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SUMMARY:Thomas Karam (University of Oxford)
DTSTART:20241210T190000Z
DTEND:20241210T200000Z
DTSTAMP:20260423T004822Z
UID:OARS/65
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OARS/65/">Th
 e structure of a function and small images</a>\nby Thomas Karam (Universit
 y of Oxford) as part of OARS Online Analysis Research Seminar\n\n\nAbstrac
 t\nIt is well known that if $G$ is an abelian group and $A$ is a finite su
 bset of $G$ which minimises the size of $A+A$ for a fixed size of $A$\, th
 en $A$ must essentially be as close as possible to being a coset of some s
 ubgroup of $G$: for instance\, if $G$ is finite then the set $A$ must be a
  coset in $G$ (provided that $G$ has subgroups with size $|A|$)\, and if $
 G=\\mathbb{Z}$ then the set $A$ must be an arithmetic progression. One may
  wish to describe an extension of this phenomenon to more arbitrary functi
 ons than addition on abelian groups: if $X$ is a set and $F: X \\times X \
 \rightarrow X$ is a function\, then can the structure of $F$ (to the exten
 t that there is some) be read from the structure of the sets $A$ minimisin
 g the size of the image $F(A\,A)$ for a fixed size of $A$ ? We will discus
 s the progress that has been made in this direction and what remains to be
  understood.\n
LOCATION:https://researchseminars.org/talk/OARS/65/
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