1,697 research outputs found
Non-Commutative Chern Numbers for Generic Aperiodic Discrete Systems
The search for strong topological phases in generic aperiodic materials and
meta-materials is now vigorously pursued by the condensed matter physics
community. In this work, we first introduce the concept of patterned resonators
as a unifying theoretical framework for topological electronic, photonic,
phononic etc. (aperiodic) systems. We then discuss, in physical terms, the
philosophy behind an operator theoretic analysis used to systematize such
systems. A model calculation of the Hall conductance of a 2-dimensional
amorphous lattice is given, where we present numerical evidence of its
quantization in the mobility gap regime. Motivated by such facts, we then
present the main result of our work, which is the extension of the Chern number
formulas to Hamiltonians associated to lattices without a canonical labeling of
the sites, together with index theorems that assure the quantization and
stability of these Chern numbers in the mobility gap regime. Our results cover
a broad range of applications, in particular, those involving
quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically
generated) lattices.Comment: 44 pages, 4 figures. v2: typos corrected and references updated. v3:
Minor changes, to appear in J. Phys. A (Mathematical and Theoretical
A walk in the noncommutative garden
This text is written for the volume of the school/conference "Noncommutative
Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in
noncommutative geometry, based on a discussion of significant examples of
noncommutative spaces in geometry, number theory, and physics. The paper also
contains an outline (the ``Tehran program'') of ongoing joint work with Consani
on the noncommutative geometry of the adeles class space and its relation to
number theoretic questions.Comment: 106 pages, LaTeX, 23 figure
The Euler multiplicity and addition-deletion theorems for multiarrangements
The addition-deletion theorems for hyperplane arrangements, which were
originally shown in [H. Terao, Arrangements of hyperplanes and their freeness
I, II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 293--320], provide
useful ways to construct examples of free arrangements. In this article, we
prove addition-deletion theorems for multiarrangements. A key to the
generalization is the definition of a new multiplicity, called the Euler
multiplicity, of a restricted multiarrangement. We compute the Euler
multiplicities in many cases. Then we apply the addition-deletion theorems to
various arrangements including supersolvable arrangements and the Coxeter
arrangement of type to construct free and non-free multiarrangements
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
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. To appear in Proc. ICM 2010.Comment: 16 pages, LaTe
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