Abstract. The central configurations of the Newtonian n-body problem are configurations of n point bodies, each with a positive mass, such that the Newtonian attraction may be exactly balanced by a centrifugal force. For example, a planar central configuration may rotate uniformly around its center of mass, thus defining a relative equilibrium of the n-body problem. Astronomical examples with n=3 are the Trojan asteroids and other bodies at some Lagrange point.
In 1995 I proved that there are only four types of central configurations of 4 bodies with equal masses, by using polynomial elimination. Here I wish to determine the central configurations of 5 bodies which have 3 bodies on a line. I will solve this problem in a symmetric case, where the existence result was published by Marian Gidea and Jaume Llibre in 2010, and independently by Kuo-Chang Chen and Jun-Shian Hsiao in 2011.

Abstract. We introduce a new family of rank metric codes. The construction of these codes relies on the "Chinese Remainder Theorem" for linearised polynomials. Linearised polynomials are polynomials in which the exponents of all the monomials are powers of q and the coefficients come from an extension field of the finite field of order q. The set of these polynomials forms a right Euclidean ring. We present the lifting of the isomorphism of the Chinese remainder theorem for linearised polynomials. We show how this lifting leads to a decoding algorithm for a special case of this family of codes: the case where the linearised polynomials have coefficients in the finite field of order q. (Joint work with Olivier Ruatta and Philippe Gaborit).

Abstract. We propose a new method for retrieving the algebraic structure of a generic alternant code given an arbitrary generator matrix, provided certain conditions are met. We then discuss how this challenges the security of the McEliece cryptosystem instantiated with this family of codes. The central object of our work is the quadratic hull related to a linear code, defined as the intersection of all quadrics passing through the columns of a given generator or parity-check matrix, where the columns are considered as points in the affine or projective space. The geometric properties of this object reveal important information about the internal algebraic structure of the code. This is particularly evident in the case of generalized Reed-Solomon codes, whose quadratic hull is deeply linked to a well-known algebraic variety called the rational normal curve. By utilizing the concept of Weil restriction of affine varieties, we demonstrate that the quadratic hull of a generic dual alternant code inherits many interesting features from the rational normal curve, on account of the fact that alternant codes are subfield-subcodes of generalized Reed-Solomon codes. If the rate of the generic alternant code is sufficiently high, this allows us to construct a polynomial-time algorithm for retrieving the underlying generalized Reed-Solomon code from which the alternant code is defined, which leads to an efficient key-recovery attack against the McEliece cryptosystem when instantiated with this class of codes. Finally, we discuss the generalization of this approach to Algebraic-Geometry codes and Goppa codes.

Abstract. Algbraic invariants plays a cental role in program analysis and verification, they provide a compact certificate for corrctness. A natural way to study them is through the algebraic closure of matrix monoids, which captures all the polynomial relations satisfied by the reachable configurations of linear or affine programs. Recent results show that given a finite set of rational matrices, the algebraic closure of the monoid they generate is effectively computable. This opens up new directions, ranging from the complexity of computing invariants to structural questions about orbit closures and algebraic groups. In this talk I will survey some of these problems and the techniques used to tackle them. I will begin with a polynomial-time algorithm for computing closures in the case of a single matrix, before moving on to the general case. I will then discuss the “reverse problem” for commutative matrices: given a set of polynomials, does their ideal define the orbit closure of an algebraic group? Finally, I will touch on connections with reachability and closure problems in one-counter systems (1-VASS). This talk is based on the following papers: arXiv:2407.09154, arXiv:2407.04626, arXiv:2507.09373.

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