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SUMMARY:Anna Giordano Bruno (University of Udine)
DTSTART:20240517T100000Z
DTEND:20240517T105000Z
DTSTAMP:20260416T215411Z
UID:GiG2024/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GiG2024/7/">
 A brief history and recent advances in the theory of characterized subgrou
 ps of the circle group</a>\nby Anna Giordano Bruno (University of Udine) a
 s part of Groups in Galway 2024\n\nLecture held in McMunn lecture theatre.
 \n\nAbstract\nA subgroup $H$ of the circle group $\\mathbb T$ is said to b
 e <em>characterized</em> by a sequence of integers $\\mathbf u = (u_n)_{n\
 \in\\mathbb N}$ if $H=\\{x\\in\\mathbb T: u_nx\\to 0\\}$. The first part o
 f the talk discusses characterized subgroups of $\\mathbb T$ and their rel
 evance in several areas of Mathematics where the behavior of the sequence 
 $(u_nx)_{n\\in\\mathbb N}$ as above is studied\, as Topological Algebra (t
 opologically torsion elements and characterized subgroups)\, Harmonic Anal
 ysis (sets of convergence of trigonometric series\, thin sets) and Number 
 Theory (uniform distribution of sequences).\n\nRecently\, generalizations 
 of the notion of characterized subgroup of $\\mathbb T$ were introduced\, 
 based on weaker notions of convergence\, starting from statistical converg
 ence and ending with $\\mathcal I$-convergence for an ideal $\\mathcal I$ 
 of $\\mathbb N$\, due to Cartan. A sequence $(y_n)_{n\\in\\mathbb N}$ in $
 \\mathbb T$ is said to <em>$\\mathcal I$-converge</em> to a point $y\\in \
 \mathbb T$\, denoted by $y_n\\overset{\\mathcal I}\\to y$\, if $\\{n\\in\\
 mathbb N: y_n \\not \\in U\\}\\in \\mathcal I$ for every neighborhood $U$ 
 of $y$ in $\\mathbb T$. A subgroup $H$ of the circle group $\\mathbb T$ is
  said to be <em>$\\mathcal I$-characterized</em> with respect to $\\mathca
 l I$ by a sequence of integers $\\mathbf u = (u_n)_{n\\in\\mathbb N}$ if $
 H=\\{x\\in\\mathbb T: u_nx\\overset{\\mathcal I}\\to 0\\}$. The second par
 t of the presentation proposes an overview on the results obtained on thes
 e new kind of characterized subgroups\, with special emphasis on $\\mathca
 l I$-characterized subgroups of $\\mathbb T$.\n\nBased on a joint work wit
 h D. Dikranjan\, R. Di Santo and H. Weber.\n
LOCATION:https://researchseminars.org/talk/GiG2024/7/
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