Reverse mathematics of matroids

Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of some basis theorems for matroids and enumerated matroids. Next, using Weihrauch reducibility, we relate the basis results to combinatorial choice principles and statements about vector spaces. Finally, we formalize some of the Weihrauch reductions to extract related reverse mathematics results. In particular, we show that the existence of bases for vector spaces of bounded dimension is equivalent to the induction scheme for \Sigma^0_2 formulas

Weak one-basedness

We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterises linearity in the setting of geometric þ-rank 1 structures and that lovely pairs of weakly one-based geometric þ-rank 1 struc- tures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite dimensional vector spaces over division rings

Combinatorial Optimization

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry

A Simple Logic of Functional Dependence

This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games.Comment: 56 pages. Journal of Philosophical Logic (2021

Expansions géométriques et ampleur

The main result of this thesis is the study of how ampleness grows in geometric and SU-rank omega structures when adding a new independent dense/codense subset. In another direction, we explore relations of ampleness with equational theories; there, we give a direct proof of the equationality of certain CM-trivial theories. Finally, we study indiscernible closed sets—which are closely related with equations—and measure their complexity in the free pseudoplaneLe résultat principal de cette thèse est l'étude de l'ampleur dans des expansions des structures géométriques et de SU-rang oméga par un prédicat dense/codense indépendant. De plus, nous étudions le rapport entre l'ampleur et l'équationalite, donnant une preuve directe de l'équationalite de certaines théories CM-triviales. Enfin, nous considérons la topologie indiscernable et son lien avec l'équationalite et calculons la complexité indiscernable du pseudoplan libr