Ph.D
Group : Graphs, ALgorithms and Combinatorics

Representation of monoids and lattice structures in the combinatorics of Weyl groups

Starts on 01/10/2015
Advisor : HIVERT, Florent

Funding : Bourse association
Affiliation : Université Paris-Saclay
Laboratory : LRI - GALaC

Defended on 25/06/2018, committee :

Research activities :

Abstract :
Algebraic combinatorics is the research field that uses combinatorial methods and algorithms to study algebraic computation, and applies algebraic tools to combinatorial problems.
One of the central topics of algebraic combinatorics is the study of permutations, interpreted in many different ways (as bijections, permutation matrices, words over integers, total orders on integers, vertices of the permutahedron...). This rich diversity of perspectives leads to the following generalizations of the symmetric group. On the geometric side, the symmetric group generated by simple transpositions is the canonical example of finite reflection groups, also called Coxeter groups. On the monoidal side, the simple transpositions become bubble sort operators that generate the $0$-Hecke monoid, whose algebra is the specialization at $q=0$ of Iwahori’s $q$-deformation of the symmetric group. This thesis deals with two further generalizations of permutations.The first one is to introduce a degeneracy at q=0 of Solomon's algebra of the q deformed rook monoid (also called monoid of partial permutations) and to study its representation theory and its equivalent in other Weyl types. The second is to generalize the notion of weak order from permutations to root systems following the recent works of G. Chatel, V. Pilaud and V. Pons.

CAUSAL LEARNING FOR DIAGNOSTIC SUPPORT


CAUSAL UNCERTAINTY QUANTIFICATION UNDER PARTIAL KNOWLEDGE AND LOW DATA REGIMES


MICRO VISUALIZATIONS: DESIGN AND ANALYSIS OF VISUALIZATIONS FOR SMALL DISPLAY SPACES
The topic of this habilitation is the study of very small data visualizations, micro visualizations, in display contexts that can only dedicate minimal rendering space for data representations. For several years, together with my collaborators, I have been studying human perception, interaction, and analysis with micro visualizations in multiple contexts. In this document I bring together three of my research streams related to micro visualizations: data glyphs, where my joint research focused on studying the perception of small-multiple micro visualizations, word-scale visualizations, where my joint research focused on small visualizations embedded in text-documents, and small mobile data visualizations for smartwatches or fitness trackers. I consider these types of small visualizations together under the umbrella term ``micro visualizations.'' Micro visualizations are useful in multiple visualization contexts and I have been working towards a better understanding of the complexities involved in designing and using micro visualizations. Here, I define the term micro visualization, summarize my own and other past research and design guidelines and outline several design spaces for different types of micro visualizations based on some of the work I was involved in since my PhD.