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Ph.D de FALQUE Justine
FALQUE Justine
Ph.D
Group : Graphs, ALgorithms and Combinatorics

Classification of P-oligomorphic groups, conjectures of Cameron and Macpherson

Starts on 01/01/1970
Advisor : THIÉRY, Nicolas

Funding :
Affiliation : Université Paris-Sud
Laboratory : l'amphithéâtre DIGITÉO du bâtiment CLAUDE SHANNON

Defended on 29/11/2019, committee :
Rapporteurs :
Peter Cameron (Queen Mary Univ. of London & Univ. of Saint Andrews)
Pascal Weil (CNRS, Université de Bordeaux)

Jury :
Peter Cameron (Queen Mary Univ. of London & Univ. of Saint Andrews)
Pascal Weil (CNRS, Université de Bordeaux)
Isabelle Guyon (Université Paris Sud)
Maurice Pouzet (Université Claude Bernard Lyon I)
Christophe Tollu (Université Paris-Nord)
Annick Valibouze (Sorbonne Universités)
Nicolas Thiéry (Université Paris Sud)

Research activities :

Abstract :
Given an infinite permutation group G, consider the function that
maps every natural integer n to the number of orbits of n-subsets,
for the induced action of G on the subsets of elements.
Cameron conjectured that this counting function, the profile of G,
is asymptotically equivalent to a polynomial if it is bounded
above by a polynomial. Another, stronger conjecture was later made
by Macpherson. It involves a certain structure of graded agebra on
the orbits of subsets, created by Cameron, and states that if the
profile of G is bounded by a polynomial, then its orbit algebra is
finitely generated.
The main achievement of the thesis is to classify the permutation
with polynomially bounded profile (up to closure), which in
particular demonstrates the two conjectures. The approach involves
studying the lattice of block systems, an experimental exploration
on computer, and tools from group theory.