Born-Rigid Flow and the AdS-CFT Correspondence

This paper reviews the concepts and assumptions of rigid flow in relativistic fluid mech- anics, particularly the generalisation of the classical Herglotz-Noether theorem, that are relevant to the fluid approximation of the AdS-CFT dual of large rotating black-holes used by Bhattacharyya et al. We then give a brief outline of the recently found proof the generalised theorem.Comment: 8 page

Towards the Andr\'e-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points

We prove in this paper the Ax-Lindemann-Weierstrass theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular we reprove a result of Silverberg in a different approach. Then combining these results we prove the Andr\'e-Oort conjecture for any mixed Shimura variety whose pure part is a subvariety of A_6^n.Comment: The arXiv version differs from the published versio

The implications of Galilean invariance for classical point particle lagrangians

We explore the implications of the requirement of Galilean invariance for classical point particle lagrangians, in which the space is not assumed to be flat to begin with. We show that for the free, time-independent lagrangian, this requirement is equivalent to the existence of gradient Killing vectors on space, which is in turn equivalent to the condition that the space is a direct product, which is totally flat in the Galilean invariant direction. We then consider more general cases and see that there is no simple generalisation to these cases.Comment: 9 page

A special point problem of Andr\'e-Pink-Zannier in the universal family of abelian varieties

The Andr\'e-Pink-Zannier conjecture concerns the intersection of subvarieties and the generalized Hecke orbit of a given point in mixed Shimura varieties. It is part of the Zilber-Pink conjecture. In this paper we focus on the universal family of principally polarized abelian varieties. We explain the moduli interpretation of the conjecture in this case and prove several different cases for this conjecture: its overlap with the Andr\'e-Oort conjecture; when the subvariety is contained in an abelian scheme over a curve and the point is a torsion point on its fiber; when the subvariety is a curve.Comment: Removed the algebraic point assumption. Comments are welcome