This major proposes an initiation to some problems and challenges encountered in climate, environment and geosciences. The whole year of this program will favours the development of a research spiritand aims at taking intou account exhanges problems between academic mathematicians and industrial risk experts, climate scientists and model users. It will allow student folloxing this major to bring innovative and valued answers, both from theoretical and practical point of views.

Courses of this major are associated to

– refresher courses at the begining of the academic year

– a curriculum core during the first semester

– two additional course to be chosen among those proposed in the other majors (see the welcome page for a complet list).

## Risk assesment modelling and climate change

Presentation of some probabilistic models, allowing to deal with spatial dependency, and spatio-temporal dependency when possible.

Investigation of some random phenomenon spatially localized and organized.

Statistical treatment of these models: parameter inference and simulation methods.

Climate change specificities.

Keywords : spatial dependency models, géostatistics, krigging, extrme-values models, meta-models, non stationnarity.

## Fluid mechanics equations and their numerical approximations

1) Fluids mechanics equation and their properties:

– Navier-Stokes (compressible, uncompressible, strong solutions, weak solutions)

– Euler compressible (weak solution, entropy condition)

– Saint-Venant for mid-deep water flows.

2) Finite volume methods for fluid mechanics equations.

– VF schemes for Euler and Saint-Venant : in 1 dimension : conservative schemes, Lax-Wendroff theorem, Riemann solvers. Robustness and discrete entropy condition. Extension of VF schemes to a multidimensionnal framework.

– VF schemes for diffusion.

3) Applications : to be picked among

- (i) Navy hydrographic services ;
- (ii) Flood barrier (EDF) propagation uncertainty flows… ;
- (iii) Avalanche models
- (iv) Pratical implementation and code optimisation
- (v) Diphasic flow models : accidental flows in civil nuclear industry ;

## Spatial ecology modelling

Keywords : stochastic and deterministic mathematical modelling, spatial ecology, microscopic-macroscopic scale changes, reaction-diffusion, propagation phenomenon, patterns

0) Presentation of the applications and/or external talks.

1) Stochastic modelling of spatial population (3h), introduction to micro/macro scale changes (2h).

2) Fisher-KPP (maximum principle, progressives waves, propagation of solutions associated to initial conditions with compact support) (3h), extension to bistable equations (1h), extension to competitive Lotka-Volterra (1h).

3) Turing instability (5h)

Description :

The goal of this course is to present a stochastic and deterministic family of models used in particular in spatial ecology but also in other domain of mathematical biology as Darwinian evolution or morphogenesis. We will begin with microscopic stochastic models describing in a sharp way individual behaviors and life cycles in a given population. After a scale change, we will investigate macroscopical reaction-diffusion equation systems describing, in a first time, territory invasion and, in a second time, the apparition of spatially heterogeneous patterns.

## Graphs and ecological networks

A graph, whose first use are mentionned in the 16th century, is a mathematical object widely used from the first appearance of network investigations, namely investigation of relationship between individual in wide sense. Ranging from social network to the internet, graphs are leading objects for the analysis of several data sets. Ecosystem relationships, from species relationship (prédation, interaction between plants and pollinating insects , etc…) social relationship between individuals (sociality between primates, etc…), offers several different possible applications of graphs modelling and network investigation.

In this course, we will investigate the framework of graph theory and network science. We will provide an introduction to modern research problems regarding ecosystems studies. We will use alternatively discrete mathematics, statistics and machine learning.

We will adress both theoretical and practical (case studies in ecology) questions.

Theretical keywords: Bases / definitions (graphs, path, etc…) – Metrics – Clustering methods – Spectral methods – Random graphs models – Graphical models (graphs inference) – Signal processing on graphs – Multi-level graphs (time, space, link types) – Embedding methods (optional)

Case studies : Contact network between animals. Interaction network between species in a marine and/or alpine environment. Consideration about the relevance of a graph for biodiversity support.