December 22, 2014

Operator Algebras Seminars
July 2014 - June 2015

Seminars are generally held every Tuesday and Thursday at 2:10 pm in Room 210.
Hosted by the Fields Institute
Fr For more information about this program please contact
George Elliott
Upcoming Seminars: every Tuesday and Thursday at 2 pm Room 210
December 18, 2014

David Barmherzig
Functional Map Methods in Computational Geometry

Non-rigid shape matching is an important problem in computational geometry that has many emerging applications in computer graphics, computer vision, and image processing. In 2012, Guibas et al. introduced functional map methods as a new powerful tool for shape matching and many other related problems. This introduction of functional maps has also introduced many interesting mathematical questions which are currently being researched.

Past Seminars
December 2, 2014

Anatoly Vershik
Notion of the standardness in the theory of AF-algebras, and iterations of Kantorovich transport metric

The notion of the standardness came from the theory of filtration (=decreasing sequences) of sigma-fields in 70-th. In the theory of AF-algebras we also have the natural filtration in the space of paths of Bratelli diagrams ---"tail-filtration" (the sets of n-th sigma-filed consist with the classes of paths up to first n levels of the path.) So by definition the standard $AF C^*$ -algebra is algebra with standard tail filtration.

The criteria of standardness uses so called iteration of Kantorovich metric/ which leads to the notion of intrinsic metric on the paths. More exactly, the criteria reduces to the uniform compactness of the levels of graph. The following theorem shows the role of the standardness

The set of the indecomposable traces of standard AF-algebra is a compactification under the intrinsic semimetric n the space of paths.

November 26, 2014

Alessandro Vignati
The total reduction property

I will give an explanation of the results of Gifford about the Total Reduction property, a property shared by all amenable operator algebras, relevant to the purpose of attacking the isomorphism problem. References can be found here.

November 18, 2014

James Lutley
Inner quasidiagonality and AF Embeddings

We isolate the obstruction to AF embeddings of nuclear UCT QD algebras via the path we have previously discussed and propose a solution to this problem.

November 12, 2014

Alessandro Vignati
Abelian amenable operator algebras are isomorphic to C*-algebra

I will present the proof of Marcoux and Popov that amenable abelian operator algebras are isomorphic to a C*-algebras.

October 16, 2014
Stewart Library

Robin Deeley
Correspondences for Smale spaces

We discuss joint work with Brady Killough and Michael Whittaker. This work centers around the functorial properties of the homology for Smale spaces introduced by Ian Putnam. In the case of a shift of finite type this homology theory is Krieger's dimension group; this case will be discussed in detail.

The fundamental object of study are correspondences between Smale spaces; the precise definition will be given in the talk. However, the idea is to encode both types of functorial properties of Smale spaces (with respect to Putnam's homology theory) into a single object. No knowledge of Smale spaces or Putnam's homology is required for the talk.

October 14, 2014

Robin Deeley
Relative constructions in geometric K-homology

The Baum-Douglas model realizes K-homology using geometric cycles (i.e., via data from manifold theory). The isomorphism between this realization of K-homology and analytic definition of Kasparov naturally leads to a proof of the Atiyah-Singer index theorem.

We discuss the process of proving other index theorems using the framework of "relative geometry K-homology". Example of such theorems include the Freed-Melrose index theorem and theorems from R/Z-valued index theory.

September 30, 2014

Magdalena Georgescu
Characterization of spectral flow in a type II factor

I will start with an introduction to spectral flow. In B(H) (the set of bounded operators on a Hilbert space), the spectral flow counts the net number of eigenvalues which change sign as one travels along a path of self-adjoint Fredholm operators. It is possible to generalize the concept of spectal flow to a semifinite von Neumann algebra, as we can use a trace on the algebra to measure the amount of spectrum which changes sign. I will give a characterization of spectral flow in a type II factor, and a sketch of the proof.

September 29, 2014

Alessandro Vignati
On what I learnt in Glasgow about the UCT (Part 2)

After the reviewing on the basic definitions related to TAF algebras, I will give a proof of the fact that, in the TAF simple case, homotopic *-homomorphisms are necessarily approximately unitarily equivalent. If there is a time I will sketch an idea of how to prove that KK-equivalent *-homs are approximately unitarily equivalent, passing through the mod n KK-theory.

September 25, 2014
Juno Jung
September 23, 2014
Jeffery Im
Finite nuclear dimension from Z-stability in the unique trace case

We report on a proof of the main result in a paper of Sato, White, and Winter which establishes finite nuclear dimension from Z-stability for the usual Toms-Winter C*-algebras with a unique tracial state. A key idea here, which also appeared in an earlier paper of Matui and Sato, is the two-colored approximately inner flip. The end result is the estimate dr(A) < 4 -obtained by using UHF-stable approximations twice, with each approximately inner flip contributing two colors.

September 22, 2014

Alessandro Vignati
On what I learnt in Glasgow about the UCT (Part 1)

I will present an introduction on the UCT and KK-theory and a collection of results related to the UCT in the TAF case. This seminar will follow the lectures of Marius Dadarlat and Christian Voigt in a recent Masterclass in Glasgow.

August 21, 2014

Jeffrey Im,
Nuclear dimension and Z-stability

We will sketch the original construction of the Jiang-Su algebra and discuss some of its properties. Apart from serving as the first (and only) example of its kind, understanding its role in K- theoretic classification of simple separable nuclear of C*-algebras has been of great interest in recent years. A number of results indicate that the largest class of C*-algebras for which Elliott's classification conjecture can hold is for those which are also Z- stable. Various disparate regularity properties, each of which is helpful for classification, arise as a consequence of being Z-stable. A well-known conjecture characterizing manifestations of Z-stability in a certain class of C*-algebras is the Toms-Winter conjecture. We discuss one of the properties, finite nuclear dimension, and progress on the conjecture.

August 21, 2014
3:15 pm

Juno Jung,
The necessity of quasidiagonality and unique tracial states

On Matui and Sato's paper, titled "Decomposition rank of UHF- absorbing C*-algebras'", proves that if A is a unital separable simple C*-algebra with a unique tracial state, then if A is nuclear, quasidiagonal and has strict comparison or projections, then the decomposition rank of A is at most 3. As a corollary, a partial affirmation of the Toms and Winters conjecture can be proven. In this talk, we focus on how quasidiagonality and having a unique tracial state are necessary conditions for this theorem.

August 19, 2014

James Lutley, University of Toronto
AF algebras of higher rank graphs
We will introduce sufficient and necessary conditions to ensure that the algebras associated with a higher rank graph are AF. These algebras are the Toeplitz extension, which has a representation on finite paths, and the main algebra, represented on infinite paths, which correspond to ultrafilters of the finite path space. We will discuss what obstructions exist, and whether or not the quotient can be AF when the Toeplitz extension is not.






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