On considère un corps $K$ muni d'une mesure et d'une distance. La densité locale d'un ensemble $X$ dans $K^n$ en un point est définie comme la limite, si elle existe, de volumes locaux normalisés. On peut généraliser cette notion à des corps valués pour lesquels il n'existe pas de théorie de la mesure classique, comme $\mathbb{C}(!(t)!)$, en utilisant l'intégration motivique. Le but de cet exposé est de présenter une formule permettant le calcul de la densité locale motivique de singularités isolées de surfaces, en utilisant une donnée supplémentaire : les taux internes reliés à la géométrie bilipschitz de la singularité, introduits par Birbair, Neumann et Pichon.
Humans are pretty good at coming up with heuristics that solve hard problems almost optimally. In the 1980s and 1990s, researchers began asking whether it was possible to design efficient algorithms that approximately solve hard problems, such as NP-hard problems. For some problems, there was success, for others, not so much. This led to a fundamental question: Can we rule out the existence of good approximation algorithms?
There were some lower bounds, but a general methodology for proving such results was lacking. Then came the PCP theorem, which provided a powerful new way to prove hardness of approximation and has since become a go-to hammer for establishing limits on approximation algorithms.
In this mini course, we will start with the definition of a NP, see why it naturally leads to inapproximability results, and then prove the PCP theorem (or at least a weaker version of it). The course will consist of 2 talks, with the following rough outline.
Talk 2: A strategy towards proving the PCP theorem, abstracting out the steps, testing zeroeness of a polynomial is enough, zero-on-variety test. This talk will be based on parts from https://eccc.weizmann.ac.il/report/2025/165/ and https://eccc.weizmann.ac.il/report/2026/134/. These are joint works with Prashanth Amireddy, Srikanth Srinivasan, Madhu Sudan, and Sophus Valentin Willumsgaard.
Humans are pretty good at coming up with heuristics that solve hard problems almost optimally. In the 1980s and 1990s, researchers began asking whether it was possible to design efficient algorithms that approximately solve hard problems, such as NP-hard problems. For some problems, there was success, for others, not so much. This led to a fundamental question: Can we rule out the existence of good approximation algorithms?
There were some lower bounds, but a general methodology for proving such results was lacking. Then came the PCP theorem, which provided a powerful new way to prove hardness of approximation and has since become a go-to hammer for establishing limits on approximation algorithms.
In this mini course, we will start with the definition of a NP, see why it naturally leads to inapproximability results, and then prove the PCP theorem (or at least a weaker version of it). The course will consist of 2 talks, with the following rough outline.
Talk 1: Introduction to PCP, Gap Problems, Hardness of approximation for Gap problems, Intuition behind why even the PCP theorem is true. Finally, if time permits, then a strategy towards the proof of PCP theorem.
The goal of this talk is to introduce skeletal semantics, a software framework for specifying and analysing programming languages, and provide a mathematical foundation for it. To this end, we will first introduce SKI calculus as an example of a target programming language and we will show how skeletal semantics can be used to specify it. In a second part, we will motivate our mathematical setting by introducing initial algebra semantics and Lawvere theories, before studying virtual double theories. They are a virtual-double-categorical extension of the latter, allowing for the interpretation of some morphisms as relations. Finally, we show that a skeletal specification (in particular the one of the SKI calculus) may be interpreted as a presentation of a virtual double theory, so that its category of models provides the intended language.
In 1948 Tarski posed the Following question: let $\mathbb{R}{\text{exp}}$ be the expansion of the ordered ring of the reals with the exponential function. Is the complete theory $T$ decidable?}}$ of $\mathbb{R}_{\text{exp}
In 1984, attempting to tackle this question geometrically led van den Dries to introduce the notion now known as o-minimality, with Wilkie later proving that $\mathbb{R}{\text{exp}}$ is model complete, and thus, by earlier results of Khovanskii, that $\mathbb{R}$ is decidable, provided that the real Schanuel's conjecture is true. Thus, the problem has been reduced to a far-reaching conjecture in transcendental number theory which seems unlikely to be solved any time soon.}}$ is indeed o-minimal. Finally Macintyre and Wilkie proved in 1996 that $T_{\text{exp}
However, in the late forties, Kleene and Mostowski independently introduced a hierarchy of undecidabilities now known as the arithmetical hierarchy. We will place $T_{\text{exp}}$ in the arithmetical hierarchy providing a low upper bound by using the tools of effective o-minimality. We will contrast the case of $\texy{exp}$ with that of other real analytic functions and real computable functions. The latter case provably exhibits a great variety of possible computational complexities.
(This talk is based on a paper in preparation by the speaker)
We will study sums of squares of regular functions on real algebraic curves and surfaces. Notion of a weakly factorial variety will be introduced and its usefulness in the study of Pythagoras number will be shown. We will then show that if X is a nonsingular rational real algebraic surface then the Pythagoras number of the ring of regular functions on X is bounded from above by 12.
TBA
I will sketch some themes and results related to real integrals and their connections to geometry, analysis, and number theory. Using real geometry and real semi-algebraic sets, I will sketch classes of functions which are stable under (parametric) integration, Fourier transform, Mellin transform, and Laplace transform (Laplace still being work in progress). This has connections to classes of distributions and their properties (like holonomicity), to periods and exponential periods and families thereof, and questions around (functional) transcendence. The directions I will focus most on comprise work by many people, in particular by Aizenbud, (my PhD student) Buggenhout, Comte, Kaiser, Lion, Miller, Raibaut, Rolin, Servi, Stout, (my PhD student) Vandebrouck. I will raise some open questions for future research as well.
Let X be a nonsingular real algebraic set and let M be a compact smooth submanifold of X. I will discuss approximation of M by nonsingular algebraic subsets of X. There is a new (2024) result by Olivier Benoist concerning the case 2dimM<dimX and involving an appropriate bordism condition. I will interpret the bordism condition in terms of homology and present some applicationsLet X be a nonsingular real algebraic set and let M be a compact smooth submanifold of X. I will discuss approximation of M by nonsingular algebraic subsets of X. There is a new (2024) result by Olivier Benoist concerning the case 2dimM<dimX and involving an appropriate bordism condition. I will interpret the bordism condition in terms of homology and present some applications.
Les équations aux dérivées partielles dispersives sont des équations d'évolution (c'est-à-dire comportant la variable temporelle) dont les solutions préservent l'énergie, mais peuvent néanmoins décroître en temps long parce que les différentes fréquences se propagent avec des vitesses distinctes. Dans certains cas, il existe des solutions spéciales appelées « solitons » qui ne changent pas de forme au fil du temps. La conjecture de résolution en solitons prédit que les solitons sont le seul obstacle à la décroissance des solutions. Plus précisément, toute solution se décompose en une superposition de solitons et d'un terme évanescent appelé « radiation ». Nous présenterons cette conjecture dans le contexte de l'équation des applications d'ondes critique, qui est l'analogue de l'équation des ondes pour les applications de R^2 dans S^2. Nous considérons les solutions « équivariantes », qui sont des solutions ayant une certaine symétrie préservée par le flot. Dans ce cas, les solitons sont centrés à l'origine, mais ils peuvent néanmoins se découpler si leurs échelles caractéristiques sont très différentes. Dans un travail commun avec Andrew Lawrie, nous démontrons que la résolution en solitons est vérifiée. À la lumière de ce résultat, il est naturel d'examiner le comportement à long terme des échelles de plusieurs solitons en interaction. Dans cette direction, dans un travail récent avec Joachim Krieger, nous construisons des solutions développant une singularité par concentration simultanée de deux solitons à l'origine.
Les équations aux dérivées partielles dispersives sont des équations d'évolution (c'est-à-dire comportant la variable temporelle) dont les solutions préservent l'énergie, mais peuvent néanmoins décroître en temps long parce que les différentes fréquences se propagent avec des vitesses distinctes. Dans certains cas, il existe des solutions spéciales appelées « solitons » qui ne changent pas de forme au fil du temps. La conjecture de résolution en solitons prédit que les solitons sont le seul obstacle à la décroissance des solutions. Plus précisément, toute solution se décompose en une superposition de solitons et d'un terme évanescent appelé « radiation ». Nous présenterons cette conjecture dans le contexte de l'équation des applications d'ondes critique, qui est l'analogue de l'équation des ondes pour les applications de R^2 dans S^2. Nous considérons les solutions « équivariantes », qui sont des solutions ayant une certaine symétrie préservée par le flot. Dans ce cas, les solitons sont centrés à l'origine, mais ils peuvent néanmoins se découpler si leurs échelles caractéristiques sont très différentes. Dans un travail commun avec Andrew Lawrie, nous démontrons que la résolution en solitons est vérifiée. À la lumière de ce résultat, il est naturel d'examiner le comportement à long terme des échelles de plusieurs solitons en interaction. Dans cette direction, dans un travail récent avec Joachim Krieger, nous construisons des solutions développant une singularité par concentration simultanée de deux solitons à l'origine.
In this talk, I will present some results obtained during my PhD, here at LAMA. I will talk about shape optimization problems under inradius constraint, focusing on the 2D case. I will discuss two approaches to tackle these problems, one analytical and the other numerical. First, I present a result obtained in collaboration with D. Bucur and G. Buttazzo. We show that the honeycomb structure is optimal for the Cheeger constant and the torsional rigidity. I will focus my presentation on the Cheeger case. Secondly, I explain a new numerical scheme implemented with B. Bogosel. We aim to numerically solve these problems with the added constraint of simple connectivity, in the framework of the levelset method.
We study the problem of answering connectivity queries on real solution sets of polynomial systems, a central question in computational real algebraic geometry with applications ranging from geometry to robotics. Following Canny’s roadmap framework, connectivity queries on a high-dimensional solution set are reduced to connectivity queries on a one-dimensional algebraic curve intersecting all connected components.
We present an algorithm that, under generic assumptions, computes such roadmaps in subquadratic time with respect to the output size, thereby extending the best known nearly optimal complexity bounds to non-compact situations. We also show that the connectivity of the roadmap itself, namely of a real algebraic curve, can be analyzed in time cubic in its size.
Finally, we illustrate the practical applicability of these methods through original implementations leveraging recent advances in polynomial system solving software.
In this seminar, I will focus on bijective digitized rotations in 2D and 3D. In the first part, I will present several new methods to approximate digitized rotations with bijective transformations. These methods, along with several classical approaches, are compared both in terms of accuracy with respect to Euclidean rotations and in terms of computational complexity. In the second part, I will focus on the characterization of bijective digitized rotations in 3D. While this characterization is well known in 2D, its extension to 3D remains an open problem. To address this, I will particularly focus on 3D non-surjective digitized reflections. Finally, some experimental results obtained with DGtal will be presented.
En géométrie intégrale, la formule cinématique additive exprime le volume moyen de la somme de Minkowski de deux compacts convexes placés aléatoirement dans l'espace euclidien. Que se passe-t-il si on ne les suppose plus convexes ? Et si l'on remplace l'espace euclidien par un autre groupe de Lie ? Dans un travail en collaboration avec Andreas Bernig, nous montrons une formule cinématique additive pour les sous-analytiques de l'espace euclidien et de la sphère de dimension 3. La clé est de généraliser la somme de Minkowski par la convolution des fonctions constructibles définie par Viro et Schapira dans les années 80. Cette convolution, fondée sur des calculs de caractéristique d'Euler, est définie à l'aide d'un formalisme des opérations sur les fonctions constructibles.
Gallina, the programming language underlying Rocq, has relatively efficient evaluation mechanisms. But sometimes this is not enough: we would like to write some computationally intensive part in a more suitable, off-the-shelve language. I will demo and discuss our current prototype which extends the Rocq kernel to allow declaring and calling native foreign functions. On the user-facing side, it rests on a monad similar to Haskell's IO, featuring unbounded iteration, error, imperative buffers and, well, foreign function calls. To allow reasoning on this monad we provide a weakest precondition program logic defined using Iris. This logic is already flexible enough to axiomatize the specification of most integer functions in GMP[0] and we hope to interface it with existing Rocq semantics for other languages such as C with VST[1] or Capla[2].
[0] https://gmplib.org/
[1] https://vst.cs.princeton.edu/
[2] https://fresco.gitlabpages.inria.fr/capla/language/index.html
During the talk, I will discuss the classical Hilbert 17th problem. This problem can be divided into two smaller ones. First question is, if we have a nonnegative function, is it necessarily a sum of squares? And if it is a sum of squares, how many squares are needed? We will be mainly interested in the second problem. In particular, I will discuss recent results concerning the Pythagoras number of a ring for various rings occuring in real algebraic geometry, such as coordinate rings of a real algebraic set, rings of regular functions on an algebraic set, fields of rational functions, and possibly the rings of k-regulous functions.