Résumé complet disponible à http://www.math.sciences.univ-nantes.fr/~viola-j/expNantes.pdf

We introduce a new functional summary statistic called the accumulative persistence function (APF) and having several attractive properties: It is a one-dimensional function easier to handle than the two-dimensional functions usually considered in persistence homology; for example, confidence regions are easier to plot and more visually appealing for a one-dimensional function than for a two-dimensional function; often, at least with probability one, there will be a one-to-one correspondence between the APF and the persistent diagram (for each fixed dimension) the APF is a natural way of constructing a monotonic function, and this will ease the proof of e.g. convergence theorems; contrary to the so-called dominant landscape function λ_1 or the silhouette, the APF provides information about topological features without distinguishing between long and short lifetimes. For instance, for application in spatial statistics, short lifetime topological features are relevant. In the talk we focus on extreme rank envelopes, a useful concept to make goodness-of-fit test associated to a confidence region for the APF, while other applications will be briefly discussed, including functional boxplots for APFs, the confidence region for the mean of APFs, and comparing groups of persistence diagrams: the two sample problem, clustering, and supervised classification.

Families of groups such as symmetric groups, braid groups, general linear groups, mapping class groups of 2- or 3-dimensional manifolds, or Higman-Thompson groups share the following stability phenomenon: the homology of the nth group in the sequence is isomorphic to that of the (n+1)st group in a range of degrees increasing with n. This phenomemon is called homological stability.

In this series of talks, I will give an introduction to homological stability, showing what the above examples have in common. I'll explain through the framework of homogeneous categories how the question of stability boils down to the question of high connectivity of certain simplicial complexes and give an idea of how these connectivity results are proved in different examples.

Families of groups such as symmetric groups, braid groups, general linear groups, mapping class groups of 2- or 3-dimensional manifolds, or Higman-Thompson groups share the following stability phenomenon: the homology of the nth group in the sequence is isomorphic to that of the (n+1)st group in a range of degrees increasing with n. This phenomemon is called homological stability.

In this series of talks, I will give an introduction to homological stability, showing what the above examples have in common. I'll explain through the framework of homogeneous categories how the question of stability boils down to the question of high connectivity of certain simplicial complexes and give an idea of how these connectivity results are proved in different examples.

Abstract: We present several classification results for Lagrangian tori, all proven using the splitting construction from symplectic field theory. Notably, we classify Lagrangian tori in the symplectic vector space up to Hamiltonian isotopy; they are either product tori or rescalings of the Chekanov torus. The proof uses the following results established in a recent joint work with E. Goodman and A. Ivrii. First, there is a unique torus up to Lagrangian isotopy inside the symplectic vector space, the projective plane, as well as the monotone S2 x S2. Second, the nearby Lagrangian conjecture holds for the cotangent bundle of the torus.

Understanding interactions between microbial communities and their environment well enough to be able to predict diversity on the basis of physicochemical parameters is a fundamental pursuit of microbial ecology that still eludes us. However, modeling microbial communities is a complicated task, because (i) communities are complex, (ii) most are described qualitatively, and (iii) quantitative understanding of the way communities interacts with their surroundings remains incomplete. Within this seminar, we will illustrate recent and complementary computational modelings that aim to overcome these points in different manners, promoting the recent field called systems ecology.

First, we will present a network analysis that focus on the biological carbon pump in the global ocean. The biological carbon pump is the process by which photosynthesis transforms CO2 to organic carbon sinking to the deep-ocean as particles where it is sequestered. While the intensity of the pump correlate to plankton community composition, the underlying ecosystem structure and interactions driving this process remain largely uncharacterized. We will show that the abundances of just a few bacterial and viral genes elucidate ecosystem behaviors and present a case study for scaling biological modelings from genes-to-ecosystems. Second, we will emphasize the functional role of bacteria within a natural community by proposing a graph-based modeling combined with a combinatorial optimization technique. Such an approach depicts from genome-scale knowledge, the respective role of microbial strains to catalyze environmental processes. Finally, we will show preliminary results on a probabilistic modeling that predicts microbial community structure across observed physicochemical data, from a putative network and partial quantitative knowledge. This modeling shows that, despite distinct quantitative environmental perturbations, the constraints on a community structure could remain stable.

Related references: Guidi, L., Chaffron, S., Bittner, L., Eveillard, D., Larhlimi, A., Roux, S., et al. (2016). Plankton networks driving carbon export in the oligotrophic ocean. Nature, 532, 465–470. Bordron, P., Latorre, M., Cortés, M.P., González, M., Thiele, S., Siegel, A., et al. (2016). Putative bacterial interactions from metagenomic knowledge with an integrative systems ecology approach. Microbiologyopen, 5, 106–117. Bourdon, J., Eveillard, D. & Siegel, A. (2011). Integrating quantitative knowledge into a qualitative gene regulatory network. PLoS Comput Biol, 7, e1002157.