Abstract
<p> This is an expository report on the recent development of continuous representations and their applications in data analysis. On one hand, we recall the technical foundations of homological persistence theory which originates from quiver representations. We discuss <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -representations and their applications, as well as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb S^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -representations with an approach of covering theory. On the other hand, we focus on the recent development of the applications of representation theory in topological data analysis, including rank decompositions and homological invariants. We also show a generalized result about morphisms between thin modules over locally finite posets. Lastly, some further problems in the trends are summarized. </p>