<div dir="ltr">PROXIMO ENCUENTRO: Miércoles 8 de Mayo, 12:00hs.<div class="gmail_extra"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">
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<div><span style="font-size:13px;font-family:arial,sans-serif">EXPOSITOR: Quinn Culver, University of Notre Dame.</span></div><div><span style="font-size:13px;font-family:arial,sans-serif"><br></span></div><div><span style="font-size:13px;font-family:arial,sans-serif">TITULO: Algorithmically Random Measures</span><div class="gmail_extra">
<div class="gmail_quote"><div class="im"><div> </div><div><span style="font-size:13px;font-family:arial,sans-serif">LUGAR: Departamento de Matemática, Aula de</span><span style="font-size:13px;font-family:arial,sans-serif"> seminarios</span><span style="font-size:13px;font-family:arial,sans-serif">, 2do piso, Pabellón 1.</span></div>
<div><span style="font-size:13px;font-family:arial,sans-serif"><br></span></div></div><div><span style="font-size:13px;font-family:arial,sans-serif">RESUMEN: </span><b style="font-size:medium;font-family:'Times New Roman';font-weight:normal"><p dir="ltr" style="line-height:1.15;margin-top:0pt;margin-bottom:0pt;text-align:justify;display:inline!important">
<span style="font-size:13px;font-family:Arial;color:rgb(34,34,34);vertical-align:baseline;white-space:pre-wrap">Algorithmic randomness attempts to answer the question "What exactly is a random real number?" It does so by declaring that a real is nonrandom if it can be captured in a null set that can be effectively (aka computably) approximated from without by open sets. There is a rich interplay between the theory of algorithmic randomness, the theory of computation (Turing degrees), and probability theory. Recently, the theory of algorithmic randomness with respect to other (i.e. non-uniform) measures has been developed. This development, coupled with the fact that the space of probability measures on the unit interval a nice enough space on which to do computability, motivates the questions "Is there a natural measure on the space of measures? What do its algorithmically random elements look like?" </span></p>
</b></div><b style="font-size:medium;font-family:'Times New Roman';font-weight:normal"><br><span style="font-size:13px;font-family:Arial;color:rgb(34,34,34);vertical-align:baseline;white-space:pre-wrap"></span><span style="font-size:13px;font-family:Arial;color:rgb(34,34,34);vertical-align:baseline;white-space:pre-wrap">In attempt to answer these questions, we define a natural, computable map that associates to each real a Borel probability measure, so that we can talk about algorithmically random measures. We show that such random measures are atomless and mutually singular with respect to Lebesgue. We introduce a certain information-theoretic-ish property that lies strictly between atomlessness and absolute continuity (with respect to the Lebesgue measure) that we conjecture random measures satisfy. We then discuss other maps, ask the question "Why is the first map more natural?", and discuss some ways in which that question might be precisifiable.</span></b></div>
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