Domain decomposition methods are widely employed for the numerical solution of partial differential equations on parallel computers. We develop an adjoint-based a posteriori error analysis for overlapping multiplicative Schwarz domain decomposition and for overlapping additive Schwarz. In both cases the numerical error in a user-specified functional of the solution (quantity of interest), is decomposed into a component that arises due to the spatial discretization and a component that results from of the finite iteration between the subdomains. The spatial discretization error can be further decomposed in to the errors arising on each subdomain. This decomposition of the total error can then be used as part of a two-stage approach to construct a solution strategy that efficiently reduces the error in the quantity of interest.

The development of effective solvers for high frequency wave propagation problems, such as those described by the Helmholtz equation, presents significant challenges. One promising class of solvers for such problems are parallel domain decomposition methods, however, an appropriate coarse space is typically required in order to obtain robust behaviour (scalable with respect to the number of domains, weakly dependant on the wave number but also on the heterogeneity of the physical parameters). In this talk we introduce a coarse space based on generalised eigenproblems in the overlap (GenEO) for the Helmholtz equation. Numerical results within FreeFEM demonstrate convergence that is effectively independent of the wave number and contrast in the heterogeneous coefficient as well as good performance for minimal overlap.

Can we get rigorous answers when computing with real and complex numbers? There are now many applications where this is possible thanks to a combination of tools from computer algebra and traditional numerical computing. I will give an overview of such methods in the context of two projects I'm developing. The first project, Arb, is a library for arbitrary-precision ball arithmetic, a form of interval arithmetic enabling numerical computations with rigorous error bounds. The second project, Fungrim, is a database of knowledge about mathematical functions represented in symbolic form. It is intended to function both as a traditional reference work and as a software library to support symbolic-numeric methods for problems involving transcendental functions. I will explain a few central algorithmic ideas and explain the research goals of these projects.

We consider CFTs with large gap in the spectrum of operators and a large number of degrees of freedom (large central charge). We analytically study a Heavy-Heavy-Light-Light correlation function, where Heavy, refers to an operator with conformal dimension which scales like the central charge and Light, refers to an operator whose dimension is of order unity in the large central charge limit. In certain regimes, the correlation function can be examined analytically leading to very simple and suggestive expressions.

The classical Universal Approximation Theorem certifies that the universal approximation property holds for the class of neural networks of arbitrary width. Here we consider the natural `dual' theorem for width-bounded networks of arbitrary depth, for a broad class of activation functions. In particular we show that such a result holds for polynomial activation functions, making this genuinely different to the classical case. We will then discuss some natural extensions of this result, e.g. for nowhere differentiable activation functions, or for noncompact domains.
 

The hard Lefschetz theorem is a fundamental statement about the symmetry of the cohomology of algebraic varieties. In nearly all cases that we systematically understand it, it comes with a geometric meaning, often in form of Hodge structures and signature data for the Hodge-Riemann bilinear form.

Nevertheless, similar to the role the standard conjectures play in number theory, several intriguing combinatorial problems can be reduced to hard Lefschetz properties, though in extreme cases without much geometric meaning, lacking any existence of, for instance,  an ample cone to do Hodge theory with.

I will present a way to prove the hard Lefschetz theorem in such a situation, by introducing biased pairing and perturbation theory for intersection rings. The price we pay is that the underlying variety, in a precise sense, has itself to be sufficiently generic. For instance, we shall see that any quasismooth, but perhaps nonprojective toric variety can be "perturbed" to a toric variety with the same equivariant cohomology, and that has the hard Lefschetz property.

Finally, I will discuss how this applies to prove some interesting theorems in geometry, topology and combinatorics. In particular, we shall see a generalization of a classical result due to Descartes and Euler: We prove that if a simplicial complex embeds into euclidean 2d-space, the number of d-simplices in it can exceed the number of (d-1)-simplices by a factor of at most d+2.

Congruence monoids in the ring of integers are given by certain unions of arithmetic progressions. To each congruence monoid, there is a canonical way to associate a semigroup C*-algebra. I will explain this construction and then discuss joint work with Xin Li on K-theoretic invariants. I will also indicate how all of this generalizes to congruence monoids in the ring of integers of an arbitrary algebraic number field.