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14
A WALK IN THE NONCOMMUTATIVE GARDEN
"... 2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9 ..."
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2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9
Spin-Hall effect with quantum group symmetry
- Lett. Math. Phys
, 2006
"... We construct a model of spin-Hall effect on the noncommutative sphere S 4 θ with isospin degrees of freedom (coming from a noncommutative instanton) and invariance under a quantum group SOθ(5). The corresponding representation theory allows to explicitly diagonalize the Hamiltonian and construct the ..."
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We construct a model of spin-Hall effect on the noncommutative sphere S 4 θ with isospin degrees of freedom (coming from a noncommutative instanton) and invariance under a quantum group SOθ(5). The corresponding representation theory allows to explicitly diagonalize the Hamiltonian and construct the ground state; there are both integer and fractional excitations. Similar models exist on higher dimensional spheres S N Θ and projective spaces CPN Θ.
On twisted Fourier analysis and convergence of Fourier series on discrete groups
, 2006
"... We study convergence and summation processes of Fourier series in reduced twisted group C ∗-algebras of discrete groups. ..."
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We study convergence and summation processes of Fourier series in reduced twisted group C ∗-algebras of discrete groups.
Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid
- Annals of Physics
, 2012
"... Abstract. Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and classify level crossings, aka. singularities in the spectrum of a family of Hamiltonians. The approach we use is singularity theory. Using this language, we obtain a characterization of Dirac poin ..."
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Abstract. Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and classify level crossings, aka. singularities in the spectrum of a family of Hamiltonians. The approach we use is singularity theory. Using this language, we obtain a characterization of Dirac points and also show that the branching behavior of the level crossings is given by an unfolding of An type singularities. Which type of singularity occurs can be read off a characteristic region inside the miniversal unfolding of an Ak singularity. We then apply these methods in the setting of families of graph Hamiltonians, such as those for wire networks. In the particular case of the Double Gyroid we analytically classify its singularities and show that it has Dirac points. This indicates that nanowire systems of this type should have very special physical properties.
The geometry of the double gyroid wire network: quantum and classical
- Journal of Noncommutative Geometry
, 2012
"... Abstract. Quantum wire networks have recently become of great interest. Here we deal with a novel nano material structure of a Double Gyroid wire network. We use methods of commutative and non-commutative geometry to describe this wire network. Its non–commutative geometry is closely related to non- ..."
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Abstract. Quantum wire networks have recently become of great interest. Here we deal with a novel nano material structure of a Double Gyroid wire network. We use methods of commutative and non-commutative geometry to describe this wire network. Its non–commutative geometry is closely related to non-commutative 3-tori as we discuss in detail.
NONCOMMUTATIVE GEOMETRY AND ARITHMETIC
"... This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the classical theory of elliptic curves with complex multiplication for imaginary quadratic fields. This talk ..."
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This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the classical theory of elliptic curves with complex multiplication for imaginary quadratic fields. This talk concentrates on two main aspects: the relation of Stark numbers to the geometry of noncommutative tori with real multiplication, and the shadows of modular forms on the noncommutative boundary of modular curves, that is, the moduli space of noncommutative tori.
8. Shimizu L-function and Lorentzian geometry 28
"... 6. Twisted index theorem, K-theory, and the range of the trace 16 ..."
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TWISTED INDEX THEORY ON ORBIFOLD SYMMETRIC PRODUCTS AND THE FRACTIONAL QUANTUM HALL EFFECT
"... Abstract. We extend the noncommutative geometry model of the fractional quantum Hall effect, previously developed by Mathai and the first author, to orbifold symmetric products. It retains the same properties of quantization of the Hall conductance at integer multiples of the fractional Satake orbif ..."
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Abstract. We extend the noncommutative geometry model of the fractional quantum Hall effect, previously developed by Mathai and the first author, to orbifold symmetric products. It retains the same properties of quantization of the Hall conductance at integer multiples of the fractional Satake orbifold Euler characteristics. We show that it also allows for interesting composite fermions and anyon representations, and possibly for Laughlin type wave functions. 1.
