<div dir="ltr"><span class="" style="border-collapse:collapse;font-family:arial,sans-serif;font-size:14px"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)"><span class="" style="background-image:initial;background-color:rgb(255,255,204);color:rgb(34,34,34)">Kobe</span> <span style="background-image:initial"><font color="#000000"><span><span style="background-image:initial;background-color:rgb(255,255,204);color:rgb(34,34,34)"><span class="" style="background-image:initial;background-color:rgb(255,255,204);color:rgb(34,34,34)">Colloquium</span></span></span></font></span> on Logic, Statistics and Informatics <div>
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$B9V1i<T!'(BRod Downey (Victoria University of Wellington, New Zealand)</div><div><br></div><div>========================================================================</div><div><br></div><div>$BBjL\!'(B<span style="font-size:14px">Effectively in Abelian Group Theory</span></div>
<div><span style="font-size:14px"><br></span></div></span></span></font></div></span><span class="" style="font-size:13px">$B%"%V%9%H%i%/%H!'(BThe effectiveness of the theory of abelian groups has been </span></span><span class="" style="border-collapse:collapse;font-family:arial,sans-serif;font-size:13px">long </span><div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif;font-size:13px">studied beginning with the work of Mal'cev in the 60's. </span><span class="" style="border-collapse:collapse;font-family:arial,sans-serif;font-size:13px">Nevertheless many problems </span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif;font-size:13px">remain. In this lecture I will discuss </span><span class="" style="border-collapse:collapse;font-family:arial,sans-serif;font-size:13px">ongoing work on questions of effectiveness of </span></div>
<div><div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif;font-size:13px">categoricity and presentability for </span><span class="" style="border-collapse:collapse;font-family:arial,sans-serif;font-size:13px">computable torsion-free abelian groups and for p-groups. </span></div>
<div><div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span class="" style="font-size:13px">For example, for categoricity, general problem is </span><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">impossible; for example Downey and </span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">Montalban showed that the isomorphism problem for torsion-free abelian groups is </span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">Sigma_1^1-complete. The principle difficulty lies in the lack of invariants. However, </span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">where there are  some invariants there we can salvage some effectiveness. </span><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">The groups </span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">we look at are the completely decomposable ones, which have decompositions of the </span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">form oplus_{i in omega} G_i with G_i a subgroup of the additive group of the rationals. </span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">Such groups are called homogeneous if G_i=H for all i. </span><span class="" style="font-size:13px">Alexander Melnikov and the </span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span class="" style="font-size:13px">author have shown that homogeneous </span><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">computable completely decomposable groups are </span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">always Delta_3^0 categorical, this bound is sharp, and have classified when the groups </span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">are Delta_2^0 categorical in terms of what are called semilow sets. In more recent work, </span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">we have shown that every computable completely decomposable group is Delta_5^0</span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">categorical and that this bound is sharp. Additionally we can show that the index set</span></span></div>
<div><span class="" style="border-collapse:collapse;font-family:arial,sans-serif"><span style="font-family:arial,sans-serif;font-size:13px;border-collapse:collapse;color:rgb(34,34,34)">of such groups is Sigma_7^0. I will also describe ongoing work on Ulm's Theorem.<div>
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