Noncommutative Geometry & Operator Algebras Seminar
Fall 2002

Organizers: Dietmar Bisch and Guoliang Yu

Tuesdays 3:00 pm - 4:30 pm, 1403 Stevenson Center

• Date: 9/3/02
  • organizational meeting

• Date: 9/10/02
  • Speaker: Nick Wright, Vanderbilt University
  • Title: An Introduction to the Roe Algebra
  • Abstract: The Roe Algebra is a non-commutative "blurring" (or coarsening) of a space. Originally this blurring was based on the metric of the space, however just as there are non-metrizable topologies, there are also non-metrizable coarse structures. This talk will introduce and describe the Roe Algebra in the metric case, and in the C₀ case which generally is non-metrizable.

• Date: 9/17/02
  • Speaker: Nick Wright, Vanderbilt University
  • Title: C₀ Coarse Geometry
  • Abstract: Coarse geometry is a form of blurry geometry. The C₀ version is a "long-sighted" geometry: the further away something is, the more sharply it is seen. As such C₀ coarse geometry is intermediate between topology and bounded coarse geometry. I will discuss the definition of C₀ coarse geometry, some properties, and some applications. The original application of C₀ coarse geometry was to scalar curvature obstructions; it also has applications to the coarse Baum-Connes conjecture.

• Date: 9/24/02
  • Speaker: Nick Wright, Vanderbilt University
  • Title: C₀ Coarse Geometry, continued

• Date: 10/1/02
  • Speaker: Bruce Hughes, Vanderbilt University
  • Title: Trees, ultrametrics and noncommutative geometry
  • Abstract: I will explain how the micro-geometry of ultrametric spaces is categorically equivalent to the geometry of trees at infinity. Then I will show how ideas from noncommutative geometry can be used to study these equivalent theories. This is done by introducing the groupoid of local isometries of an ultrametric space. In certain cases, the K-theory of the C*-algebra of this groupoid can be computed from a diagram of the tree.

• Date: 10/3/02 Guest lecture in Bisch's Quantum Information Theory course, 9:35-10:50am in SC 1403
  • Speaker: Manny Knill, Los Alamos National Laboratory
  • Title: Overview of Efficient Linear Optics Quantum Computation
  • Abstract: Efficient linear optics quantum computation (eLOQC) is a scheme for optically realizing quantum computers with beam splitters (e.g. partially-silvered mirrors), single photon sources and photodetectors. That this is possible was a bit of a surprise, because linear optics was thought to do no more than compute with classical waves. I will describe the four steps needed to implement eLOQC, emphasizing the distinction between the physical systems and the logical qubits used to carry quantum information.

• Date: 10/3/02 Mathematics Colloquium at 4:10pm in SC 1431
  • Speaker: Manny Knill, Los Alamos National Laboratory
  • Title: Algebraic Methods for Quantum Noise Control
  • Abstract: To be useful, a model of information processing (computation, communication) needs to be robustly realizable using physical devices. The fundamental theorem of quantum information processing (QIP) is that QIP can in principle be realized in the presence of constant physical error rates. The unifying idea underneath most approaches for controlling quantum errors is that of a subsystem, defined as a tensor factor of a subspace of a Hilbert space. There is a close relationship between subsystems and properties of matrix algebras.
    
    I will start with a hands-on introduction to quantum computing. After a short explanation of "everything you need to know about robust realization of information", I will describe a few of the ways in which elementary algebra and representation theory are contributing toward understanding and using noisy quantum systems.

• Date: 10/8/02
  • Speaker: Bruce Williams, University of Notre Dame
  • Title: Lefschetz Fixed Point Theory
  • Abstract: In the original papers of Atiyah-Singer the index of an elliptic complex $E$ on a manifold $M^n$ was just a number. A later refinement was an index which lived in the topological K-theory of a $C^*$-algebra associated to $\pi_1(M).$ This refinement was important for geometric questions such as the Novikov conjecture and the existence of positive scalar curvature metrics. If the elliptic complex $E$ is equipped with a geometric endomorphism $\phi,$ then Atiyah-Bott-Segal studied a Lefschetz invariant $L(E,\phi)$ which is also a number. In this talk we consider a refinement $\hat L(E,\phi)$ when $E$ is the deRahm complex. This refinement lives in a group defined using a twisted version of the free loop space of $M.$ There is also a family version for fiber bundles equipped with a fiberwise endomorphism $\phi$. When dim (fibers)-dim(base) is greater than 2, then the vanishing of the family Lefschetz invariant implies $\phi$ is fiberwise homotopic to an endomorphism with no fixed points.

• Date: 10/15/02
  • Speaker: Bruce Hughes, Vanderbilt University
  • Title: Trees, ultrametrics and noncommutative geometry, continued

• Date: 10/17/02 Mathematics Colloquium at 4:10pm in SC 1431
  • Speaker: Guihua Gong, University of Puerto Rico
  • Title: C*-algebras and classification
  • Abstract: A C*-algebra is a closed selfadjoint subalgebra of the algebra of all bounded linear operators on a Hilbert space. C*-algebras can be considered as noncommutative topological spaces and have significant applications in geometry, topology and physics. In this talk, we will survey some recent developments in the classification of C*-algebras.

• Date: 10/22/02
  • no meeting (fall break)

• Date: 10/29/02 This talk will be in Garland 101, 3:10pm-4:00pm
  • Speaker: Mark Ellingham, Vanderbilt University
  • Title: The Four Color Theorem, Part 1
  • Abstract: The famous Four Color Theorem states that four colors suffice to color a planar map so that no two regions that share a nontrivial boundary are the same color. This can be reformulated as 4-coloring the vertices of a planar graph. It was first proved by Appel and Haken in the 1970's, and in the 1990's a new proof was given by Robertson, Sanders, Seymour and Thomas. We survey the techniques used in the proof. There are two main steps: discharging and reducibility. Discharging is used to find a finite set of unavoidable configurations (1476 for A&H, 633 for RSS&T) and then reducibility is used to show that any graph that contains one of them is 4-colorable. Both the discharging argument and the reducibility argument require extensive use of computers. Discharging, which is based on Euler's formula, has become an important method for graphs embedded on surfaces, and has been applied to other coloring problems and to other problems such as graph reconstruction.

• Date: 10/31/02 Mathematics Colloquium at 4:10pm in SC 1431
  • Speaker: Ron Douglas, Texas A&M University
  • Title: Multivariate Operator Theory and Complex Geometry
  • Abstract: In considering the study of multivariate operator theory on Hilbert space, a module approach is useful in bringing to bear concepts and techniques from complex and algebraic geometry. In the talk I will demonstrate instances of such applications to the study of submodules and quotient modules determined by algebraic objects. The emphasis will be on concrete examples that illustrate the general results. Holomorphic hermitian bundles along with curvature and other spectral invariants will be shown to be relevant.

• Date: 11/5/02
  • no meeting

• Date: 11/7/02 Mathematics Colloquium at 4:10pm in SC 1431
  • Speaker: John Roe, Penn State University
  • Title: Topology at infinity
  • Abstract: Algebraic topology distils structural information about a topological space from the combinatorics of the intersections of small open sets therein (the Cech construction). I will talk about a variant of topology in which one replaces the word "small" by "(uniformly) large" in the previous sentence. The resulting theory reveals topological structure behind many seemingly discrete objects.

• Date: 11/12/02
  • Speaker: Geoffrey Price, US Naval Academy
  • Title: Spin Systems and Toeplitz Matrices
  • Abstract: In the 1980's Robert Powers introduced a family of endomorphisms on the C*-algebras generated by irreducible spin systems. Powers' binary shifts have recently been classified up to conjugacy. We discuss this result and show how its proof has led to some new results on Toeplitz matrices and polynomials over finite fields.

• Date: 11/15/02 (Friday). 3:10pm-4:30pm in SC 1404
  • Speaker: James McKernan, UC Santa Barbara
  • Title: The cone of curves of M_g,n
  • Abstract: The moduli space M_g,n of n-pointed compact Riemann surfaces of genus g is perhaps one of the most intensively studied vartieties in algebraic geometry. On the other hand, by work of Mori, an interesting invariant of any variety is its cone of curves, since this controls the morphisms of this variety to other varieties.
    
    Mumford conjectured that the cone of curves of M_g,n is generated by the 1-strata, the locus of those nodal curves with 3g-2+n nodes.
    
    Gibney, Keel and Morrison proved that Mumford's conjecture follows from the case g=0, which is known as Fulton's conjecture. We review some recent progress on Fulton's conjecture.

• Date: 11/19/02 This talk will be in Garland 101, 3:00pm-4:00pm
  • Speaker: Mark Ellingham, Vanderbilt University
  • Title: The Four Color Theorem, Part 2

• Date: 11/26/02
  • no meeting (Thanksgiving break)

• Date: 12/3/02
  • Speaker: Dechao Zheng, Vanderbilt University
  • Title: Compact operators on the Bergman space via the Berezin transform
  • Abstract: A common intuition is that for operators on the Bergman space of the unit disk, "closely associated with function theory'', compactness is equivalent to having vanishing Berezin transform on the boundary of the unit circle. I will discuss what "closely associated with function theory'' means.