“The essence of mathematics lies in its freedom” -- Georg Cantor

Biography

Thierry De Pauw was born and grew up in Brussels, Belgium. In 1998, at the national Belgian mathematical olympiad he was awarded the *Prix Vanhamme* for the most elegant proof. He graduated from the Université Catholique de Louvain with a bachelor’s degree in mathematics in 1993 and from the same university with a PhD degree in 1998. He subsequently held temporary post-doctoral appointments at University College London, Rice University, and Université de Paris Sud, Orsay (now Université Paris Saclay). He worked as a research associate with the FNRS, Belgium, at the Université Catholique de Louvain, starting from 2002 and now holds the title of honorary senior researcher of the FNRS. In 2008, he was awarded the *Prix Jacques Deruyts* in mathematical analysis, for the period 2004 – 2008, by the Belgian Royal Academy of Science. The same year, he accepted a professorship at the Université Paris VII (now Université Paris Cité) which he held until 2024 when he joined the Institute for Theoretical Sciences at Westlake University.

Research

Professor De Pauw’s scientific research belongs to mathematical analysis, with a geometric flavor. Specifically, he contributes to the field of Geometric Measure Theory, a branch of mathematics of which the paradigm is Plateau’s problem. It consists of studying the geometrical complexity of soap films and soap bubbles, including those in infinite dimensional spaces.

He started his career studying non absolutely convergent integration theories. These generalize Lebesgue’s celebrated theory and integrate highly oscillating derivatives that are not Lebesgue integrable. Ideas originating from his early mathematical days allowed the following subsequent developments. With W.F. Pfeffer, he identified removable singularities for a class of partial differential equations including the minimal surface equation and the Laplace equation [12]. Together, they also characterized those distributions F such that the ill-posed linear partial differential equation *div(v) = F* admits a continuous solution *v*, a kind of optimal regularity result [10]. In fact, the equation is ill-posed to the extent that it is impossible to choose a solution *v* that depends in a uniformly continuous way upon the data *F* even if that dependence is allowed to be nonlinear! Further developments, together with R. Hardt and W.F. Pfeffer, led to the creation of a new homology and cohomology theory, with real coefficients, of compact metric spaces *X* that reflects (at least some) *metrical* aspects of *X* rather than merely *topological* ones [5].

In 1960, H. Federer and W.H. Fleming published their seminal paper *Normal and Integral Currents*, setting the scene for many decades of contributions to the study of Plateau’s problem in Euclidean spaces and compact Riemannian manifolds. Many interesting cases of Plateau’s problem were not encompassed in this original context. For instance, the mass functional does not model some familiar soap films that are best described by means of the size functional: Even though the existence problem is still open in general, professor De Pauw contributed to the solution in a slightly different formalism [9] and, with R. Hardt, in regard to approximating problems [13]. Building on previous work of L. Ambrosio – B. Kirchheim, and B. White, De Pauw and Hardt introduced in [8] a new theory providing the same useful tools as Federer-Fleming’s, yet valid in a more general realm regarding coefficients group and ambient spaces even though some basic theorem upon which the original proofs rely does not hold in this generality [6]. This opened up many new paths of investigations, for instance: The study of partial regularity in infinite dimensional spaces [11, 4], the study of partial regularity in infinite dimensional Banach spaces [7], and the study of isoperimetric-type inequalities in singular ambient spaces [3].

Professor De Pauw has also contributed to some questions arising in classical real analysis, for instance pertaining to derivation bases for the Lebesgue density theorem [2]. Recently, together with Ph. Bouafia, he has proved the existence in a categorical sense of a smallest version of any measure space for which the Radon-Nikodým theorem holds [1] and they identified explicitly this *Radon-Nikodýmification *in the case of integral geometric measures.

Representative Publications

1. Ph. Bouafia, Th. De Pauw. Radon-Nikodýmification of arbitrary measure spaces. To appear in Extracta Math.

2. Th. De Pauw. Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation. J.Ec. polytech. Math., 9, 2022, 1473-1512.

3. Th. De Pauw, R. Hardt. Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets. J. Singul., 24, 2022, 145-168.

4. Th. De Pauw, R. Züst. Partial regularity of almost minimising rectifiable G chains in Hilbert space. Amer. J. Math., 141(6), 2019, 1591-1705.

5. Th. De Pauw, R. Hardt, W.F. Pfeffer. Homology of normal chains and cohomology of charges. Memoirs Amer. Math. Soc. 247 n° 1172, 2017, v+115pp.

6. Th. De Pauw. An example pertaining to the failure of the Besicovitch-Federer structure Theorem in Hilbert space. Publ. Mat., 61(1), 2017, 153-173.

7. Th. De Pauw, A. Lemenant, V. Millot. On sets minimising their weighted length in uniformly convex separable Banach spaces. Adv. Math., 305, 2017, 1268-1319.

8. Th. De Pauw, R. Hardt. Rectifiable and flat G chains in metric spaces. Amer. J. Math., 134(1), 2012, 1-69.

9. Th. De Pauw. Size minimising surfaces. Ann. Sci. Ecole Norm. Sup., 42(1), 2009, 37-101.

10. Th. De Pauw, W.F. Pfeffer. Distributions for which div v = F has a continuous solution. Commun. Pure Appl. Math., 61(2), 2008, 230-260.

11. Th. De Pauw. Concentrated, nearly monotonic, epiperimetric measures in Euclidean space. J. Differential Geom., 77(1), 2007, 77-134.

12. Th. De Pauw, W.F. Pfeffer. The Gauss-Green Theorem and removable sets for 2nd order PDEs in divergence form. Adv. Math., 183(1), 2004, 155-182.

13. Th. De Pauw, R. Hardt. Size minimisation and approximating problems. Calc. Var. Partial Diff. Eq., 17(4), 2003, 405-442.

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