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.

Part of the series 'What do historians of mathematics do?'

Ada Lovelace (1815-1852) is famous as "the first programmer" for her prescient writings about Charles Babbage's unbuilt mechanical computer, the Analytical Engine. Biographers have focused on her tragically short life and her supposed poetic approach – one even dismissed her mathematics as "hieroglyphics". This talk will focus on how she learned the mathematics she needed to write the paper – a correspondence course she took with Augustus De Morgan – which is available in the Bodleian Library. I'll also reflect more broadly on things I’ve learned as a newcomer to the history of mathematics.

Part of the series 'What do historians of mathematics do?'

"In this talk I present the history and proof of the four-colour theorem: Can every map be coloured with just four colours so that neighbouring countries are coloured differently?  The proof took 124 years to find, and used 1200 hours of computer time. But what did it involve, and is it really a proof?"

Part of the series 'What do historians of mathematics do?'

I'm currently working on a biography of Charles Hutton (1737–1823): pit lad, FRS, and professor of Mathematics. No-one much has heard of him today, but to his contemporaries he was "one of the greatest mathematicians in Europe". I'll give an outline of his remarkable story and say something about why he's worth my time.