Mathematical Sciences Institute (MSI)
Algebras, Operators and Noncommutative
Geometry
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Mathematical Sciences Institute (MSI)
Algebras, Operators and Noncommutative
Geometry
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Program (Updated)Talks are in Room G35, John Dedman Building Monday, Dec. 1:
Tuesday, Dec. 2:
Wednesday, Dec. 3:
Thursday, Dec. 4:
Friday, Dec. 5:
AbstractsJohannes AastrupMunster UniversityIndex theory of boundary value problems via continuous fields of $C^*$-algebras We use methods from noncommutative geometry to give an index formula for elliptic operators in Boutet de Monvels calculus. Joint work with E. Schrohe and R. Nest. Astrid an HuefUniversity of NSWProper actions on $C^*$-algebras In 1990, Rieffel formulated the notion of a proper action $alpha$ of a group on a $C^*$-algebra $A$. Under reasonable hypotheses, the reduced crossed-product $C^*$-algebra $Atimes_{alpha,r}G$ is Morita equivalent to a ``generalized fixed point algebra'' $A^{alpha}$ in the multiplier algebra $M(A)$. In this talk I will discuss examples of proper actions and recent work of Kaliszewski, Quigg and Raeburn which shows that Rieffel's construction is natural in a categorical sense. If time permits I'll briefly talk about work in progress with Iain Raeburn (University of Wollongong), Steven Kaliszewski (Arizona State University) and Dana Williams (Dartmouth College) which extends the work of Kaliszewski, Quigg and Raeburn. Partha Sarathi ChakrabortyUniversity of AdelaideNoncommutative Geometry of Odd Dimensional Quantum Spheres We will report our atempts to understand noncommutative geometry in the lights of the example of quantum spheres. We will see how to produce an equivariant fundamental class and also indicate some of the limitations of isospectral deformations. Ruy ExelUniversidade Federal de Santa CatarinaC*-algebras of associated to irreversible dynamical systems I plan to review some of the recent developments in the theory of crossed products of C*-algebras by endomorphisms or semigroups thereof, leading to the concept of "interaction groups". I also plan to present examples associated to pairs of commuting maps on a topological space which lead to interaction groups. Jesper GrimstrupNiels Bohr InstituteOn spectral triples of holonomy loops In my talk I will show how a semifinite spectral triple is obtained from a rearrangement of central elements of Loop Quantum Gravity. The triple is based on a countable set of graphs and the algebra consists of holonomy loops in this set. The Dirac type operator resembles a global functional derivation operator. The interaction between the algebra of holonomy loops and the Dirac type operator reproduces the structure of the Poisson bracket of General Relativity. In the talk I will argue how one might obtain a Hamilton constraint from the spectral triple construction. Hendrik GrundlingUniversity of NSWUnderstanding the Bochner-Minlos theorem through group algebras. The Bochner-Minlos theorem (which we only consider on R^infty) seems like a straightforward generalization to infinite dimensions of the state space decomposition of the group algebra of R^n. However, as R^infty is not locally compact, it has no group algebra, hence it is a mystery how to understand it from the C*-algebraic point of view. Here we will define "partial group algebras" for R^infty, and show that the Bochner-Minlos theorem can be proven with the help of these alone. Vaughan JonesUniversity of California, Berkeley.An orthogonal approach to the graded algebra of a planar algebra. The Voiculescu trace on the graded algebra of a planar algebra is inspired by large random matrices and may be used to define a II_1 factor and subfactor whose standard invariant is the planar algebra in question. But a change of basis suggested by Walker makes the von Neumann algebra results much easier to prove. This is joint work with Guionnet, Shlyakhtenko and Walker. The change of basis and a similar proof were discovered independently by Kodiyalam and Sunder. Gus LehrerUniversity of SydneyQuantum Endomorphisms and Cellularity I'll show how cellular structure may be used to determine presentations of commutants of tensor powers of quantum group representations. Motoko KotaniTohoku UniversityGeometric Aspects of random walks on a crystal lattice We will discuss about the relation between the long time behavior of random walk on a crystal lattice and its geometry. Yoshiaki MaedaKeio UniversityNonformal noncommutative calculus By introducing a family of product formulae, we will present a plot type of noncommutative transcendental calculus for the case of (complex) Weyl algebra. In particular, we propose an algebraic approach to the spectral analysis from a point of deformation quantization by using a regularization technique. Ryszard NestCopenhagenUniversal Coefficient Theorems for Kirchberg's KK-theory We will describe a general approach to construct Universal Coefficient Theorems for Kirchberg's KK-theory for C*-algebras over finite topological spaces and discuss two examples: one where filtrated K-theory is enough to get a UCT, and one where it is not and where we have to add another invariant to get a UCT. David PaskUniversity of WollongongGraph algebras and noncommutative dynamical systems In this talk I shall describe the relationship between certain shift spaces and various flavours of graph C*-algebras. I shall begin by introducing shift spaces, their properties and their graphical presentations. A brief overview of certain graph C*-algebras will then be given, with emphasis given to their dynamical aspects. Denis PotapovFlinders UniversitySpectral flow is the integral of one form The talk discusses recent progress in spectral flow theory representing the spectral flow of a path of self adjoint Fredholm operators as an integral of one form along the path. It also discusses some noncommutative analytical methods behind this result. Ashwin PandeAustralian Natioanl UniversityTopological T-duality and T-folds We explicitly construct the C*-algebras arising in the formalism of Topological T-duality due to Mathai and Rosenberg from string-theoretic data in several key examples. We construct a continuous-trace algebra with an action of ${\mathbb R}^d$ unique up to exterior equivalence from the data of a smooth ${\mathbb T}^d$-equivariant gerbe on a trivial bundle $X = W \times {\mathbb T}^d$. We argue that the `noncommutative T-duals' of Mathai and Rosenberg, should be identified with the nongeometric backgrounds well-known in string theory. We identify the charge group of D-branes on T-fold backgrounds in the C*-algebraic formalism of Topological T-duality. We also study D-branes on T-fold backgrounds. We show that the $K$-theory bundles studied by Echterhoff, Nest and Oyono-Oyono give a natural description of these objects. Iain RaeburnUniversity of WollongongThe Toeplitz algebra of an ax+b semigroup Cuntz has recently constructed a very interesting simple C*-algebra from the ax+b semigroup over the natural numbers. We will discuss joint work with Marcelo Laca in which we study the (much larger) Toeplitz algebra of this semigroup, which turns out to be quasi-lattice ordered in the sense of Nica. We find a presentation of this Toeplitz algebra, show that Cuntz's algebra is the boundary quotient of the Toeplitz algebra, and calculate the KMS states for a canonical dual action of the real numbers. Aidan SimsUniversity of WollongongGraph C*-algebras, Exel-Laca algebras and ultragraph C*-algebras We will summarize a series of recent joint papers with Katsura, Muhly and Tomforde regarding the structure of three important classes of $C^*$-algebras and the relationships between them. Fyodor SukochevUniversity of NSWSpectral Flow and Spectral Shift Function We shall discuss recently discovered connection between spectral flow and spectral shift function and present some new formulae for spectral flow. This talk is the precursor to the talk by D. Potapov and is based on several joint works with N. Azamov, A. Carey, P. Dodds and D. Potapov. Varghese MathaiUniversity of AdelaideThe index of projective families of elliptic operators An index theory for projective families of elliptic pseudodifferential operators is developed, when the corresponding Azumaya bundle can be realized in terms of smoothing operators. The topological and the analytic index of a projective family of elliptic operators both take values in twisted $K$-theory of the parameterizing space, $X.$ The main result is the equality of these two notions of index. The twisted Chern character of the index class is then computed by a variant of Chern-Weil theory. This is joint work with Melrose and Singer. Ruibin ZhangUniversity of SydneyQuantum group actions on associative algebras and noncommutative invariant theory We discuss a non-commutative analogue of the first fundamental theorem of classical invariant theory. For a quantum group associated with each classical Lie algebra, we construct a non-commutative module algebra, that is, an associative algebra such that the underlying vector space forms a module for the quantum group and the algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is proved to be finitely generated. We construct the generators of the subalgebra of invariants and determine their commutation relations. In the limit q going to 1, the results recover the first fundamental theorem of classical invariant theory. This is joint work with Gus Lehrer and H. Zhang.
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