Gregoire Fournier

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Research & Academic Projects

  • Master Thesis: A Tropical Interpretation of Neural Networks

    This thesis explores the application of tropical geometry to neural networks, providing a framework to analyze their expressivity and information capacity. By interpreting neural networks as tropical varieties, the research offers insights into their structural properties and potential optimizations.

  • Ultrafilters and Tychonoff's Theorem (Functional Analysis)

    This project delves into the concept of ultrafilters and their role in the proof of Tychonoff's theorem, a fundamental result in topology stating that any product of compact spaces is compact. The study examines the interplay between ultrafilters and compactness, highlighting their significance in functional analysis.

  • The Iteration Number of the Weisfeiler-Lehman Algorithm

    Investigating the Weisfeiler-Lehman algorithm, a prominent method in graph isomorphism testing, this project focuses on determining the iteration number required for stable results. Understanding this iteration count is crucial for assessing the algorithm's efficiency and applicability in various graph-theoretical contexts.

  • Graphs 0-1 Law

    This study explores the 0-1 law in the context of graph theory, which asserts that certain properties in random graphs appear with probability approaching either 0 or 1 as the number of vertices increases. The project analyzes the implications of this law for understanding the behavior of large-scale networks.

  • Information Bound

    Focusing on the concept of information bounds, this project examines the theoretical limits of information retrieval and processing. It provides insights into the constraints and capabilities of various information systems, contributing to the broader field of information theory.

  • Dinur's Proof of the PCP Theorem

    This project reviews Irit Dinur's proof of the Probabilistically Checkable Proofs (PCP) theorem, a cornerstone in computational complexity theory. The PCP theorem has profound implications for the hardness of approximation problems, and Dinur's combinatorial proof offers a new perspective on its understanding.

  • Ladner's Theorem (Computational Complexity)

    Exploring Ladner's theorem, this project addresses the existence of problems in NP that are neither in P nor NP-complete, assuming P ≠ NP. The theorem has significant implications for classifying computational problems and understanding the landscape of computational complexity.

  • Bilu-Linial Stability (Computer Science)

    This study investigates the Bilu-Linial stability condition in the context of metric embeddings and clustering. The project analyzes how slight perturbations in input data affect the stability of clustering solutions, contributing to the development of robust algorithms in computer science.