What is the problem/issue being addressed?
I am willing to solve several well-known models from combinatorics, probability theory and statistical mechanics: the Ising model on isoradial graphs, dimer models, spanning forests, random walks in cones, occupation time problems, etc. Although completely unrelated a priori, these models have the common feature of being presumed "exactly solvable" models, for which surprising and spectacular formulas should exist for quantities of interest. This is captured by the title "Elliptic Combinatorics", the wording elliptic referring to the use of special functions, in a broad sense.
Besides the exciting nature of the models which we aim at solving, one main strength of our project lies in the variety of modern methods and fields that we cover: combinatorics, probability, algebra (representation theory), computer algebra, algebraic geometry, with a spectrum going from applied to pure mathematics. We propose in addition one major application, in population biology (evolution of multitype populations). We plan to work in close collaborations with researchers from this field, to eventually apply our results (study of extinction probabilities for selfincompatible flower populations, for instance).
Several of our works are about the transcendental nature of the generating functions counting walks in cones (in particular in the quarter plane). This goal is actually part of a bigger project, consisting in studying transcendence in combinatorics and probability theory. The key idea is that many combinatorial or probabilistic models may be studied via generating functions, whose transcendence appears as a measurement of the complexity of the associated model.
Why is it important for society?
I would like to briefly describe the numerous impacts of our project. To summarize, they are triple:
.The first group impacted by our results will be, indisputably, the French and international mathematical communities: indeed, we aim at obtaining various results from quite different fields.
.The second community which will benefit from this project is the group of PhD students and postdocs who I have hired: they are receiving a strong formation at the forefront of mathematics, with the opportunity to stay in the academic world or to work in companies.
.Finally, considering the applied part of our results, we plan to work in close collaborations with researchers from these fields. Researchers in biology will be the last group impacted by our results.
What are the overall objectives?
Our many scientific objectives may be classified within three thematic sections: random walks in cones, statistical mechanics, applications in population biology. In each of these domains, we have many concrete mathematical questions, which together look in the same direction of a better understanding of these exactly solvable models.
