Godeaux Lecture Prize

The Belgium Mathematical Society's "Godeaux Lecture Prize" is named in honour of the Belgian mathematician Lucien Godeaux. The prize is awarded every year, on a proposal from a Belgium Mathematical Society board member, to a prominent Belgian or international mathematician who is invited to give a talk at a conference in Belgium. It was first awarded in 2007.

These lectures honouring the memory of Lucien Godeaux are organised with the assets of the Belgian Centre for Mathematical Studies which were transferred to the Belgium Mathematical Society after the dissolution of this Centre. Lucien Godeaux (1887-1975) was one of the world's most prolific mathematicians (with more than 700 papers published) and took many initiatives to encourage young mathematicians to communicate their research. He was the founder of the Belgian Centre for Mathematical Studies in 1949. The Godeaux lecture is aimed at a broad audience with interest in mathematics.

Recipients of the Godeaux Lecture Prize.

2007 Godeaux Lecture.

Guido Vanden Berghe (Universiteit Gent, Belgium).

He delivered the lecture Simon Stevin (1548-1620), Mathematician, physicist, ..., Uomo universale at PhD-Day on Monday, 10 September 2007 at the Université Libre de Bruxelles.

Abstract.

In this talk we shall give in first instance attention to the family and the life of Simon Stevin. Born in Bruges his grandparents were original wealthy inhabitants of Ypres and Veurne. In the second place we shall present a comprehensive picture of the activities and the creative heritage of Simon Stevin, who made outstanding contributions to various fields of science in particular, physics and mathematics and many more. Among the striking spectrum of his ingenious achievements, it is worth emphasising, that Simon Stevin is rightly considered as the father of the system of decimal fractions as it is in use today. Stevin also urged the universal use of decimal fractions along with standardisation in coinage, measures and weights. This was a most visionary proposal. Stevin was the first since Archimedes to make a significant new contribution to statics and hydrostatics. His activities as an engineer will be discussed; in particular the construction of fortifications, windmills and the famous sailing chariot will be illustrated. He truly was ìuomo universalisî.

2009 Godeaux Lecture.

Cédric Villani (Ecole normale supérieure de Lyon, France).

He gave the lecture Mathematical problems arising from classical plasma physics at Campus Arenberg III in Heverlee, Leuven on 4 December 2009.

Abstract.

Messy as it is, classical plasma physics has triggered the discovery of remarkable mathematical phenomena. I will address some of these recent developments in this lecture, including the study of degenerate dissipative equations and the once mysterious Landau damping.

2010 Godeaux Lecture.

Gilles Godefroy (Institut de Mathématiques de Jussieu, France).

He gave the lecture Linear chaos and invariant subsets at PhD-Day on Monday, 13 September 2010 at the Université Libre de Bruxelles.

Abstract.

Many linear operators on the Hilbert space (and more generally on Banach spaces or Frechet spaces) behave in a chaotic way, in the sense of dynamical systems: a dense set of vectors have dense orbits, but there are also e.g. many eigenvectors. We will provide natural examples of such linear chaos (with differential operators for instance) and investigate it from the point of view of ergodic theory. We will also survey some recent techniques for showing the existence of closed invariant subspaces and comment the important problems which remain open. The relevant arguments are quite simple and technicalities will be avoided.

2012 Godeaux Lecture.

Ana Vargas Rey (Universidad Autónoma de Madrid, Spain).

She gave the lecture Multilinear Restriction, Multipliers and Waves on 6 June 2012 at the Mathematical Societies Conference at Liège.

Abstract.

In the seventies, C Fefferman proved that the spherical partial Fourier integrals of an $L^{p}$ function in $\mathbb{R}^{n} (n ≥ 2)$ do not converge in norm to the function, unless $p = 2$. After that, the study of other summation methods, such us Césaro or Bochner-Riesz sums, became the object of very active research in Harmonic Analysis. Those are examples of oscillatory integrals. After almost forty years, the problem of their boundedness is still open. Other examples of oscillatory integral are the adjoint Fourier restriction operators.

The problem of restriction of the Fourier transform to hypersurfaces (or more generally to submanifolds in $\mathbb{R}^{n}$ was posed by Stein in the seventies. This operator (in its adjoint form) gives the solution of dispersive equations (Schrödinger, wave, etc...) in terms of the Fourier transform of the initial data. Somehow the restriction operator is simpler than the Bochner-Riesz multiplier operators, and can be studied a model case. Moreover, there are many open problems about dispersive equations for which it can be used a powerful tool. As an example of those, we will introduce the problem of smoothing of the solution of the equations after local integration in time. In the case of the wave equation, it is a strong form of a version of the Bochner-Riesz sums, known as the cone multiplier.

The $L^{2}$ restriction estimates were proven on the seventies. It was Bourgain in the nineties who was first able to deal with other exponents. After his work there was a big development of the theory via the so-called bilinear method (Lee, Moyua, Tao, V Vega, Wolff ...). Bennett-Carbery-Tao proved a sharp multilinear version of the restriction theorem. Quite recently, Bourgain and Guth used their result to improve on the restriction problem. Their method can be also used to deal with multiplier operators. In particular, it has being used to obtain new bounds for the multiplier of the cone. This is a joint work with Sanghyuk Lee.

2016 Godeaux Lecture.

Jean Van Schaftingen (Université Catholique de Louvain, Belgium).

He delivered the lecture Sobolev mappings: from liquid crystals to irrigation via degree theory at the 9th Brussels Summer School of Mathematics in August 2016 at the Université Libre de Bruxelles.

Abstract.

Sobolev spaces are a natural framework for the analysis of problems in partial differential equations and calculus of variations. Some physical and geometric contexts, such as liquid crystals models and harmonic maps, lead to consider Sobolev maps, that is, Sobolev vector functions whose range is constrained in a surface or submanifold of the space. This additional nonlinear constraint provokes the appearance of finite-energy topological singularities. These singularities are characterised by a nontrivial topological invariant such as the topological degree, they represent an obstruction to the strong approximation by smooth maps and they become source and sink terms in an optimal transportation or irrigation problem of topological charges arising in the study of the weak approximation and of the relaxed energy.

2017 Godeaux Lecture.

Davy Paindaveine (Université Libre de Bruxelles, Belgium).

He gave the lecture Hypothesis testing in non-standard situations at the Joint VVWL-BMS-SBPMef conference of mathematics in Brussels in May 2017.

Abstract.

Hypothesis testing, that allows to check the validity of a model, the efficiency of a drug, the appropriateness of an economic policy, etc., is at the very heart of daily statistical practice. In standard situations, the mathematical theory of hypothesis testing is essentially complete and the performance of the corresponding statistical procedures is well understood. In this lecture, we will consider hypothesis testing in two non-standard situations. The first one is associated with the "high-dimensional" case, where many variables are recorded on a relatively small number of observations. The second one relates to a setup where the underlying distribution is close to a singularity of the model. We will identify the mathematical/statistical challenges raised by such cases and see how they can be addressed. We focus mainly on illustrations in directional statistics, that is, in problems where observations are on unit hyperspheres.

2018 Godeaux Lecture.

Laure-Saint Raymond (Ecole normale supérieure de Lyon, France).

She gave the lecture Internal waves in a domain with topography at PhD-Day at Ghent University on Friday 25 May 2018.

Abstract.

Stratification of the density in an incompressible fluid is responsible for the propagation of internal waves. In domains with topography, these waves exhibit interesting properties. In particular, numerical and lab experiments show that in 2D these waves concentrate on attractors for some generic frequencies of the forcing (see Dauxois et al). At the mathematical level, this behaviour can be analysed with tools from spectral theory and microlocal analysis.

2019 Godeaux Lecture.

John Guaschi (Université de Caen Normandie, France).

He gave the lecture Fixed point theory of n-valued maps, configuration spaces and braid groups at the conference Nielsen Theory and Related Topics at KU Leuven Campus Kulak Kortrijk on 3 June 2019.

2022 Godeaux Lecture.

Tim Gowers (Collège de France, University of Cambridge, UK).

He gave the lecture Can computers find interesting proofs of interesting theorems? at the University of Liège on Friday 13 May 2022.

Abstract.

Automatic theorem proving is a branch of computer science that has as its main goal to develop computer programs that can discover proofs of theorems. This turns out to be a challenging problem, and there are many examples of proofs that humans can find easily but that are well beyond the capabilities of the best programs. I shall discuss a project that I am just starting, which can be described as "extreme human-oriented" automatic theorem proving. Roughly, the idea is not to allow a program to use methods that a human mathematician would not use, such as big searches or training on vast datasets. This may seem like an unnecessary restriction, but I shall try to explain why I believe that it could be the key to making serious progress in the field.

2023 Godeaux Lecture.

Sophie Grivaux (Centre national de la recherche scientifique, Université de Lille, France).

She gave the lecture On the dynamics of the $\times _{p}$ and $\times _{q}$maps on the unit circle at the "Young Scholar Day" held on the Vrije Universiteit Brussel-campus in Brussels on 20 December 2023.

Abstract.

For every integer $n ≥ 1$, denote by $T_{n}$ the map $x \mapsto nx \mod 1$ from the circle group $T = \mathbb{R}/\mathbb{Z}$ into itself. Let $p, q ≥ 2$ be two multiplicatively independent integers. I will present an overview of Furstenberg's $\times _{p} - \times _{q}$ conjecture, which states that any continuous Borel probability measure on $T$ which is simultaneously $T_{p}$- and $T_{q}$-invariant must be the Lebesgue measure on $T$. Using Baire Category arguments, I will then show that generically, a continuous $T_{p}$-invariant probability measure $\mu$ on $T$ is such that $(T_{q}^{n} \mu)_{n≥0}$ does not converge $w^{∗}$ to the Lebesgue measure on $T$. This disproves Conjecture (C3) from a 1988 paper by R Lyons, which is a stronger version of Furstenberg's rigidity conjecture.

The talk will be based on a joint work with Catalin Badea (Lille).

2024 Godeaux Lecture.

Mirna Džamonja (Centre national de la recherche scientifique, Institut de recherche en informatique fondamentale, Université de Paris, France).

She gave the lecture The universality problem in mathematics at PhD-Day 2024 held on Campus Middelheim at the University of Antwerp on 24 May 2024.

Abstract.

We discuss the following problem: let $K$ be a class of objects equipped with a reflexive transitive relation ≤ . For example, $K$ could be the class of all graphs of a given infinite size $k$, with the relation $L ≤ L'$ being that there exists an order-preserving embedding from $L$ to $L'$. We discuss methods that have been developed for recognising situations when $K$ has or does not have a dominant element, meaning an element $X^{*}$ such that for all $X \in K$ we have $X ≤ X^{*}$. Such an element is called a universal element of $K$. For example, when $K$ is the class of all countable graphs, there is a universal element in $K$, namely the random graph. Already in this example and at the next infinite cardinal, the situation is much more complex and the answer is independent of the axioms of set theory.