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