Singularity dominated strong fluctuations for some random matrix averages

The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval $(0,1)$ of the real line respectively. The averaged value of the modulus of the corresponding characteristic polynomial raised to the power $2 \mu$ diverges, for $2\mu \le -1$, at points approaching the eigenvalue support. Using the theory of generalized hypergeometric functions based on Jack polynomials, the functional form of the leading asymptotic behaviour is established rigorously. In the circular ensemble case this confirms a conjecture of Berry and Keating.Comment: 11 pages, to appear Commun. Math. Phy

Rethinking Sovereignty : Independence-lite, devolution-max and national accommodation

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Wintenberger's Functor for Abelian Extensions

Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$-adic Lie extensions $E/F$, where $F$ is a local field with residue field $k$, and a category whose objects are pairs $(K,A)$, where $K\cong k((T))$ and $A$ is an abelian $p$-adic Lie subgroup of \Aut_k(K). In this paper we extend this equivalence to allow \Gal(E/F) and $A$ to be arbitrary abelian pro-$p$ groups.Comment: 12 page

Will They Come? Get Out The Word About Going Mobile

To be effective, libraries must promote, market, and advertise mobile initiatives. When libraries introduce services that use new tools and modes of thought, they must demonstrate what is possible, how services are relevant, and how new resources can help

Extensions of local fields and truncated power series

Let $K$ be a finite tamely ramified extension of \Q_p and let $L/K$ be a totally ramified $(\Z/p^n\Z)$-extension. Let $\pi_L$ be a uniformizer for $L$, let $\sigma$ be a generator for \Gal(L/K), and let $f(X)$ be an element of \O_K[X] such that $\sigma(\pi_L)=f(\pi_L)$. We show that the reduction of $f(X)$ modulo the maximal ideal of \O_K determines a certain subextension of $L/K$ up to isomorphism. We use this result to study the field extensions generated by periodic points of a $p$-adic dynamical system.Comment: 29 page

Lagrangian tori in four-dimensional Milnor fibres

The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of Milnor fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities, which are one level of complexity up from the simple ones.Comment: v2: 66 pages, 37 figures; minor change