In 1995 N. Hitchin constructed explicit algebraic solutions to the Painlevé VI (1/8,-1/8,1/8,3/8) equation starting with any Poncelet trajectory, that is a closed billiard trajectory inscribed in a conic and circumscribed about another conic. In this talk I will show that Hitchin's construction is the Okamoto transformation between Picard's solution and the general solution of the Painlevé VI (1/8,-1/8,1/8,3/8) equation. Moreover, this Okamoto transformation can be written in terms of an Abelian differential of the third kind on the associated elliptic curve, which allows to write down solutions to the corresponding Schlesinger system in terms of this differential as well. This is a joint work with V. Dragovic.

Puzzling things happen in human perception when ambiguous or incomplete information is presented to the eyes. Rivalry occurs when two different images, presented one to each eye, lead to alternating percepts, possibly of neither image separately. Illusions, or multistable figures, occur when a single image can be perceived in several ways. The Necker cube is the most famous example. Impossible objects arise when a single image has locally consistent but globally inconsistent geometry. Famous examples are the Penrose triangle and etchings by Maurits Escher.

In this lecture Ian Stewart will demonstrate how these phenomena provide clues about the workings of the visual system, with reference to recent research in the field which has modelled simplified, systematic methods by which the brain can make decisions. In these models a neural network is designed to interpret incoming sensory data in terms of previously learned patterns. Rivalry occurs when different interpretations are confused, and illusions arise when the same data have several interpretations.

The lecture will be non-technical and highly illustrated, with plenty of examples.

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To predict the behaviour of a dynamical system using a mathematical model, an accurate estimate of the current state of the system is needed in order to initialize the model. Complete information on the current state is, however, seldom available. The aim of optimal state estimation, known in the geophysical sciences as ‘data assimilation’, is to determine a best estimate of the current state using measured observations of the real system over time, together with the model equations. The problem is commonly formulated in variational terms as a very large nonlinear least-squares optimization problem. The lack of complete data, coupled with errors in the observations and in the model, leads to a highly ill-conditioned inverse problem that is difficult to solve.

To understand the nature of the inverse problem, we examine how different components of the assimilation system influence the conditioning of the optimization problem. First we consider the case where the dynamical equations are assumed to model the real system exactly. We show, against intuition, that with increasingly dense and precise observations, the problem becomes harder to solve accurately. We then extend these results to a 'weak-constraint' form of the problem, where the model equations are assumed not to be exact, but to contain random errors. Two different, but mathematically equivalent, forms of the problem are derived. We investigate the conditioning of these two forms and find, surprisingly, that these have quite different behaviour.

Question: Given a smooth compact manifold $M$ without boundary, does $M$
 admit a Riemannian metric of positive scalar curvature?

 We focus on the case of spin manifolds. The spin structure, together with a
 chosen Riemannian metric, allows to construct a specific geometric
 differential operator, called Dirac operator. If the metric has positive
 scalar curvature, then 0 is not in the spectrum of this operator; this in
 turn implies that a topological invariant, the index, vanishes.

  We use a refined version, acting on sections of a bundle of modules over a
 $C^*$-algebra; and then the index takes values in the K-theory of this
 algebra. This index is the image under the Baum-Connes assembly map of a
 topological object, the K-theoretic fundamental class.

 The talk will present results of the following type:

 If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has
 non-trivial index, what conditions imply that $M$ does not admit a metric of
 positive scalar curvature? How is this related to the Baum-Connes assembly
 map? 

 We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$),
 Engel and new generalizations. Moreover, we will show how these results fit
 in the context of the Baum-Connes assembly maps for the manifold and the
 submanifold. 
 

I will summarize recent (arXiv:1603.05262) and upcoming work with Igor Buchberger and Oliver Schlotterer. We construct a map from n-point 1-loop string amplitudes in maximal supersymmetry to n-3-point 1-loop amplitudes in minimal supersymmetry. I will outline a few implications for the quantum string effective action.

In this talk, a concrete realization of the Bern-Carrasco-Johansson (BCJ) duality between color and kinematics in non-abelian gauge theories is presented. The method of Berends-Giele to package Feynman diagrams into currents is shown to yield classical solutions to the non-linear Yang-Mills equations. We describe a non-linear gauge transformation of these perturbiner solutions which reorganize the cubic-diagram content such that the kinematic dependence obeys the same Jacobi identities as the accompanying color factors. The resulting tree-level subdiagrams are assembled to kinematic numerators of tree-level and one-loop amplitudes which satisfy the BCJ duality.