Fall 2005

**Organizers: Dietmar Bisch, Guoliang Yu**

**Tuesdays, 4:00pm-5:00pm in SC 1432**

- Date:
**9/6/05****Organizational meeting for NCGOA and RTG seminars**

- Date:
**9/22/05 (Thursday), Mathematics Colloquium, 4:10-5:00pm in SC 5211**- Speaker:
**Paula Cohen, Texas A&M University**Title:**Subvarieties of Shimura varieties, special values of classical functions, and monodromy** - Abstract: Inspired by Hilbert's 7th problem, Siegel (1932) and Schneider (1937) obtained the first significant results about the transcendence of periods of doubly periodic functions and special values of modular functions. In particular, they found a relation to the class fields occurring in algebraic number theory. Siegel formulated similar problems for G-functions, a special case of which is the classical hypergeometric function. The modern development of this circle of ideas was made possible by the work of Alan Baker and its outgrowth. In our lecture we focus on recent results, for example, the characterization of points at which hypergeometric functions take algebraic values. We describe the surprising role, first noticed by Wolfart, played by non-arithmetic monodromy groups acting on the complex ball. We also show how such problems are related to questions on subvarieties of Shimura varieties. The lecture will be self-contained and accessible to a general audience.

- Speaker:
- Date:
**9/23/05 (Friday), Special NCGOA Seminar, 3:00-4:00pm in SC 1432**- Speaker:
**Paula Cohen, Texas A&M University** - Title:
**Non-commutative boundaries of modular curves and varieties** - Abstract: In this talk we put forward some ideas on how recent work of Manin-Marcolli on non-commutative boundaries (or compactifications) of modular curves might be extended to more general curves or to higher dimension. No background in Number Theory is required to follow the talk.

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- Date:
**9/27/05**- Speaker:
**Brett Wick, Vanderbilt University** - Title:
**A New Proof of the Carleson Embedding Theorem** - Abstract: In this talk, we will give a new proof of Carleson's Embedding Theorem. The proof gives the best known constant in the theorem. This is joint work with S. Petermichl and S. Treil.

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- Date:
**10/7/05 (Fr), 10/8/05 (Sat) and 10/9/05 (Sun), Special Lecture Series (see below for time and place)**- Speaker:
**Yan Soibelman, Kansas State University** - Title:
**Commutative and noncommutative geometry of mirror symmetry** - Abstract:
We will explain the approach to Mirror Symmetry suggested
in a joint
project with Maxim Kontsevich. The aim is to explain the phenomenon of
Mirror Symmetry in terms of homological algebra and non-commutative
geometry. Discovered by physicists as a duality on a certain class of
string theories, Mirror Symmetry turned out to be related to many deep
questions of algebraic and symplectic geometry, algebra, number theory
and differential equations. Non-commutative geometry provides an
appropriate framework for study of what is called "D-branes" in the
String Theory.

It combines physical idea of degenerating Conformal Field Theories with mathematical idea of Gromov-Hausdorff collapse of Calabi-Yau manifolds, as well as with unexpected relation to rigid analytic geometry. We suggest to view a given Conformal Field Theory as a kind of non-commutative space. Such non-commutative spaces can degenerate "at infinity". Mirror symmetry can be explained in terms of the residual commutative geometry. On the algebraic side we will meet homotopy categories associated with compact symplectic manifolds. I am going to explain non-commutative formal geometry of those homotopy categories. There is another kind of non-commutative geometry of Mirror Symmetry. It is geometry of deformed Calabi-Yau manifolds (a kind of deformation quantization). I plan to discuss the way to construct such spaces starting with real manifolds equipped with an integral affine structure. This part of my lectures is also related to the so-called "tropical geometry".

The lectures will be accessible to graduate students. Schedule:

**Friday, October 07, 2005**

4:00 pm - 6:00 pm BUTTRICK 201

**Saturday, October 08, 2005**

10:00 am - 12:30 pm BUTTRICK 201

**Sunday, October 09, 2005**

10:00 am - 12:30 pm BUTTRICK 201

**Organizer and contact: Mark Sapir**

- Speaker:
- Date:
**10/11/05**- Speaker:
**Guihua Gong, University of Puerto Rico (visiting Vanderbilt)** - Title:
**Invariants and Classification of C*-algebras** - Abstract: In this talk, I will discuss K-theory invariants of C*-algebras and some basic ideas in classification theory.

- Speaker:
- Date:
**10/13/05 (Thursday), Mathematics Colloquium, 4:10-5:00pm in SC 5211**- Speaker:
**William Johnson, Texas A&M University** - Title:
**A survey of non linear Banach space theory** - Abstract:
There are three reasonable notions of geometric equivalence for metric spaces: Lipschitz equivalence, uniform equivalence, and Gromov's notion of coarse equivalence (which has recently attracted interest from geometric analysts because of its relation to the Novikov and Baum-Connes conjectures). I'll survey what is known about these types of equivalences when at least one of the metric spaces is a Banach space. When both spaces are Banach spaces, a fundamental question is: When does the existence of one of these non linear equivalences between the spaces imply the existence of a linear equivalance (i.e., an isomorphism)? If there is time I'll also discuss the recently introduced concept of Lipschitz quotient maps, which are closely related to the non collapsing maps studied by David and Semmes.

The talk is suitable for graduate students as well as faculty.

- Speaker:
- Date:
**10/18/05**- Speaker:
**Guihua Gong, University of Puerto Rico (visiting Vanderbilt)** - Title:
**Invariants and Classification of C*-algebras, Part II**

- Speaker:
- Date:
**10/25/05****no meeting, fall break**

- Date:
**10/31/05 (Monday), Special NCGOA Seminar, 4:10pm-5:00pm in SC 1307**- Speaker:
**Paul Baum, Penn State University** - Title:
**The Extended Quotient** - Abstract: Let G be a finite group acting on a topological space X. Then we can form the quotient space X/G. This talk presents a different kind of quotient space which will be referred to as the extended quotient. Applications are to equivariant Chern character and to the representation theory of p-adic groups. This talk is intended for non-experts and should be accessible to graduate students.

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- Date:
**11/1/05**- Speaker:
**Magdalena Musat, University of Memphis** - Title:
**Boundedness of noncommutative martingale transforms and an approximation result for hyperfinite martingales** - Abstract:
Inspired by the classical theory, noncommutative probability is
motivated by quantum physics.
Gilles Pisier and Quanhua Xu proved that, under certain conditions
on the filtration, a noncommutative martingale can be transferred
to a commutative vector-valued martingale for which classical
theory applies.

We will show that in the setting of a hyperfinite von Neumann algebra, an $L_p$-martingale can be approximated in the p-norm ($1 < p < \infty$) by martingales with respect to finite dimensional filtrations, for which the argument of Pisier and Xu applies. The proof relies on a perturbation argument using approximate matrix units techniques. We use this result to study the operator space UMD property, introduced in this context by Pisier, and establish connections with the Banach space property.

Click here for a pdf version of the abstract.

- Speaker:
- Date:
**11/8/05**- Speaker:
**Dechao Zheng, Vanderbilt University** - Title:
**Quantum Douglas Algebras** - Abstract:
A Douglas algebra $B$ is a closed subalgebra of $L^{\infty}$
containing $H^{\infty}$. Let $\mathcal B$ be the algebra on the disk
generated by the harmonic extensions of the functions in $B$. The
``quantum Douglas algebra" is the Toeplitz algebra
generated by Toeplitz operators (on the weighted Bergman
space) with symbols in $\mathcal B$.

In this talk I will discuss some properties of quantum Douglas algebras.

- Speaker:
- Date:
**11/15/05**- Speaker:
**Qin Wang, Dong Hua University (visiting Vanderbilt)** - Title:
**Ideal Structure of the Roe Algebras** - Abstract: The Roe algebras are $C^*$-algebras associated to metric spaces. These algebras have important applications to geometry, topology and analysis, due to the fact that their $K$-theory groups are receptacles of higher indices of elliptic differential operators on noncompact spaces. In this talk, I will discuss ideal structure of the Roe algebras. If a metric space satisfies a certain nice condition, i.e. Yu's property A, then any ideal of the Roe algebra can be geometrically constructed by a bunch of subspaces of the metric space. In general, however, this is not the case. I will discuss a counterexample as well. This talk is based on joint work with Xiaoman Chen.

- Speaker:
- Date:
**11/22/05****no meeting, Thanksgiving break**

- Date:
**11/29/05**- Speaker:
**Lin Shan, Vanderbilt University** - Title:
**An equivariant index theory and non positively curved space** - Abstract: In this talk we define an equivariant version of index theory from the K-homology of the space X to the K-theory of the group invariant Roe algebra C*(X)^Gamma, and prove that the equivariant index map is injective when the manifold is non positively curved.

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- Date:
**12/2/05 (Fr) and 12/3/05 (Sat), Special Lecture Series (see below for time and place)**- Speaker:
**Jean Bellissard, Georgia Tech** - Title:
**Informal Talks about the Quantum Hall Effect and its Mathematical Aspects** - Abstract:
The first lecture will review the physics of the integer and of
the fractional quantum Hall effect. In the second, the Noncommutative
Geometry framework will be described together with the main theorem
concerning the Integer Quantum Hall Effect. If the speaker is still alive,
the third lecture will be a discussion of the fractional effect which, to
a large extent, still lacks a first principle description.

Warning: The three lectures will be based on the 1994 paper on Noncommutative Geometry of the Quantum Hall effect. However open discussions will be welcome and the emphasis might be changed depending on the interest of the participants. A sample of these lectures can be found on http://www.math.gatech.edu/~jeanbel/talksjbE.html (look at either May 3-6,2005 or August 16-20 1999)

**Schedule:**

**Friday, December 2, 2005**

2:10 pm - 3:30 pm in SC 1312

**Saturday, December 3, 2005**

11:00 am - 12:30 pm in SC 1432

2:30 pm - 4:00 pm in SC 1432

**Organizer and contact: Dietmar Bisch**

- Speaker:
- Date:
**12/6/05**- Speaker:
**Tavan Trent, University of Alabama** - Title:
**Maximal Ideals in Algebras of Functions** - Abstract: We give a brief survey of ``corona problems'' for various algebras of functions. Along the way, several open problems are mentioned. We conclude by discussing some recent work on one of these problems, the corona problem for H$^\infty(D^n)$.

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