Several scholars of evolutionary biology have suggested that functional redundancy (also known as "biological degener-
acy") is important for robustness of biological networks. Structural redundancy indicates the existence of structurally
similar subsystems that can perform the same function. Functional redundancy indicates the existence of structurally
di erent subsystems that can perform the same function. For networks with Ornstein--Uhlenbeck dynamics, Tononi et al.
[Proc. Natl. Acad. Sci. U.S.A. 96, 3257{3262 (1999)] proposed measures of structural and functional redundancy that are
based on mutual information between subnetworks. For a network of n vertices, an exact computation of these quantities
requires O(n!) time. We derive expansions for these measures that one can compute in O(n3) time. We use the expan-
sions to compare the contributions of di erent types of motifs to a network's functional redundancy.

In this talk, we will discuss some recent progress on the study of six-dimensional S-matrices as well as their applications. Six-dimensional theories we are interested include the world-volume theories of single probe M5-brane and D5-brane, as well as 6D super Yang-Mills and supergravity. We will present twistor-string-like formulas for all these theories, analogue to that of Witten’s twistor string formulation for 4D N=4 SYM. 
As the applications, from the 6D results we also deduce new formulas for scattering amplitudes of theories in lower dimensions, such as SYM and supergravity in five dimensions, and 4D N=4 SYM on the Columbo branch. 
 

It is an old problem in number theory to count number fields of a fixed degree and having a fixed Galois group for its Galois closure, ordered by their absolute discriminant, say. In this talk, I shall discuss some background of this problem, and then report a recent work with Stanley Xiao. In our paper, we considered quartic $D_4$-fields whose ring of integers has a certain nice algebraic property, and we counted such fields by their conductor.

Boundary layers control the transport of momentum, heat, solutes and other quantities between walls and the bulk of a flow. The Prandtl-Blasius boundary layer was the first quantitative example of a flow profile near a wall and could be derived by an asymptotic expansion of the Navier-Stokes equation. For higher flow speeds we have scaling arguments and models, but no derivation from the Navier-Stokes equation. The analysis of exact coherent structures in plane Couette flow reveals ingredients of such a more rigorous description of boundary layers. I will describe how exact coherent structures can be scaled to obtain self-similar structures on ever smaller scales as the Reynolds number increases.

A quasilinear approximation allows to combine the structures self-consistently to form boundary layers. Going beyond the quasilinear approximation will then open up new approaches for controlling and manipulating boundary layers.

Structure from Motion (SfM) is a problem which asks: given photos of an object from different angles, can we reconstruct the object in 3D? This problem is important in computer vision, with applications including urban planning and autonomous navigation. A key part of SfM is bundle adjustment, where initial estimates of 3D points and camera locations are refined to match the images. This results in a high-dimensional nonlinear least-squares problem, which is typically solved using the Gauss-Newton method. In this talk, I will discuss how dimensionality reduction methods such as block coordinates and randomised sketching can be used to improve the scalability of Gauss-Newton for bundle adjustment problems.

In this talk we address the numerical approximation of Mean Field Games with local couplings. For finite difference discretizations of the Mean Field Game system, we follow a variational approach, proving that the schemes can be obtained as the optimality system of suitably defined optimization problems. In order to prove the existence of solutions of the scheme with a variational argument, the monotonicity of the coupling term is not used, which allow us to recover general existence results. Next, assuming next that the coupling term is monotone, the variational problem is cast as a convex optimization problem for which we study and compare several proximal type methods. These algorithms have several interesting features, such as global convergence and stability with respect to the viscosity parameter. We conclude by presenting numerical experiments assessing the performance of the proposed methods. In collaboration with L. Briceno-Arias (Valparaiso, CL) and F. J. Silva (Limoges, FR).

Many problems that involve the propagation of time-harmonic waves are naturally posed in unbounded domains. For instance, a common problem in the are a of acoustic scattering is the determination of the sound field that is generated when an incoming time-harmonic wave (which is assumed to arrive ``from infinity'') impinges onto a solid body (the scatterer). The boundary
conditions to be applied on the surface of the scatterer (most often of Dirichlet, Neumann or Robin type) tend to be easy to enforce in most numerical solution schemes. Conversely, the imposition of a suitable decay condition (typically a variant of the Sommerfeld radiation condition), which is required to ensure the well-posedness of the solution, is considerably more involved. As a result, many numerical schemes generate spurious reflections from the outer boundary of the finite computational domain.

Perfectly matched layers (PMLs) are in this context a versatile alternative to the usage of classical approaches such as employing absorbing boundary conditions or Dirichlet-to-Neumann mappings, but unfortunately most PML formulations contain adjustable parameters which have to be optimised to give the best possible performance for a particular problem. In this talk I will present a parameter-free PML formulation for the case of the two-dimensional Helmholtz equation. The performance of the proposed method is demonstrated via extensive numerical experiments, involving domains with smooth and polygonal boundaries, different solution types (smooth and singular, planar and non-planar waves), and a wide range of wavenumbers (R. Cimpeanu, A. Martinsson and M.Heil, J. Comp. Phys., 296, 329-347 (2015)). Possible extensions and generalisations will also be touched upon.