Oxford

I will describe work exploring the spectrum of two-dimensional unitary conformal field theories(CFT) with no extended chiral algebra and central charge larger than one. I will revisit a classical result from analytic number theory by Rademacher, which provides an exact formula for the Fourier coefficients of modular forms of non-positive weight. Generalizing this, I will explain how we employed Rademacher's idea to study the spectral density of two-dimensional CFT of our interest. The expression is given in terms of a Rademacher expansion, which converges for nonzero spin. The implications of our spectral density to the pure gravity in AdS3 will be discussed.

The asymptotically locally Euclidean Ricci-flat self-dual 4-manifolds were classified and constructed by Kronheimer as hyperkahler quotients. Each belongs to a finite-dimensional family and a particularly interesting subfamily consists of manifolds with a circle action which can be identified with the minimal resolution of a quotient singularity C^2/G where G is a finite subgroup of SU(2). The resolved singularity is a configuration of rational curves and there is a distinguished one which is pointwise fixed by the circle action. The talk will give an explicit description of the induced metric on this central sphere, and involves twistor theory and the geometry of the lines on a cubic surface.
 

A subset $D$ of a finite cyclic group $\mathbb{Z}/m\mathbb{Z}$ is called a "perfect difference set" if every nonzero element of $\mathbb{Z}/m\mathbb{Z}$ can be written uniquely as the difference of two elements of $D$. If such a set exists, then a simple counting argument shows that $m=n^2+n+1$ for some nonnegative integer $n$. Singer constructed examples of perfect difference sets in $\mathbb{Z}/(n^2+n+1)\mathbb{Z}$ whenever $n$ is a prime power, and it is an old conjecture that these are the only such $n$ for which $\mathbb{Z}/(n^2+n+1)\mathbb{Z}$ contains a perfect difference set. In this talk, I will discuss a proof of an asymptotic version of this conjecture.

There is a class $\mathcal{B}$ of analytic Besov functions on a half-plane, with a very simple description.   This talk will describe a bounded functional calculus $f \in \mathcal{B} \mapsto f(A)$ where $-A$ is the generator of either a bounded $C_0$-semigroup on Hilbert space or a bounded analytic semigroup on a Banach space.    This calculus captures many known results for such operators in a unified way, and sometimes improves them.   A discrete version of the functional calculus was shown by Peller in 1983.

In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime.  We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. Modular groups and modular forms arise in relation to these attractor points in a way that, to a physicist, is unexpected. This is a report on joint work with Xenia de la Ossa, Mohamed Elmi and Duco van Straten.

Given a number field K, it is natural to ask whether it has a finite extension with ideal class number one. This question can be translated into a fundamental question in class field theory, namely the class field tower problem. In this talk, we are going to discuss this problem as well as its solution due to Golod and Shafarevich using methods of group cohomology.
 

A well-known theorem of Choquet-Bruhat and Geroch states that for given smooth initial data for the Einstein equations there exists a unique maximal globally hyperbolic development. In particular, time evolution of globally hyperbolic solutions is unique. This talk investigates whether the same result holds for quasilinear wave equations defined on a fixed background. After recalling the notion of global hyperbolicity, we first present an example of a quasilinear wave equation for which unique time evolution in fact fails and contrast this with the Einstein equations. We then proceed by presenting conditions on quasilinear wave equations which ensure uniqueness. This talk is based on joint work with Harvey Reall and Felicity Eperon.
 

Reconstruction of 3D images from a set of 2D X-ray projections is a standard inverse problem, particularly in medical imaging. Improvements in imaging technologies have enabled the development of a flat-panel X-ray source, comprised of an array of low-power emitters that are fired in quick succession. During a complete firing sequence, there may be shifts in the patient’s resting position which ultimately create artifacts in the final reconstruction. We present a method for correcting images with respect to unknown body motion, focusing on the case of simple rigid body motion. Image reconstructions are obtained by solving a sparse linear inverse problem, with respect to not only the underlying body but also the unknown velocity. Results find that reconstructions of a moving body can be much better than those obtained by measuring a stationary body, as long as the underlying motion is well approximated.