Universality of directed polymers in the intermediate disorder regime

In 2018, Krishnan and Quastel showed that the fluctuations of Sepp\"al\"ainen's log-gamma polymer converge in law to the Tracy--Widom GUE distribution in the intermediate disorder regime, which corresponds to taking the inverse temperature $\beta$ to depend on the length of the polymer $2n$, with $\beta=n^{-\alpha}$ for some $\alpha<1/4$. They also conjectured that this should hold for directed polymers with general i.i.d weights. We prove that their conjecture is true for any $1/5<\alpha<1/4$.Comment: 40 pages, 3 figure

Automatic differentiation for Fourier series and the radii polynomial approach

In this work we develop a computer-assisted technique for proving existence of periodic solutions of nonlinear differential equations with non-polynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the augmented system, and prove the existence of a true solution nearby using an a-posteriori validation scheme (the radii polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is analytic in a neighborhood of the periodic orbit) hence the computer-assisted proofs are formulated in a Banach space of sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a number of computer-assisted existence proofs for periodic orbits of the Planar Circular Restricted Three-Body Problem (PCRTBP

Les espaces de Hilbert à noyau reproduisant et leurs applications en analyse complexe

Tableau d'honneur de la Faculté des études supérieures, 2018-2019Dans un article récent de Aleman, Hartz, McCarthy et Richter, les auteurs ont montré que toute fonction dans un espace avec la propriété de Pick complète peut s’écrire comme un quotient de deux multiplicateurs. Ce résultat était un des deux points clé manquant dans la démonstration d’une version du théorème de Gleason–Kahane–Zelazko pour l’espace de Dirichlet. Le but de ce mémoire est de développer la théorie des espaces de Hilbert à noyau reproduisant et d’utiliser celle-ci afin d’étudier trois espaces importants de fonctions holomorphes sur D, soit l’espace de Hardy, l’espace de Dirichlet et l’espace de Bergman, et de bien comprendre le résultat de Aleman, Hartz, McCarthy et Richter. On est par la suite en mesure de démontrer le théorème GKZ pour l’espace de Dirichlet. ii

Imaging exocytosis of ATP-containing vesicles with TIRF microscopy in lung epithelial A549 cells

Nucleotide release constitutes the first step of the purinergic signaling cascade, but its underlying mechanisms remain incompletely understood. In alveolar A549 cells much of the experimental data is consistent with Ca2+-regulated vesicular exocytosis, but definitive evidence for such a release mechanism is missing, and alternative pathways have been proposed. In this study, we examined ATP secretion from A549 cells by total internal reflection fluorescence microscopy to directly visualize ATP-loaded vesicles and their fusion with the plasma membrane. A549 cells were labeled with quinacrine or Bodipy-ATP, fluorescent markers of intracellular ATP storage sites, and time-lapse imaging of vesicles present in the evanescent field was undertaken. Under basal conditions, individual vesicles showed occasional quasi-instantaneous loss of fluorescence, as expected from spontaneous vesicle fusion with the plasma membrane and dispersal of its fluorescent cargo. Hypo-osmotic stress stimulation (osmolality reduction from 316 to 160 mOsm) resulted in a transient, several-fold increment of exocytotic event frequency. Lowering the temperature from 37°C to 20°C dramatically diminished the fraction of vesicles that underwent exocytosis during the 2-min stimulation, from ~40% to ≤1%, respectively. Parallel ATP efflux experiments with luciferase bioluminescence assay revealed that pharmacological interference with vesicular transport (brefeldin, monensin), or disruption of the cytoskeleton (nocodazole, cytochalasin), significantly suppressed ATP release (by up to ~80%), whereas it was completely blocked by N-ethylmaleimide. Collectively, our data demonstrate that regulated exocytosis of ATP-loaded vesicles likely constitutes a major pathway of hypotonic stress-induced ATP secretion from A549 cells