86 research outputs found
o-minimal Flows on Abelian Varieties
Let A be an abelian variety over CC of dimension n and π:Cn⟶Aπ:Cn⟶A be the complex uniformization. Let X be an unbounded subset of CnCn definable in a suitable o-minimal structure. We give a description of the Zariski closure of π(X)π(X)
o-minimal Flows on Abelian Varieties
Let A be an abelian variety over CC of dimension n and π:Cn⟶Aπ:Cn⟶A be the complex uniformization. Let X be an unbounded subset of CnCn definable in a suitable o-minimal structure. We give a description of the Zariski closure of π(X)π(X)
Heights of pre-special points of Shimura varieties
Let s be a special point on a Shimura variety, and x a pre-image of s in a fixed fundamental set of the associated Hermitian symmetric domain. We prove that the height of x is polynomially bounded with respect to the discriminant of the centre of the endomorphism ring of the corresponding ZZ -Hodge structure. Our bound is the final step needed to complete a proof of the André–Oort conjecture under the conjectural lower bounds for the sizes of Galois orbits of special points, using a strategy of Pila and Zannier
Algebraic flows on abelian varieties
Let A be an abelian variety. The abelian Ax–Lindemann theorem shows that the Zariski closure of an algebraic flow in A is a translate of an abelian subvariety of A. The paper discusses some conjectures on the usual topological closure of an algebraic flow in A. The main result is a proof of these conjectures when the algebraic flow is given by an algebraic curve
Mesoscopic Anderson Box: Connecting Weak to Strong Coupling
Both the weakly coupled and strong coupling Anderson impurity problems are
characterized by a Fermi-liquid theory with weakly interacting quasiparticles.
In an Anderson box, mesoscopic fluctuations of the effective single particle
properties will be large. We study how the statistical fluctuations at low
temperature in these two problems are connected, using random matrix theory and
the slave boson mean field approximation (SBMFA). First, for a resonant level
model such as results from the SBMFA, we find the joint distribution of energy
levels with and without the resonant level present. Second, if only energy
levels within the Kondo resonance are considered, the distributions of
perturbed levels collapse to universal forms for both orthogonal and unitary
ensembles for all values of the coupling. These universal curves are described
well by a simple Wigner-surmise type toy model. Third, we study the
fluctuations of the mean field parameters in the SBMFA, finding that they are
small. Finally, the change in the intensity of an eigenfunction at an arbitrary
point is studied, such as is relevant in conductance measurements: we find that
the introduction of the strongly-coupled impurity considerably changes the wave
function but that a substantial correlation remains.Comment: 17 pages, 7 figure
Algebraic flows on Shimura varieties
In this paper we formulate some conjectures about algebraic flows on Shimura varieties. In the first part of the paper we prove the `logarithmic Ax-Lindemann theorem'. We then prove a result concerning the topological closure of the images of totally geodesic subvarieties of symmetric spaces uniformising Shimura varieties. This is a special case of our conjectures
From Weak- to Strong-Coupling Mesoscopic Fermi Liquids
We study mesoscopic fluctuations in a system in which there is a continuous
connection between two distinct Fermi liquids, asking whether the mesoscopic
variation in the two limits is correlated. The particular system studied is an
Anderson impurity coupled to a finite mesoscopic reservoir described by random
matrix theory, a structure which can be realized using quantum dots. We use the
slave boson mean field approach to connect the levels of the uncoupled system
to those of the strong coupling Nozi\`eres Fermi liquid. We find strong but not
complete correlation between the mesoscopic properties in the two limits and
several universal features.Comment: 6 pages, 3 figure
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