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every Tuesday and Thursday at 2 pm |
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Monday June 22, 2015
4:00pm
BA6183
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Ron Douglas
An Analytic Grothendieck-Riemann-Roch Theorem
The Arveson Conjecture would provide an analytic object for every
homogenous polynomial ideal. The BDF theory yields an odd K-homology
class for such an object and the correspondence between ideals and
such classes can be viewed as a generalization of the Grothendieck-Riemann-Roch
Theorem. In my talk I will discuss these matters including some recent
results of Tang, Yu and myself for ideals with smooth zero variety.
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Friday May 29, 2015
2:10pm
BA026
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James Lutely
The Generalized Goodearl Construction
Goodearl's original construction is an AH algebra that uses "diagonal"
maps in its direct limit. We will define a construction from general
RFD algebras which subsumes several constructions which have appeared
since Goodearl in the work of Villadsen, Lin and others. We will discuss
the difficulty in determining anything meaningful in the general RFD
case when using non-trivial projections. Supposing that all projections
are trivial we will prove stable rank one in full generality. Provided
that the construction uses a unique inclusion at each stage, we will
show a dichotomy which yields TAF algebras in one case and approximately
divisible algebras in the other. Hence, Z-stability and unperforation
are universal under this assumption.
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Tuesday May 5, 2015
2:10pm
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James Lutely
QD and AF algebras of higher rank graphs
A higher rank graph is a category consisting of different "coloured"
morphisms which we think of as edges which obey rigid factorization
rules. Two C*-algebras are associated to each graph, one which permutes
finite paths, and a quotient which permutes infinite paths, which
correspond to ultrafilters. Both may also be described as groupoid
algebras. Here we characterize when the first algebra is QD, and we
give necessary and sufficient conditions for each algebra to be AF.
For 2-graphs we show that the two algebras are AF simultaneously,
and assuming that the underlying 2-coloured graph is well behaved,
we characterize when they are AF.
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Thursday April 9, 2015
2:10pm
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Working Seminar
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Friday April 2, 2015
2:10pm
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Stuart White
Coloured Classification of Maps
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Monday March 30, 2015
4:00pm
Location: BA6183
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Raphael Ponge (Seoul National University)
Noncommutative geometry, equivariant cohomology, and conformal invariants
We will explain how to apply the framework of noncommutative geometry
in the setting of conformal geometry. We plan to describe three main
results. The first result is a reformulation of the local index formula
of Atiyah-Singer in conformal geometry, i.e., in the setting of the
action of a group of conformal-diffeomorphism. The second result is
the construction of new conformal invariants out of equivariant characteristic
classes. The third result is a version in conformal geometry of the
Vafa-Witten inequality for eigenvalues of Dirac operators. This is
joint work with Hang Wang (University of Adelaide).
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Wednesday March 11, 2015
2:10pm
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James Lutely
The Structure of Essential Quasidiagonal Representations
We will review some of Dadarlat's work on quasidiagonal C*-algebras.
He gave a proof that any QD C*-algebra may be embedded into one which
is a limit of RFD algebras which uses the structure of essential representations
to give more information on the limit sequence than the Blackadar
Kirchberg approach (although it applies to a smaller class of algebras).
It is also stronger in the sense that it does not require nuclearity
or even exactness, but does pass these properties as well as the UCT
to the RFD algebras when present in the original algebra. We will
show how this additional information can be used in TAF and ultimately
AF embeddings.
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Tuesday March 3, 2015
2:10pm
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Michael Hartz
Classification of multiplier algebras of Nevanlinna-Pick spaces
Nevanlinna-Pick spaces are Hilbert function spaces for which an analogue
of the Nevanlinna-Pick interpolation theorem from complex analysis
holds. We will consider their multiplier algebras, which are commutative
semi-simple non-selfadjoint operator algebras. The investigation of
the classification problem for these algebras was initiated by Davidson,
Ramsey and Shalit.
I will report on the current state of this problem and talk about
recent work which uses a somewhat different perspective on these algebras.
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Thursday February 26, 2015
2:10pm
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Emily Redelmeier
Real and Quaternionic Second-Order Freeness
Free probability (a noncommutative analogue of probability) provides
a method for studying the behaviour of large random matrices, in particular
in the several-matrix context. Second-order freeness was developed
in order to study the second-order behaviour (fluctuations around
limits) of random matrices. However, unlike in the first-order case,
real and quaternionic matrices behave differently from complex matrices.
I will focus on the behaviour of quaternionic matrices, which show
surprising behaviour rooted in the asymmetry which appears in the
in the several-matrix context due to the non-cyclic trace.
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Thursday February 19, 2015
2:10pm
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Working Seminar
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Tuesday February 17, 2015
2:10pm
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Working Seminar
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February 11, 2015
3:30pm
Stewart Library
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Alessandro Vignati
Set theory and amenable operator algebras
I will present my past and present work on logic and operator algebras.
First I will show the construction of a nonseparable amenable operator
algebra A with the property that every nonseparable subalgebra of
A is not isomorphic to a C*-algebra, yet A is an inductive limit of
algebras isomorphic to C*-algebras. Secondly, I will sketch possible
techniques, associated to Model Theory in a continuous setting, that
can be applied to operator algebras.
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February 5, 2015
2:10pm |
James Lutley
Direct Limits of RFD Algebras
We will discuss a variety of connected results about limits of residually
finite dimensional C*-algebras and compare them with Kirchberg and
Blackadar's work on generalized inductive limits.
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Tuesday January 27, 2015
2:10pm
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Working Seminar
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Friday January 23, 2015
11:00am
Stewart Library
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Martino Lupini
Operator algebras and abstract classification
We present applications of Borel complexity theory and Fraisse theory
to the study of C*-algebras, operator spaces, and their automorphisms.
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January 15, 2015
2:10pm |
James Lutley
Traces and AF Embeddings
We will revisit Lin's construction of an AF embedding from a sequence
of RFD algebras. We will discuss the freedom we have in choosing an
embedding and which traces may be extended to an AF algebra.
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December 18, 2014
2:10pm |
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.
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December 2, 2014
2:10pm
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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.
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November 26, 2014
11:00am
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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.
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November 18, 2014
2:10pm
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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.
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November 12, 2014
11:00am
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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.
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October 16, 2014
2:10pm
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.
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October 14, 2014
2:10pm |
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.
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September 30, 2014
2:10pm |
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.
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September 29, 2014
11:00am-1:00pm
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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.
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September 25, 2014
2:10pm |
Juno Jung |
September 23, 2014
2:10pm |
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.
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September 22, 2014
11:00am-1:00pm
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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.
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August 21, 2014
2:10pm
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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.
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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.
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August 19, 2014
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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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