-Connes system to imaginary quadratic fields K of
arbitrary class number and fully incorporates the explict class field
theory for such fields. This system admits the Dedekind zeta function as
partition function and the Idele class group as group of symmetries. The
extremal KMS states at zero temperature intertwine this symmetry with the
Galois action on the values of the states on the arithmetic subalgebra. We
also give an interpretation of the original BC system and of the GL(2)
system in terms of Shimura varieties, which motivates the construction for
imaginary quadratic fields. The geometric notion underlying the
construction is that of commensurability of K-lattices.
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Remus Nicoara, Vanderbilt University

On II_1 factors arising from 2-cocycles of w-rigid groups

Abstract: (jointly with Sorin Popa and Roman Sasyk)
We consider $II_1$ factors $M_\nu=L_\nu (G)$, where $G$ is a discrete 
ICC group with a relative property (T) subgroup $H$, and $\nu$ are 
scalar 2-cocycles of $H$. We show that for each fixed $(G,H)$ there 
exists no separable $II_1$ factor that contains $M_\nu$ for uncountably 
many $\nu\in \text{\rm H}^2(H)$. We consider several classes of examples,
such as the irrational rotation $HT$ factors 
$M_\alpha(\Gamma)=L_{\nu_\alpha}(\mathbb Z^2\rtimes \Gamma)$, 
where $\Gamma$ is a non-amenable subgroup of $SL(2,\mathbb Z)$
and $\nu_\alpha$ are cocycles of $\mathbb{Z}^2$, $\alpha=e^{2\pi it},
t \notin \mathbb{Q}$.
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Sergey Neshveyev, University of Oslo

Types of von Neumann algebras arising from boundary actions of
hyperbolic groups

Abstract: A non-degenerate finitely supported probability measure on a
non-elementary hyperbolic group defines a quasi-invariant harmonic measure
on the boundary of the group. The corresponding crossed product algebra is
well-known to be a factor of type III. In my talk I shall discuss what can
be said about the S-invariant of the factor, or in other words, about the
ratio set of the orbit equivalence relation on the boundary. This is a joint
work with Masaki Izumi and Rui Okayasu.
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Narutaka Ozawa, University of Tokyo and UCLA

Hyperbolic Groups and Type II_1 Factors,

Abstract: Most of discrete groups admit compact boundaries on which
they act amenably, even if groups themselves are not amenable.
In particular, groups with a geometric structure, e.g. hyperbolic groups,
naturally come with such boundaries. We can read the degree of
non-amenability of a group from its amenable boundary.
I will explain how to use amenable action to the classification of
group von Neumann algebras.
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Jesse Peterson, UCLA

Amalgamated Free Products of w-rigid Factors and Calculation of
their Symmetry Groups, part I

Abstract:  We present several unique decomposition results (\'a la
Bass-Serre) for amalgamated free products of w-rigid factors.  We apply
this to prove rigidity results for isomorphisms between such factors and
to explicitly calculate their symmetry groups, i.e. their fundamental
groups and automorphism groups.  This is joint work with Adrian Ioana
and Sorin Popa.
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Roman Sauer, University of Muenster

Another approach to Gaboriau's L^2-Betti numbers of equivalence relations

Abstract: We present an alternative approach to Gaboriau's L^2-Betti numbers
of equivalence relations using Lueck's dimension theory. Moreover,
we explain the relation between L^2 Betti numbers and the minimal
volume (based on an idea of Gromov) which shows how useful
the viewpoint of equivalence relations can be -- even if one is only
interested in L^2 Betti numbers of manifolds.
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Hanne Schultz, University of Southern Denmark

Brown measures of sets of commuting operators in a II_1 factor

Abstract: In recent work by U. Haagerup and the speaker it was shown that
for any operator T in a II_1 factor M and for any Borel set B in the
complex plane there is a maximal T-invariant socalled spectral subspace
K_T(B) affiliated with M such that the Brown measure of T|_K_T(B) is
concentrated on B. Moreover, K_T(B) is T-hyperinvariant (i.e. S-invariant
for every S in {T}'). This enables us to prove the existence of a Brown
measure for any finite set T=(T_1,..., T_n) of commuting operators in M.
The Brown measure \mu_T is a probability measure on C^n with certain nice
properties.
We also show that exactly as in the case of a single operator, one can
associate to every Borel set in C^n a T_1-,...,T_n-invariant spectral
subspace.
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Roland Speicher, Queen's University

Free probability and random matrices

Abstract: I will report on recent results around the relation between
free probability theory and random matrices.
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Yoshimichi Ueda, Kyushu University

HNN extensions of von Neumann algebras

Abstract: I'll present my work on reduced HNN extensions of von Neumann
algebras (as well as C^*-algebras). If time permits, I'll talk about 
my recent attempt of analyzing ascending HNN extensions (i.e., HNN 
extensions by endomorphisms).
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Alain Valette, Universite de Neuchatel

Group II_1-factors with trivial fundamental group and all L^2-Betti 
numbers zero

Abstract: Popa introduced a class $HT$ of $II_1$-factors displaying a mixture
of rigidity (relative property (T)) and non-rigidity (Haagerup property).
Using Gaboriau's work, he defined the $L^2$-Betti numbers of factors in this
class, and showed that, as soon as some $L^2$-Betti number is non-zero (and
finite) then the fundamental group of the factor is zero. This is the case
for the factor of $\bbZ^2\rtimes SL(2,\bbZ)$, which was the first example of
a factor with trivial fundamental group. We construct new examples of group
$II_1$-factors in $HT$, with all $L^2$-Betti numbers zero and with trivial
fundamental group. Triviality of the fundamental group follows from recent
results of Monod-Shalom.
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