Fall 2003

**Organizer: Dietmar Bisch **

**Tuesdays, 3:00pm-4:30pm in SC 1420**

- Date:
**9/18/03 (Thursday) Mathematics Colloquium at 4:10pm in SC 1206**- Speaker:
**Vaughan Jones, UC Berkeley** - Title:
**In and Around the Origin of Quantum Groups** - Abstract: Quantum groups were invented largely to provide solutions of the Yang-Baxter equation and hence solvable models in 2-dimensional statistical mechanics and one-dimensional quantum mechanics. They have been hugely successful. But not all Yang-Baxter solutions fit into the framework of quantum groups. We shall explain how other mathematical structures, especially subfactors, provide a language and examples for solvable models. The prevalence of the Connes tensor product of Hilbert spaces over von Neumann algebras leads us to speculate concerning its potential role in describing entangled or interacting quantum systems.

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- Date:
**Saturday & Sunday, 9/20/03 & 9/21/03, SC 4309**

**First East Coast Operator Algebras Symposium (at Vanderbilt University)** - Date:
**9/23/03**- Speaker:
**Narutaka Ozawa, University of Tokyo (visiting UC Berkeley)** - Title:
**On solid von Neumann algebras**

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- Date:
**9/30/03**- Speaker:
**Doug Drinen, University of the South** - Title:
**An introduction to C*-algebras of directed graphs** - Abstract:
In 1980, Cuntz and Krieger introduced a class of C*-algebras
generated by a collection of partial isometries satisfying certain
conditions determined by a square 0-1 matrix. In the mid-90s,
Kumjian, Pask, Raeburn, and Renault defined, given a directed
graph, a C*-algebra generated by collections of partial isometries
and projections satisfying relations arising from the structure of
the graph. They did so in a way that generalized the Cuntz-Krieger
algebras. Further, they proved
that certain properties of the C*-algebra could be deduced from the
structure of the graph. This made it possible to rephrase theorems
about Cuntz-Krieger algebras, the statements of which had been
quite complicated in some cases, in much more friendly graphical
terms.

This talk will cover in very broad general terms the past, present, and future of graph algebras. We will start with Cuntz algebras, move on to graph algebras, and then touch upon the many ways in which graph algebras have been generalized.

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- Date:
**10/7/03**- Speaker:
**Dietmar Bisch, Vanderbilt University** - Title:
**Exotic subfactors I** - Abstract:
A subfactor is a group-like object which can encode what one might call
the ``generalized symmetries'' of a quantum physical or mathematical
situation, from which the subfactor was constructed, much like a group does.
However a subfactor is a highly noncommutative, infinite dimensional
mathematical object and the symmetries it captures are more general than
group symmetries. Operator algebraic methods, which are both analytical
and algebraic-combinatorial in nature, can then be used to analyze these
symmetries.

As in group theory there are many exotic subfactors some of which are not very well understood. Planar algebra techniques shed some light on the structure of theses subfactors.

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- Date:
**10/14/03**- Speaker:
**Dietmar Bisch, Vanderbilt University** - Title:
**Exotic subfactors II**

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- Date:
**10/21/03****no meeting, fall break**

- Date:
**10/28/03**- Speaker:
**Dechao Zheng, Vanderbilt University** - Title:
**Toeplitz algebras on the disk** - Abstract: Let $B$ be a Douglas algebra and let $\cal B$ be the algebra on the disk generated by the harmonic extensions of the functions in $B$. In this talk I will show that $\cal B$ is generated by $H^{\infty}(D)$ and the complex conjugates of the harmonic extensions of the interpolating Blaschke products invertible in $B$. Every element $S$ in the Toeplitz algebra ${\cal T}_{B}$ generated by Toeplitz operators with symbols in $\cal B$ on the Bergman space has a canonical decomposition $S=T_{\tilde{S}}+\R$ for some $\R$ in the commutator ideal ${\mathcal C}_{{\cal T}_{B}}$; and $S$ is in ${\mathcal C}_{{\cal T}_{B}}$ iff the Berezin transform $\tilde{S}$ vanishes identically on the union of the maximal ideal space of the Douglas algebra $B$ and the set ${\cal M}_{1}$ of trivial Gleason parts. This is a joint work with S. Axler.

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- Date:
**10/31/03 (Friday) Special Talk, 3:00pm-4:00pm, SC 1313**- Speaker:
**Adebisi Agboola, UCSB** - Title:
**Rational points on elliptic curves** - Abstract:
Mathematicians have been interested in finding integer solutions to
polynomial equations in several variables for thousands of years e.g.
Euclid gave a complete solution to the equation x^2 + y^2 = z^2. It is now
known that, (thanks to the negative resolution of Hilbert's
10th Problem due to Matiyasevich (building on earlier work of Davis,
Putnam and Robinson)), it is impossible to do this sort of thing in
general.

In special cases, however, one can nevertheless hope to say a great deal. For instance, when the equation in question defines an elliptic curve, e.g.

y^2 = x^3 + 19x

then an amazing conjecture due to Birch and Swinnerton-Dyer implies that the behaviour of the solutions is governed by the properties of an analytic object, namely the L-function associated to the elliptic curve. The mere existence of this L-function is itself an extremely deep problem which was only recently resolved by work of Wiles, Taylor, Diamond, Conrad, and Breuil.

In my talk I shall explain some of the ideas that go into formulating the conjecture of Birch and Swinnerton-Dyer, and I shall describe some aspects of our current state of knowledge. I shall not assume any previous knowledge of the subject.

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- Date:
**11/6/03 (Thursday) Mathematics Colloquium at 4:10pm in SC 1206**- Speaker:
**Nicolas Monod, University of Chicago** - Title:
**Orbit Equivalence Rigidity** - Abstract: Geometric group theory leads naturally to the notion of measure equivalence. This concept generalizes the classical orbit equivalence studied in ergodic theory. I will introduce these topics and present new superrigidity statements. We are in particular interested in "negatively curved groups", e.g. hyperbolic in Gromov's sense. This illustrates a new approach initiated in collaboration with Y. Shalom.

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- Date:
**11/20/03 (Thursday) Mathematics Colloquium at 4:10pm in SC 1206**- Speaker:
**Nigel Higson, Penn State University** - Title:
**Index Theorems and Noncommutative Geometry** - Abstract: This talk will be aimed at graduate students and others seeking an introductory view of an important topic in noncommutative geometry. I will discuss the Fredholm index theory of elliptic differential operators and some of the basic tools of noncommutative geometry - cyclic cocycles and groupoid algebras - which can play a role in the formulation and proof of the Atiyah-Singer index theorem.

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- Date:
**11/25/03****no meeting, Thanksgiving break**

- Date:
**12/3/03 (Wednesday) Topology & Group Theory Seminar, 4:00pm-5:00pm, SC 1403**- Speaker:
**Guoliang Yu, Vanderbilt University** - Title:
**Novikov type conjectures and uniform convexity** - Abstract: I will explain why Novikov type conjectures are interesting and how the geometric condition of uniform convexity can be used to study these conjectures. This talk should be accessible to nonexperts. This is joint work with Gennadi Kasparov.

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