Spring 2008

**Organizer: Dietmar Bisch**

**Mondays, 4:10-5:30pm in SC 1432**

- Date:
**1/14/08****no meeting due to recruitment**

- Date:
**1/21/08****no meeting due to recruitment**

- Date:
**1/28/08**- Speaker:
**Paramita Das, Vanderbilt** - Title:
**A new construction of subfactors from random matrix theory I** - Abstract:
This series of talks will present the details of a new construction of subfactors due to
Guionnet, Jones and Shlyakhtenko. They use a graded algebra structure of a planar algebra and
certain traces inspired by random matrix theory to construct II
_{1}factors and subfactors whose standard invariant realizes the given planar algebra. Thus they obtain an alternative proof of Popa's theorem that every (appropriate) planar algebra can be realized by a subfactor.

- Speaker:
- Date:
**2/4/08****no meeting due to recruitment**

- Date:
**2/11/08**- Speaker:
**Paramita Das, Vanderbilt** - Title:
**A new construction of subfactors from random matrix theory II**

- Speaker:
- Date:
**2/18/08**- Speaker:
**Paramita Das, Vanderbilt** - Title:
**A new construction of subfactors from random matrix theory III**

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- Date:
**2/25/08**- Speaker:
**Paramita Das, Vanderbilt** - Title:
**A new construction of subfactors from random matrix theory IV**

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- Date:
**3/3/08****no meeting, Spring break**

- Date:
**3/10/08, Special Time, 3:10-4:00pm in SC 1432**- Speaker:
**Alan Wiggins, Vanderbilt** - Title:
**A new construction of subfactors from random matrix theory V**

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- Date:
**3/17/08**- Speaker:
**Alan Wiggins, Vanderbilt** - Title:
**A new construction of subfactors from random matrix theory VI**

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- Date:
**3/21/08 (Friday), Special Subfactor Seminar, 4:10-5:30pm in SC 1432**- Speaker:
**Cyril Houdayer, UCLA** - Title:
**Prime factors and amalgamated free products** - Abstract: I will show that a non-amenable II_1 factor arising as an amalgamated free product over an abelian von Neumann algebra is prime. I will moreover discuss some applications and generalizations of this result. This is joint work with Ionut Chifan.

- Speaker:
- Date:
**3/24/08****no meeting (Banff workshop on von Neumann algebras)**

- Date:
**3/31/08**- Speaker:
**Stuart White, University of Glasgow** - Title:
**Groupoid normalisers and tensor products** - Abstract:
Given a unital inclusion $B\subset M$ of a subalgebra inside
a finite von Neumann algebra, the normalisers of $B$ in $M$ are those
unitaries $u\in M$ such that $uBu^*=B$ and the two-sided groupoid
normalisers $GN_M(B)$ are those partial isometries $v\in M$ such that
$vBv^*\subset B$ and $v^*Bv\subset B$. We shall consider two such
inclusions $B_1\subset M_1$ and $B_2\subset M_2$ and examine the
normalisers and groupoid normalisers of $B_1\ \overline{\otimes}\ B_2$
inside $M_1\ \overline{\otimes}\ M_2$ and the von Neumann algebars
they generate. It is easy to construct examples inside the matrix
algebras for which the normalisers of the tensor product generate a
larger algebra than the tensor product of the algebra generated by the
normalisers (and similarly for the two-sided groupoid normalisers).
However, when each $B_i'\cap M_i\subseteq B_i$, we do have

GN_{M_1}(B_1)''\ \overline{\otimes}\ GN_{M_2}(B_2)''=GN_{M_1\ \overline{\otimes}\ M_2}(B_1\ \overline{\otimes}\ B_2)''.

This result is established by examining the behaviour of certain projections in the relative commutant of the basic construction and is joint work with Junsheng Fang, Roger Smith and Alan Wiggins,

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- Date:
**4/7/08**- Speaker:
**Romain Tessera, Vanderbilt University** - Title:
**Haagerup property and Poincare inequalities** - Abstract: We prove that a metric space does not coarsely embed into a Hilbert space if and only if it satisfies a sequence of Poincare inequalities, which can be formulated in terms of (generalized) expanders. We also give quantitative statements, relative to the compression. In the equivariant context, our result says that a group does not have the Haagerup property if and only if it has relative property T with respect to a family of probabilities whose supports go to infinity. We give versions of this result both in terms of unitary representations, and in terms of affine isometric actions on Hilbert spaces. See arXiv:0802.2541.

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- Date:
**4/14/08**- Speaker:
**Richard Burstein, UC Berkeley** - Title:
**Subfactors and Hadamard Matrices** - Abstract:
A II_1 subfactor may be obtained from a symmetric commuting square
via iteration of the basic construction. One class of commuting squares
is obtained from generalized Hadamard matrices. The standard invariant
of such a Hadamard subfactor may be computed to any level in finite time,
but their general classification remains intractable.

I will discuss how a certain twisted tensor product of Hadamard matrices produces a subfactor of the form M^G \in (M \rtimes H). G and H generate a group K in Out(M), with an associated 3-cocycle \lambda. The principal graph may then be computed from K using the methods of Bisch and Haagerup. By considering \lambda as well, we may sometimes obtain a classification up to subfactor isomorphism.

I will give several examples, including a full classification of Hadamard subfactors of index 4.

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- Date:
**4/17/08, Mathematics Colloquium, 4:10-5:00pm, SC 5211**- Speaker:
**Yasuyuki Kawahigashi, University of Tokyo** - Title:
**Moonshine and Operator Algebras** - Abstract: ``Moonshine'' is a name for mysterious relations between elliptic modular functions and the largest finite simple group among the 26 sporadic groups, the Monster. The Moonshine vertex operator algebra is a mathematical object to understand these relations, and I will present its operator algebraic counterpart in the framework of conformal field theory.

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- Date:
**4/18/08 - 4/20/08, Shanks workshop at Vanderbilt University** - Date:
**4/21/08**- Speaker:
**Jesse Peterson, UC Berkeley** - Title:
**Group cocycles and the ring of affiliated operators** - Abstract: I will present some results (joint work with Andreas Thom) on cocycles from a group into its left regular representation and also into the ring of affiliated operators of the group von Neumann algebra. Specifically I will be interested in when a group $\Gamma$ has positive first $\ell^2$-Betti number. I will include a strong generalization of a result of L\"{u}ck and Gaboriau which states that if $\Lambda$ is a finitely generated normal subgroup of a group $\Gamma$ with $0 < \beta_1^{(2)}(\Gamma) < \infty$ then either $|\Lambda| < \infty$ or $[\Gamma : \Lambda] < \infty$. I will also include applications about the structure of $C_r^*(\Gamma)$ and as time permits.

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