Spreads of [set of prime numbers]3 over finite fields can yield geproci sets. We study the existence of transversals to such spreads, proving that spreads with two transversals exist for all finite fields, before further considering the groupoids coming from spreads when transversals do or do not exist. This is further considered for spreads of hig...
We first show a simple but striking result in bilevel optimization: unconstrained $C^\infty$ smooth bilevel programming is as hard as general extended-real-valued lower semicontinuous minimization. We then proceed to a worst-case analysis of box-constrained bilevel polynomial optimization. We show in particular that any extended-real-valued semi-al...
Dans cette thèse de doctorat, nous avons, dans un premier temps, étudié la décomposition en composantes irréductibles du lieu des points fixes sous l’action d’un sous-groupe fini Γ de SL2(C) de la variété de carquois de Nakajima du carquois de Jordan. La variété de carquois associé au carquois de Jordan est isomorphe soit au schéma ponctuel de Hilb...
Belmans, Oberdieck, and Rennemo asked whether all unnatural automorphisms of Hilbert schemes of points on surfaces, i.e. those automorphisms which do not arise from the underlying surface, can be characterized by the fact that they do not preserve the diagonal of non-reduced subschemes. Sasaki recently published examples, independently discovered b...
In this paper, we exhibit a formula relating punctured Gromov-Witten invariants used by Gross and Siebert in [GS2] to 2-point relative/logarithmic Gromov-Witten invariants with one point-constraint for any smooth log Calabi-Yau pair (W, D). Denote by Na,b the number of rational curves in W meeting D in two points, one with contact order a and one w...
An affine surface (over an algebraically closed field) is a variety of dimension 2 defined by polynomial equations. Given an endomorphism of such a surface, we can ask the following questions: Are there Zariski dense orbits ? If the orbit of a point goes to infinity, can we control the speed of divergence ? Are there a lot of periodic orbits ? Can ...
We present a computational approach to the classical Schottky problem based on Fay's trisecant identity for genus ggeq4. For a given Riemann matrix \Omega\in\mathbb{H}^g , the Fay identity establishes linear dependence of secants in the Kummer variety if and only if the Riemann matrix corresponds to a Jacobian variety as shown by Krichever. The the...
In 1978, McEliece introduced a new public-key cryptosystem, based on error-correcting codes. Since then, it has demonstrated to have a lot of advantages, such as a fast encryption and decryption, in addition to the fact that it is a good candidate for post-quantum cryptography. The main constraint is that it imposes large keys sizes compared with o...
This thesis is situated at the interface between theoretical physics, in particular string theory and supersymmetric field theories, and mathematics, in particular algebraic geometry and representation theory. We study the counting of BPS ie supersymmetry-preserving) states in four-dimensional N=2 supersymmetric theories, obtained by compactifying ...
