Évènements

The ovarian follicles are the basic anatomical and functional units of the ovaries, which are renewed from a
quiescent pool all along reproductive life. Follicular development involves a finely tuned sequence of growth and
maturation processes, involving complex cell dynamics. In their early stages of development, ovarian follicles are made up of a germ cell (oocyte), whose diameter increases steadily, and of surrounding proliferating somatic
cells, which are layered in a globally spherical and compact structure.
Here, we present two complementary modeling approaches dedicated to the first stages of a follicle develop-
ment, starting with the exit from the pool of quiescent (primordial) follicles leading to growth initiation, and
ending up just before the breaking of the spherical symmetry induced by the follicle cavitation (formation of
the antrum cavity).
The initiation phase is described by joint stochastic dynamics accounting for cell shape transitions (from
a flattened to a cuboidal shape) and proliferation of reshaped cells. We can derive the mean time elapsed before all cells have changed shapes and the corresponding increment in the total cell number, which is fitted
to experimental data retrieved from primordial follicles (single layered follicle with only flattened cells) and primary follicles (single layered follicles with only cuboidal cells).
The next stages, characterized by the accumulation of cell layers around the oocyte, are described by
multi-type structured models in either a stochastic or deterministic framework. We have designed a linear age-structured stochastic (Bellman-Harris branching) process ruling the changes in the number of follicular cells and their distribution into successive layers, which is inspired from the nonlinear model initially introduced in [1], as well as is deterministic counterpart (multi-dimensional Mc Kendrick Von Foerster). We have studied the large-time behavior of the models and derived explicit analytical formulas characterizing an exponential growth
of the population (Malthus parameter, asymptotic cell number moments and stable age distribution). We have
compared the theoretical and numerical outputs of the models with experimental biological data informing on follicle morphology in the ovine species (follicle and oocyte diameters, layer number and total cell number) from the primary to the pre-antral stage. In addition, in the case of age independent division rates, we have established the structural identifiability of the parameters, and estimated the parameter values fitting the cell numbers in each layer during the early stages of follicle development.

[1 ] Clément F., Michel P., Monniaux D., Stiehl T., Coupled somatic cell kinetics and germ cell growth:
mutliscale model-based insight on ovarian follicular development,
Multiscale et Modeling & Simulation
, 11(3), 719-746, 2013.

[2 ] Clément F., Robin F., Yvinec R., Analysis and calibration of a linear model for structured cell populations with unidirectional motion : Application to the morphogenesis of ovarian follicles,
Submitted. https://arxiv.or/abs/1712.05372

In this talk I will present some recent results for the resolvent norm of linear operators and their implication for the pseudospectrum of matrices. In the presentation I restrict myself to matrices, even though most statements also hold, at least locally, for a certain class of closed linear operators on a separable Hilbert space. As the main theorem we have that for any point in the resolvent set there are directions in which the norm grows at least quadratically in the distance from this point. Besides others this directly implies the well-known fact that level sets of the resolvent norm cannot have interior points. Moreover, I will show how the main theorem can be used to construct a finite polygonal contour inside the pseudospectrum linking a given arbitrary point in the pseudospectrum to an eigenvalue of the matrix. This talk is based on joint work with H. Cornean, H. Garde and A. Jensen.

In this talk, I first review the method of layer potentials, with the emphasis on the double layer potential operator (also called Neumann-Poincar ́e operator) associated to the Laplace operator and a domain. Then I show that layer potential groupoids for conical domains constructed in an earlier paper (Carvalho-Qiao, Central European J. Math., 2013) are Fredholm groupoids, which enables us to deal with many analysis problems on singular spaces in a unified treatment. As an application, we obtain Fredholm criteria for operators on layer potential groupoids. This is joint with Catarina Carvalho.