Mon, 18 Jan 2010

16:00 - 17:00
SR1

An Round-Up of the Circle Problem

Timothy Trudgian
(Oxford)
Abstract

How many integer-points lie in a circle of radius $\sqrt{x}$?

A poor man's approximation might be $\pi x$, and indeed, the aim-of-the-game is to estimate

$$P(x) = \sharp\{(m, n) \in\mathbb{Z}: \;\; m^{2} + n^{2} \leq x\} -\pi x,$$

Once one gets the eye in to show that $P(x) = O(x^{1/2})$, the task is to graft an innings to reduce this bound as much as one can. Since the cricket-loving G. H. Hardy proved that $P(x) = O(x^{\alpha})$ can only possible hold when $\alpha \geq 1/4$ there is some room for improvement in the middle-order.

In this first match of the Junior Number Theory Seminar Series, I will present a summary of results on $P(x)$.

Mon, 01 Feb 2010

12:00 - 13:00
L3

Twistor-Strings, Grassmannians and Leading Singularities

Lionel Mason
(Oxford)
Abstract
A systematic procedure is derived for obtaining an explicit, L-loop leading singularities of planar N=4 super Yang-Mills scattering amplitudes in twistor space directly from their momentum space channel diagrams. The expressions are given as integrals over the moduli of connected, nodal curves in twistor space whose degree and genus matches expectations from twistor-string theory. We propose that a twistor-string theory for pure N=4 super Yang-Mills, if it exists, is determined by the condition that these leading singularity formulae arise as residues when an unphysical contour for the path integral is used, by analogy with the momentum space leading singularity conjecture. We go on to show that the genus g twistor-string moduli space for g-loop N^{k-2}MHV amplitudes may be mapped into the Grassmannian G(k,n). Restricting to a leading singularity, the image of this map is a 2(n-2)-dimensional subcycle of G(k,n) of exactly the type found from the Grassmannian residue formula of Arkani-Hamed, Cachazo, Cheung and Kaplan. Based on this correspondence and the Grassmannian conjecture, we deduce restrictions on the possible leading singularities of multi-loop N^pMHV amplitudes. In particular, we argue that no new leading singularities can arise beyond 3p loops.
Mon, 25 Jan 2010

12:00 - 13:00
L3

Scanning through Heterotic Vacua

Yang-Hui He
(Oxford)
Abstract
We discuss some recent progress in obtaining the exact spectrum of the MSSM from a generalized embedding of the heterotic string. Utilizing current developments in algebraic geometry, especially algorithmic, we search through the landscape of vector bundles over Calabi-Yau manifolds for a special corner wherein such exact models may be found.
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