In quite an elementary, hands-on talk, I will discuss some Ax-Lindemann type results in the setting of modular functions.  There are some very powerful results in this area due to Pila, but in nonclassical variants we have only quite weak results, for a rather silly reason to be discussed in the talk.

Multidimensional single molecule microscopy (MSMM) generates image time series of biomolecules in a cellular environment that have been tagged with fluorescent labels. Initial analysis steps of such images consist of image registration of multiple channels, feature detection and single particle tracking. Further analysis may involve the estimation of diffusion rates, the measurement of separations between molecules that are not optically resolved and more. The analysis is done under the condition of poor signal to noise ratios, high density of features and other adverse conditions. Pushing the boundary of what is measurable, we are facing among others the following challenges. Firstly the correct assessment of the uncertainties and the significance of the results, secondly the fast and reliable identification of those features and tracks that fulfil the assumptions of the models used. Simpler models require more rigid preconditions and therefore limiting the usable data, complexer models are theoretically and especially computationally challenging.

In algebraic and arithmetic geometry, there is the ubiquitous notion of a moduli space, which informally is a variety (or scheme) parametrising a class of objects of interest. My aim in this talk is to explain concretely what we mean by a moduli space, going through the functor-of-points formalism of Grothendieck. Time permitting, I may also discuss (informally!) a natural obstruction to the existence of moduli schemes, and how one can get around this problem by taking a 2-categorical point of view.

 About 10 years ago Minahan and Zarembo made a remarkable discovery: the one-loop Dilatation Operator in the SO(6) sector of planar N=4 SYM can be identified with the Hamiltonian of an integrable spin chain. This one-loop Dilatation operator was obtained by computing a two-point correlation function at one loop, which is a completely off-shell quantity. Around the same time, Witten proposed a duality between N=4 SYM and twistor string theory, which initiated a revolution in the field of on-shell objects like scattering amplitudes. In this talk we illustrate that these techniques that have been sucessfully used for on-shell quantities can also be employed for the computation of off-shell quantities by computing the one-loop Dilatation Operator in the SO(6) sector. The first half of the talk will be dedicated to doing this calculation using MHV diagrams and the second half of the talk shows the computation in twistor space. 

These two short talks will be followed by an informal afternoon session for those interested in further details of these approaches, and in form factors in Class Room C2 from 2-4.30 pm then from 4.30pm in N3.12.  All are welcome.