Réunion d'été SMC 2015
Université du Prince Edward Island, 5 - 8 juin 2015
- MAN-DUEN CHOI, University of Toronto
How I could think of tensor products of matrices [PDF]
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In all years, I have mathematical dreams on completely positive linear maps, concerning tensor products of complex matrices. Suddenly, I wandered into the quantized world of fantasies and controversies. To release myself from Quantum Entanglements and the Principle of Locality, I need to seek the meaning of physics and the value of metaphysics. Conclusion: I THINK, THEREFORE I AM a pure mathematician.
- MAGDALENA GEORGESCU, University of Toronto
Spectral Flow and C*-algebras [PDF]
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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; in other words, spectral flow measures a change in the spectrum of the operators. The beginning of the talk will make precise the definition of spectral flow in this context, its properties and its connections to K-theory and non-commutative geometry. I will conclude the talk with a discussion of spectral flow in the context of a unital C*-algebra with a norm-closed 2-sided ideal.
- MICHEL HILSUM, Paris 7 / Jussieu
- CRISTIAN IVANESCU, University of Alberta and MacEwan University
The Cuntz semigroup of tensor product of C*-algebras [PDF]
- CRISTIAN IVANESCU, University of Alberta and MacEwan University
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In my joint work with Dan Kucerovsky, we study how the Cuntz semigroup of the
tensor product, $A \otimes B$, relates to the Cuntz semigroup of A and the Cuntz semigroup of B. The
case A = B turns out to be already very interesting. A survey of our results will be presented.
- DAN KUCEROVSKY, University of New Brunswick at Fredericton
K-theory of C*-bialgebras [PDF]
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We report on extending the Elliott classification program to certain classes of C*-bialgebras.
- GORDON MACDONALD, University of Prince Edward Island
Faster Matrix Multiplication [PDF]
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In 1969, Volker Strassen came up with an algorithm for multiplying two $2 \times 2 $ matrices using only 7 multiplications (instead of the usual 8). Using block matrices, this allows us to multiply two $n \times n$ matrices in $n^{2.81}$ multiplications. Subsequent improvements by Coppersmith and Winograd, Cohn and Umans, Stothers, and others have reduced this to $O(n^{2.38})$ multiplications, but these techniques only provide advantage for extremely large matrices.
We present some common operator-theoretic frameworks for all these results, and discuss some new results for small matrices.
- PING WONG NG, Lousiana
- SUTANU ROY, Department of Mathematics and Statistics, University of Ottawa
Slices of braided multiplicative unitaries. [PDF]
- SUTANU ROY, Department of Mathematics and Statistics, University of Ottawa
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Braided multiplicative unitaries naturally arise in the theory of semidirect product of quantum groups. Roughly, it is a unitary operator acting on two fold tensor product of Hilbert spaces with certain properties. In this talk, we show that, under certain conditions, slices of braided multiplicative unitaries generate $C^*$-algebras. This is one of the key result to study braided ($C^*$-algebraic) quantum groups using braided multiplicative unitaries as a fundamental object, following the axiomatisation of ($C^*$-algebraic) quantum groups by Baaj and Skandalis, and Woronowicz. This is a joint work with S.L. Woronowicz.
- AYDIN SARRAF, University of New Brunswick
On the classification of inductive limits of certain real circle algebras [PDF]
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In this talk, I give a classification theorem for simple unital real $C^{\ast}$-algebras that are inductive limits of certain real circle algebras. This is an attempt to provide a classification theorem similar to the well-known classification theorem of simple unital complex AT-algebras but for real $C^{\ast}$-algebras.
- ANA SAVU, University of Alberta
Spectral gap of a class of unbounded, positive-definite operators [PDF]
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The spectra of Toeplitz operators is well understood. We
use the properties of the spectra of Toeplitz operators to understand
the spectra of a class of unbounded, positive-definite operators.
- ANDREW TOMS, Purdue
- DILIAN YANG, University of Windsor
Maximal abelian subalgebras in higher rank graph C*-algebras [PDF]
- DILIAN YANG, University of Windsor
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Higher rank graphs are a natural generalization of directed graphs. The graph C*-algebra of a higher rank graph is the universal C*-algebra generated by the partial isometries associated to paths and projections associated to vertices, which satisfy the
Cuntz-Krieger relations. It turns out that the C*-subalgebra, called the diagonal subaglebra, generated by those projections is abelian, and that it is a maximal abelian subalgebra if and only if the ambient graph is aperiodic. In this talk, we will report some recent results on a natural candidate corresponding to the diagonal subalgebra for a periodic higher rank graph.
