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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.