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J’ai été interviewé pour la chronique Nos vies numériques de France Inter du 10 septembre 2026, consacrée à l’intelligence artificielle et aux équations de Navier–Stokes. Mon intervention a été fortement raccourcie — ce que le format très court de la chronique peut expliquer — et l’extrait diffusé ne reflète pas l’ensemble de ma pensée.

Oui, les progrès récents de l’IA sont passionnants. On entrevoit de nouvelles façons de faire des mathématiques, qui ouvriront très vraisemblablement la voie à des avancées majeures. Mais cet enthousiasme ne peut pas être séparé des questions qu’ils soulèvent pour la recherche et l’éducation.

Convexity is not the only route to fast optimization. The Polyak–Łojasiewicz (P–L) condition asks for something more directly connected to gradient descent: away from the minimum value, the gradient must remain large enough.

For a differentiable function with minimum value \(f_\star\), the condition is

\[\frac12\|\nabla f(x)\|^2\geq \mu\bigl(f(x)-f_\star\bigr), \qquad \mu>0.\]

It rules out stationary points with a nonoptimal value, without requiring a convex landscape.

The Mathematical Nexus brings together 800 animated vignettes and 140 accompanying Python notebooks, encompassing most of the mathematical content I have shared on social media.

The idea is to give these short explorations a more permanent home: a place where one can move from a visual intuition to the mathematics and the code behind it, and follow connections between topics.

A positive Markov matrix repeatedly mixes probability vectors until they approach a single stationary distribution. Sinkhorn’s algorithm has a similar visual story: repeated application of a nonlinear map brings positive scaling vectors toward a distinguished ray.

The analogy is particularly striking on the simplex of probability vectors with three entries — a triangle. Linear mixing shrinks straight-sided triangles; Sinkhorn bends them as it contracts. The appropriate geometry for the nonlinear picture is projective.

Take Gaussian distributions \(\nu_i=\mathcal N(m_i,\Sigma_i)\) with positive definite covariances, and weights \(\lambda_i\geq0\) summing to one. Their barycenter for the objective

\[\min_\mu\sum_i\lambda_i\,\mathrm{KL}(\mu\mid\nu_i)\]

is again Gaussian. The striking point is that we average the precision matrices \(\Sigma_i^{-1}\), rather than the covariances themselves.

How does a long product of matrices change when each factor is perturbed? There is an exact answer that replaces one factor at a time:

\[A_1\cdots A_n-B_1\cdots B_n =\sum_{k=1}^n A_1\cdots A_{k-1}(A_k-B_k)B_{k+1}\cdots B_n.\]

This discrete Duhamel formula is simply a telescoping sum — but one whose ordering is indispensable when matrices do not commute.

For a convex function \(f:\mathbb R^d\to\mathbb R\cup\{+\infty\}\), write

\[f^*(y)=\sup_x\{\langle x,y\rangle-f(x)\}\]

for its Legendre transform. A beautiful consequence of Prékopa–Leindler is that the functional

\[\Phi(f)=-\log\int_{\mathbb R^d}e^{-f^*(y)}\,\mathrm dy\]

is convex, on a class of proper closed convex functions for which the integrals below are finite and strictly positive.

A symmetric matrix can be positive in two different senses: positive semidefinite, and nonnegative entry by entry. Requiring both gives the cone of doubly nonnegative matrices

\[\mathcal D_n=\{X\in\mathbb R^{n\times n}:X=X^\top,\ X\succeq0,\ X_{ij}\geq0\}.\]

A seemingly similar condition is to ask for a Gram factorization with nonnegative vectors. This defines the cone of completely positive matrices. The two cones agree up to dimension four — but not in dimension five.

The alpha version of my new book, Optimal Transport for Machine Learners, is out. In particular, there is an online version with interactive figures, so that the mathematical exposition can be explored alongside the examples.

This is an early version: the text and its illustrations will continue to evolve. My aim is to make the geometry and computation of optimal transport accessible to readers interested in machine learning, with the mathematics and its visual intuition close together.

Given a way to measure a positive semidefinite matrix, a natural way to measure a general matrix is to apply it to its Gram matrix:

\[N(X)=\sqrt{\gamma(X^\top X)}.\]

When does this construction give a norm? The key condition is monotonicity in the positive semidefinite order: increasing a covariance should not decrease its size.

It is intuitively clear that straighter trajectories should be easier to integrate numerically. This intuition is especially appealing in flow matching, where one would like to generate samples with as few velocity evaluations as possible.

I wanted a precise estimate that exhibits this dependence directly. Rewriting the usual stability argument behind Cauchy–Lipschitz gives one: Euler’s global error is controlled by the material acceleration along the exact trajectory, multiplied by a stability factor.