Spring 2006

**Organizers: Dietmar Bisch, Guoliang Yu**

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

- Date:
**1/17/06****no meeting due to special colloquium**

- Date:
**1/24/06**- Speaker:
**Daoxing Xia, Vanderbilt University** - Title:
**Determinant formula for trace class perturbations of Heisenberg commutation relations**

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- Date:
**1/31/06**- Speaker:
**Marcin Jankiewicz, Vanderbilt University (Physics)** - Title:
**Extensions of Monster Moonshine to c=24k Conformal Field Theories** - Abstract: We present a family of conformal field theories (or candidates for CFTs) that is build on extremal partition functions. Spectra of these theories can be decomposed into the irreducible representations of the Fischer-Griess Monster sporadic group. Interesting periodicities in the coefficients of extremal partition functions are observed and interpreted as a possible extension of Monster moonshine to c=24k holomorphic field theories.

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- Date:
**2/2/06 (Thursday), Mathematics Colloquium, 4:10-5:00pm in SC 5211**- Speaker:
**Yehuda Shalom, Tel Aviv University (visiting Princeton)** - Title:
**Analytic methods in group theory** - Abstract: We shall try to explain how analytic tools, involving spectral analysis, ergodic theory and probability on groups, combine together to yield purely algebraic properties of some interesting classes of groups. We shall also use the main result, which is entirely elementary in its statement, as a motivation to discuss some aspects of the fundamental notions of amenability and property (T), assuming no prior familiarity. The talk should be accessible to every graduate student.

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- Date:
**2/3/06 (Friday), Special Seminar, SC 1432, 4:10-5:00pm**- Speaker:
**Yehuda Shalom, Tel Aviv University (visiting Princeton)** - Title:
**Factor and normal subgroups theorems for lattices in products of groups** - Abstract: I think it would be good to concentrate on one of the main results which I will describe in the colloquium talk (February 2). It will be an independent self contained talk, yet will benefit from the "big picture" motivation presented in the colloquium talk.

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- Date:
**2/7/06**- Speaker:
**Liangqing Li, University of Puerto Rico (visiting Vanderbilt)** - Title:
**Reduction from dimension three to dimension two for local spectra of simple AH algebras** - Abstract: In the classification of simple AH algebras, an important step is the reduction of thedimension of local spectra of the algebras to dimension three. In this talk I will present a reduction theorem to reduce the dimension of local spectra from three to two, using sub-homogeneous algebras. This reduction can be used to unify the classification of simple AH algebras (due to Elliott-Gong-Li) and the classification of inductive limit algebras of matrix algebras over circles with dimension drops (due to Thomsen).

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- Date:
**2/14/06**- Speaker:
**Dmitri Nikshych, University of New Hampshire** - Title:
**A rigidity theorem for semisimple tensor categories** - Abstract: This talk is based on a joint work with Pavel Etingof and Viktor Ostrik. Tensor categories arise as "non-commutative symmetries" in many areas of mathematics and physics -- conformal field theory, operator algebras, representation theory of quantum groups, and low-dimensional topology. In this talk I will describe the foundations of the algebraic theory of semisimple tensor categories and will prove that such categories and functors between them do not admit non-trivial deformations. In particular, the number of tensor categories realizing a given set of fusion rules is finite. I will also introduce the notion of a Frobenius-Perron dimension of an object in tensor category and discuss its arithmetic properties and applications to classification problems.

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- Date:
**2/21/06**- Speaker:
**Guihua Gong, University of Puerto Rico (visiting Vanderbilt)** - Title:
**AH algebras with ideal property** - Abstract: In this talk, I will discuss a class of AH algebras which including simple AH algebras and real rank zero AH algebras as subclasses. We will give a reduction theorem for this class of C*-algebras, this is a joint work with Cornel Pasnicu and Liangqing Li.

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- Date:
**2/28/06**- Speaker:
**Teodor Banica, Universite Paul Sabatier (Toulouse)** - Title:
**Spectral measures of small index graphs** - Abstract: This is a report on joint work with Dietmar Bisch. Inspired from deep results in subfactor theory - ADE classification, planar algebras, annular structure - we associate to any bipartite graph of norm \leq 2 a probability measure supported on the unit circle. We compute this measure for ADE graphs, and we get remarkably simple formulae. We also show that there is a purely analytic approach to this construction by using the equation \Delta = U+U^{-1}, where \Delta is the discrete Laplacian.

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- Date:
**3/7/06****no meeting (spring break)**

- Date:
**3/14/06**- Speaker:
**Teodor Banica, Universite Paul Sabatier (Toulouse)** - Title:
**Integration over free quantum groups** - Abstract: This is a report on joint work with Benoit Collins. It is known that the spectral measures of characters of the free quantum groups A_0,A_u,A_s are semicircular, circular, and free Poisson. These can be regarded as order 0 results, and our purpose is to get now into order 1 problems. The main result so far is a formulation of the problem in terms of meander determinants of Di Francesco et al., with a nice application to A_s(4).

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- Date:
**3/16/06 (Thursday), Mathematics Colloquium, 4:10-5:00pm in SC 5211**- Speaker:
**Richard Kadison, University of Pennsylvania** - Title:
**Re-examining the Pythagorean Theorem - A Functional Analyst's View** - Abstract:
In the colloquium talk we'll study some surprising
twists and connections with operator algebras, symmetric
spaces, Schur's work - on reviewing the basic theorem. We'll
start at the beginning.

In the second lecture, we'll study the Schur-Horn extension and Pythagoras in II_{1}factors, as well as connections with the work of Kostant, Atiyah, and Guillemin-Sternberg. This may be a bit more technical.

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- Date:
**3/17/06 (Friday), Special NCGOA Seminar, 3:10-4:00pm in SC 1431**- Speaker:
**Richard Kadison, University of Pennsylvania** - Title:
**Re-examining the Pythagorean Theorem - A Functional Analyst's View, Part II** - Abstract:
In the colloquium talk we'll study some surprising
twists and connections with operator algebras, symmetric
spaces, Schur's work - on reviewing the basic theorem. We'll
start at the beginning.

In the second lecture, we'll study the Schur-Horn extension and Pythagoras in II_{1}factors, as well as connections with the work of Kostant, Atiyah, and Guillemin-Sternberg. This may be a bit more technical.

- Speaker:
- Date:
**3/21/06**- Speaker:
**Guihua Gong, University of Puerto Rico (visiting Vanderbilt)** - Title:
**Classification of simple AH algebras: why K group together with tracial space is enough?** - Abstract: In this talk, I will explain why ordered K-group together with space of tracial states is enough to classify sipmple AH algebras by discussing a special case: simple inductive limits of matrix algebras over interval (which is due to G. Elliott).

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- Date:
**3/23/06 (Thursday), Mathematics Colloquium, 4:10-5:00pm in SC 5211**- Speaker:
**Jonathan Rosenberg, University of Maryland** - Title:
**An analogue of the Novikov Conjecture in complex algebraic geometry** - Abstract: We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain "higher Todd genera" are birational invariants, and implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). The conjecture is, in a certain sense, best possible, and unlike the usual Novikov Conjecture, it is already known to be true in all cases, though some variants are still open. An interesting biproduct of this work is a curious analogy between the homotopy category of smooth manifolds and the birational category of smooth projective varieties.

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- Date:
**3/28/06**- Speaker:
**Remus Nicoara, Vanderbilt University** - Title:
**A continuous family of non-isomorphic, irreducible, hyperfinite subfactors with the same standard invariant** - Abstract: We construct a 1-parameter family of irreducible subfactors of the hyperfinite II$_1$ factor, which are non-isomorphic, have Jones index 6 and have all the same standard invariant. This is joint work with Dietmar Bisch and Sorin Popa.

*********** Starting this week the seminar will meet at 4:10pm *********** - Speaker:
- Date:
**4/4/06**- Speaker:
**Rongwei Yang, SUNY at Albany** - Title:
**Functional spectrum of contractions** - Abstract: The idea of functional spectrum associates a closed subset of the Hardy space with a contractive operator. In generic cases, there is a cononical imbedding of the ordinary spectrum into the functional spectrum. Functional spectrum has much richer structure and they enable a finer analysis of contractive operators. In this talk, we will see some elementary facts and examples.

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- Date:
**4/11/06**- no meeting

- Date:
**4/18/06**- Speaker:
**Guoliang Yu, Vanderbilt University** - Title:
**The coarse geometric Novikov conjecture for a class of expanders** - Abstract: I will first explain what is the coarse geometric Novikov conjecture and why it is interesting. I will then outline a proof of the conjecture for a class of expanders. This is joint work with G. Gong and Q. Wang.

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- Date:
**4/25/06**- Speaker:
**Kunyu Guo, Fudan University** - Title:
**Essentially normal Hilbert modules and K-homology** - Abstract: This talk mainly concerns homogeneous and quasi-homogeneous submodules of essentially normal Hilbert modules. When the submodules are essentially normal, their spectrum, essential spectrum are described by zero varieties of the submodules. In dimensions $d=2,\,\,3$, the $C^*$-extensions determined by the corresponding quotient modules give nontrivial information of algebraic varieties--$K$-homology invariant. In dimension $d=2$, and in the case of finite multiplicity, it is proved that each homogeneous submodule is $p$-essentially normal for $p>2$. In particular, the \textbf{Arveson's Conjecture} is true when $d=2$. The talk also will describe recent progress in $p$-essential normality of submodules of Hilbert modules in general. This is a joint work with K.Wang.

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