Federer’s characterization, which is a central result in the theory of functions of bounded variation, states that a set is of finite perimeter if and only if n−1-dimensional Hausdorff measure of the set's measure-theoretic boundary is finite. The measure-theoretic boundary consists of those points where both the set and its complement have positive upper density. I show that the characterization remains true if the measure-theoretic boundary is replaced by a smaller boundary consisting of those points where the lower densities of both the set and its complement are at least a given positive constant.

Numerous researchers recently applied empirical spectral analysis to the study of modern deep learning classifiers. We identify and discuss an important formal class/cross-class structure and show how it lies at the origin of the many visually striking features observed in deepnet spectra, some of which were reported in recent articles and others unveiled here for the first time. These include spectral outliers and small but distinct bumps often seen beyond the edge of a "main bulk". The structure we identify organizes the coordinates of deepnet features and back-propagated errors, indexing them as an NxC or NxCxC array. Such arrays can be indexed by a two-tuple (i,c) or a three-tuple (i,c,c'), where i runs across the indices of the train set; c runs across the class indices and c' runs across the cross-class indices. This indexing naturally induces C class means, each obtained by averaging over the indices i and c' for a fixed class c. The same indexing also naturally defines C^2 cross-class means, each obtained by averaging over the index i for a fixed class c and a cross-class c'. We develop a formal process of spectral attribution, which is used to show the outliers are attributable to the C class means; the small bump next to the "main bulk" is attributable to between-cross-class covariance; and the "main bulk" is attributable to within-cross-class covariance. Formal theoretical results validate our attribution methodology.
We show how the effects of the class/cross-class structure permeate not only the spectra of deepnet features and backpropagated errors, but also the gradients, Fisher Information matrix and Hessian, whether these are considered in the context of an individual layer or the concatenation of them all. The Kronecker or Khatri-Rao product of the class means in the features and the class/cross-class means in the backpropagated errors approximates the class/cross-class means in the gradients. These means of gradients then create C and C^2 outliers in the spectrum of the Fisher Information matrix, which is the second moment of these gradients. The outliers in the Fisher Information matrix spectrum then create outliers in the Hessian spectrum. We explain the significance of this insight by proposing a correction to KFAC, a well known second-order optimization algorithm for training deepnets.

We study nearly singular PDEs that arise in the solution of linear elasticity and incompressible flow. We will demonstrate, that due to the nearly singular nature, standard methods for the solution of the linear systems arising in a finite element discretisation for these problems fail. We motivate two key ingredients required for a robust multigrid scheme for these equations and construct robust relaxation and prolongation operators for a particular choice of discretisation.
 

Combinatorial anabelian geometry is a modern branch of anabelian geometry which deals with those aspects of anabelian geometry which manifest themselves over algebraically closed fields of characteristic zero. The origin of combinatorial anabelian geometry is in S. Mochizuki’s pioneering papers from 2007, in which he reinterpreted and generalised some key components of his earlier famous proof of the Grothendieck conjecture. S. Mochizuki  discovered that one can separate arguments which work over algebraically closed fields from arithmetic arguments, and study the former by using combinatorial methods. This led to a very nontrivial development of the theory of combinatorial anabelian geometry by S. Mochizuki and Y. Hoshi and other mathematicians. In this talk, after introducing the theory of combinatorial anabelian geometry I will discuss  applications of combinatorial anabelian geometry to the study of the absolute Galois group of number fields and of p-adic local fields and to the study of the Grothendieck-Teichmueller group. In particular, I will talk about the recent construction of a splitting of the natural inclusion of the absolute Galois group of p-adic numbers to the (largest) p-adic Grothendieck–Teichmueller group and a splitting of the natural embedding of the absolute Galois group of rationals into the commensurator of the absolute Galois group of the maximal abelian extension of rationals in the Grothendieck–Teichmueller group.
 

There is a very simple algorithm for the inference of posteriors for probability models on trees. This algorithm, known as "Belief Propagation" is widely used in coding theory, in machine learning, in evolutionary inference, among many other areas. The talk will be devoted to the analysis of Belief Propagation in some of the simplest probability models. We will highlight the interplay between Belief Propagation, linear estimators (statistics), the Kesten-Stigum bound (probability) and Replica Symmetry Breaking (statistical physics). We will show how the analysis of Belief Propagation allowed proof phase transitions for phylogenetic reconstruction in evolutionary biology and developed optimal algorithms for inference of block models. Finally, we will discuss the computational complexity of this 'simple' algorithm.

Boundary integral equations are an elegant tool to model and simulate a range of physical phenomena in bounded and unbounded domains.

While mathematically well understood, the numerical implementation (e.g. via boundary element methods) still poses a number of computational challenges, from the efficient assembly of the underlying linear systems up to the fast preconditioned solution in complex applications. In this talk we provide an overview of some of these challenges and demonstrate the efficient implementation of boundary element methods on modern CPU and GPU architectures. As part of the talk we will present a number of practical examples using the Bempp-cl boundary element software, our next generation boundary element package, that has been developed in Python and supports modern vectorized CPU instruction sets and a number of GPU types.

Iterative algorithms for the solution of convex quadratic optimization problems are investigated, which exploit inaccurate matrix-vector products. Theoretical bounds on the performance of a Conjugate Gradients method are derived, the necessary quantities occurring in the theoretical bounds estimated and a new practical algorithm derived. Numerical experiments suggest that the new method has significant potential, including in the steadily more important context of multi-precision computations.

Everybody is familiar with the concept of eigenvalues of a matrix. In this talk, we consider the nonlinear eigenvalue problem. These are problems for which the eigenvalue parameter appears in a nonlinear way in the equation. In physics, the Schroedinger equation for determining the bound states in a semiconductor device, introduces terms with square roots of different shifts of the eigenvalue. In mechanical and civil engineering, new materials often have nonlinear damping properties. For the vibration analysis of such materials, this leads to nonlinear functions of the eigenvalue in the system matrix.

One particular example is the sandwhich beam problem, where a layer of damping material is sandwhiched between two layers of steel. Another example is the stability analysis of the Helmholtz equation with a noise excitation produced by burners in a combustion chamber. The burners lead to a boundary condition with delay terms (exponentials of the eigenvalue).

We often receive the question: “How can we solve a nonlinear eigenvalue problem?” This talk explains the different steps to be taken for using Krylov methods. The general approach works as follows: 1) approximate the nonlinearity by a rational function; 2) rewrite this rational eigenvalue problem as a linear eigenvalue problem and then 3) solve this by a Krylov method. We explain each of the three steps.