Domain decomposition research intensified in the early nineties, and there is still substantial research activity in this field. There has been however an important shift in domain decomposition, and I will explain three new interesting research directions that are pursued very actively at the moment, and give newest results:
1. Iterative solvers for time harmonic wave propagation: time harmonic wave propagation problems are very hard to solve by iterative methods. All classical iterative methods, like Krylov methods, multigrid, and also domain decomposition methods, fail for the key model problem, the Helmholtz equation. There are new, highly promising domain decomposition methods for such problems, which I will present, and I will also state precisely under which conditions they can work well, and when they still fail.
2. Coarse space components: domain decomposition analysis has lacked behind multigrid in the precise understanding of the interaction between the domain decomposition smoother and coarse space solver, and all classical domain decomposition solvers need Krylov acceleration to be effective, while multigrid does not. I will present a new spectral analysis of the Schwarz iteration operator, which allows us to achive as an accurate understanding of two level Schwarz methods as the seminal Fourier analysis of multigrid methods.
3. Time parallelization: new computing architectures have too many computing cores to parallelize only in space for evolution problems. I will present time and space-time domain decomposition methods and explain which can be effective for parabolic and hyperbolic problems.[-]

A simple group is pseudofinite if and only if it is isomorphic to a (twisted) Chevalley group over a pseudofinite field. This celebrated result mostly follows from the work of Wilson in 1995 and heavily relies on the classification of finite simple groups (CFSG). It easily follows that any simple pseudofinite group $G$ is finite-dimensional. In particular, if $\operatorname{dim}(G)=3$ then $G$ is isomorphic to $\operatorname{PSL}(2, F)$ for some pseudofinite field $F$. In this talk, we describe the structures of finite-dimensional pseudofinite groups with dimension $<4$, without using CFSG. In the case $\operatorname{dim}(G)=3$ we show that either $G$ is soluble-by-finite or has a finite normal subgroup $Z$ so that $G / Z$ is a finite extension of $\operatorname{PSL}(2, F)$. This in particular implies that the classification $G \cong \operatorname{PSL}(2, F)$ from the above does not require CFSG. This is joint work with Frank Wagner.[-]

We consider independent Hermitian heavy-tailed random matrices. Our model includes the Lévy matrices as well as sparse random matrices with O(1) non-zero entries per row. By representing these matrices as weighted graphs, we derive a large deviation principle for key macroscopic observables. Specifically, we focus on the empirical distribution of eigenvalues, the joint neighborhood distribution, and the joint traffic distribution. As an application, we define a notion of microstates entropy for traffic distribution which is additive under free traffic convolution.[-]

In a seminal paper Henry Whitehead introduced four elementary operators, collapses and expansions (the inverse of a collapse), perforations and fillings (the inverse of a perforation), which correspond to an homotopy equivalence between two simplicial complexes. In this talk, we consider some transformations which are obtained by the means of these four operators. The presentation is composed of two parts. We begin the first part by introducing a certain axiomatic approach for combinatorial topology, which is settled in the framework of completions. Completions are inductive properties which may be expressed in a declarative way and may be combined. Then, we present a transformation that is based solely on collapses and expansions. This transformation involves homotopic pairs, it may be seen as a refinement of simple homotopy, which takes as input a single object. A homotopic pair is a couple of objects (X, Y ) such that X is included in Y and (X, Y ) may be transformed to a trivial couple by collapses and expansions that keep X inside Y . Our main result states that the collection of all homotopic pairs may be fully described by four completions which correspond to four global properties. After, we consider a transformation that is based on collapses, expansions, perforations, and fillings. This transformation involves contractible pairs, which are extensions of homotopic pairs. Again we show that the collection of all contractible pairs may be fully described by four completions which correspond to four global properties. Three of these completions are the same as the ones describing homotopic pairs. In the second part of the presentation, we introduce the notion of a Morse sequence, which provides a very simple approach to discrete Morse theory. A Morse sequence is obtained by considering only expansions and fillings of a simplicial complex, or, in a dual manner, by considering only collapses and perforations. A Morse sequence may be seen as an alternative way to represent the gradient vector field of an arbitrary discrete Morse function. We introduce reference maps, which are maps that associate a set of critical simplexes to each simplex appearing in a Morse sequence. By considering the boundary of each critical simplex, we obtain a chain complex from these maps, which corresponds precisely to the Morse complex. Then, we define extension maps. We show that, when restricted to homology, an extension map is the inverse of a reference map. Also we show that these two maps allow us to recover directly the isomorphism theorem between the homology of an object and the homology of its Morse complex[-]

Engineering analysis should match an underlying designed shape and not restrict the quality of the shape. I.e. one would like finite elements matching the geometric space optimized for generically good shape. Since the 1980s, classic tensor-product splines have been used both to define good shape geometry and analysis functions (finite elements) on the geometry. Polyhedral-net splines (PnS) generalize tensor-product splines by allowing additional control net patterns required for free-form surfaces: isotropic patterns, such as n quads surrounding a vertex, an n-gon surrounded by quads, polar configurations where many triangles join, and preferred direction patterns, that adjust parameter line density, such as T-junctions. PnS2 generalize C1 bi-2 splines, generate C1 surfaces and can be output bi-3 Bezier pieces. There are two instances of PnS2 in the public domain: a Blender add-on and a ToMS distribution with output in several formats. PnS3 generalize C2 bi-3 splines for high-end design. PnS generalize the use of higher-order isoparametric approach from tensor-product splines. A web interface offers solving elliptic PDEs on PnS2 surfaces and using PnS2 finite elements.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.[-]

The classical von Neumann inequality provides a fundamental link between complex analysis and operator theory. It shows that for any contraction $T$ on a Hilbert space and any polynomial $p$, the operator norm of $p(T)$ satisfies
$\|p(T)\| \leq \sup _{|z| \leq 1}|p(z)|$
Whereas Andô extended this inequality to pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false. However, it is still not known whether von Neumann's inequality for triples of commuting contractions holds up to a constant. I will talk about this question and about function theoretic upper bounds for $\|p(T)\|$.[-]