Despite many years of intensive research, the modeling of contact lines moving by spreading and/or evaporation still remains a subject of debate nowadays, even for the simplest case of a pure liquid on a smooth and homogeneous horizontal substrate. In addition to the inherent complexity of the topic (singularities, micro-macro matching, intricate coupling of many physical effects, …), this also stems from the relatively limited number of studies directly comparing theoretical and experimental results, with as few fitting parameters as possible. In this presentation, I will address various related questions, focusing on the physics invoked to regularize singularities at the microscale, and discussing the impact this has at the macroscale. Two opposite “minimalist” theories will be detailed: i) a classical paradigm, based on the disjoining pressure in combination with the spreading coefficient; ii) a new approach, invoking evaporation/condensation in combination with the Kelvin effect (dependence of saturation conditions upon interfacial curvature). Most notably, the latter effect enables resolving both viscous and thermal singularities altogether, without needing any other regularizing effects such as disjoining pressure, precursor films or slip length. Experimental results are also presented about evaporation-induced contact angles, to partly validate the first approach, although it is argued that reality might often lie in between these two extreme cases.

Full details are available at: http://www.stcatz.ox.ac.uk/alantayler

Rational Krylov subspaces have been proven to be useful for many applications, like the approximation of matrix functions or the solution of matrix equations. It will be shown that extended and rational Krylov subspaces —under some assumptions— can be retrieved without any explicit inversion or system solves involved. Instead we do the necessary computations of $A^{-1} v$ in an implicit way using the information from an enlarged standard Krylov subspace.

It is well-known that both for classical and extended Krylov spaces, direct unitary similarity transformations exist providing us the matrix of recurrences. In practice, however, for large dimensions computing time is saved by making use of iterative procedures to gradually gather the recurrences in a matrix. Unfortunately, for extended Krylov spaces one is required to frequently solve, in some way or another a system of equations. In this talk both techniques will be integrated. We start with an orthogonal basis of a standard Krylov subspace of dimension $m+m+p$. Then we will apply a unitary similarity built by rotations compressing thereby significantly the initial subspace and resulting in an orthogonal basis approximately spanning an extended or rational Krylov subspace.

Numerical experiments support our claims that this approximation is very good if the large Krylov subspace contains $A^{-(m+1)} v$, …, $A^{-1} v$ and thus can culminate in nonneglectable dimensionality reduction and as such also can lead to time savings when approximating, e.g., matrix functions.

A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Francis's implicitly-shifted $QR$ algorithm.  A companion matrix is an upper Hessenberg matrix that is unitary-plus-rank-one, that is, it is the sum of a unitary matrix and a rank-one matrix.  These properties are preserved by iterations of Francis's algorithm, and it is these properties that are exploited here. The matrix is represented as a product of $3n-1$ Givens rotators plus the rank-one part, so only $O(n)$ storage space is required.  In fact, the information about the rank-one part is also encoded in the rotators, so it is not necessary to store the rank-one part explicitly.  Francis's algorithm implemented on this representation requires only $O(n)$ flops per iteration and thus $O(n^{2})$ flops overall.  The algorithm is described, backward stability is proved under certain conditions on the polynomial coefficients, and an extensive set of numerical experiments is presented.  The algorithm is shown to be about as accurate as the (slow) Francis $QR$ algorithm applied to the companion matrix without exploiting the structure.  It is faster than other fast methods that have been proposed, and its accuracy is comparable or better.

We present advances in several fundamental fields of computer vision: image segmentation, object tracking, stereo reconstruction for depth map estimation and full 3D multi-view reconstruction. The basic method applied to these fields is convex relaxation. Convex relaxation methods allow for global optimization of numerous energy functionals and provide a step towards less user input and more automation. We will show how the respective computer vision problems can be formulated in this convex optimization framework. Efficient parallel implementations of the arising numerical schemes using graphics processing units allow for interactive applications.

Modularity is a quality function on partitions of a network which aims to identify highly clustered components. Given a graph G, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the modularity q(G) of G is the maximum modularity of a partition of V(G). Knowledge of the maximum modularity of the corresponding random graph is important to determine the statistical significance of a partition in a real network. We provide bounds for the modularity of random regular graphs. Modularity is related to the Hamiltonian of the Potts model from statistical physics. This leads to interest in the modularity of lattices, which we will discuss. This is joint work with Colin McDiarmid.

The curve graph of a surface has a vertex for each curve on the surface and an edge for each pair of disjoint curves. Although it deals with very simple objects, it has connections with questions in low-dimensional topology, and some properties that encourage people to study it. Yet it is more complicated than it may look from its definition: in particular, what happens if we start following a 'diverging' path along this graph? It turns out that the curves we hit get so complicated that eventually give rise to a lamination filling up the surface. This can be understood by drawing some train track-like pictures on the surface. During the talk I will keep away from any issue that I considered too technical.

Cluster algebras are commutative algebras generated by a set S, obtained by an iterated mutation process of an initial seed. They were introduced by S. Fomin and A. Zelevinski in connection with canonical bases in Lie theory. Since then, many connections between cluster algebras and other areas have arisen.
This talk will focus on cluster algebras for which the set S is finite. These are called cluster algebras of finite type and are classified by Dynkin diagrams, in a similar way to many other objects.

Stallings' folding technique lets us factor a map of graphs as a sequence of "folds" (edge identifications) followed by an immersion. We will show how this technique gives an algorithm to express a free-group automorphism as the product of Whitehead automorphisms (and hence Nielsen transformations), as well as proving finite generation for some subgroups of the automorphism group of a free group.