Jekyll2021-09-02T23:42:29+00:00https://drossegger.github.io/feed.xmlDino Rossegger’s homepageLogician; Marie Curie fellow at UC Berkeley and TU Wien. Find news about my research and teaching.Dino Rosseggerdino@math.berkeley.eduPositive enumerable functors2021-02-09T00:00:00+00:002021-02-09T00:00:00+00:00https://drossegger.github.io/positiveenumerablefunctors<script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>
<p>We just submitted a new article titled “Positive enumerable functors” and you can find the preprint <a href="/assets/files/positiveenumerablefunctors.pdf">here</a> or on <a href="https://arxiv.org/abs/2011.14160">arxiv</a>. This article is joint work with Barbara Csima and Daniel Yu who proved most of the results presented there in an undergraduate research project he did with Barbara and me over the summer.</p>
<p>We investigate different effectivizations of functors using Turing operators and enumeration operators and compare these notions. The most promising of these notions are <em>positive enumerable functors</em>. A positive enumerable functors from \(\mathcal A\) to \(\mathcal B\) consists of two enumeration operators \(\Psi\) and \(\Psi_*\) such that when given an enumeration of the positive diagram \(P(\hat{\mathcal A})\) of a structure isomorphic to \(\hat{\mathcal A}\cong \mathcal A\), \(\Phi\) enumerates the positive diagram of \(F(\hat{\mathcal A})\). The operator \(\Phi_*\) will take care of the isomorphisms by enumerating the graph of \(F(f:\hat{\mathcal A}\to\tilde{\mathcal A})\) given \(P(\hat{\mathcal A})\oplus Graph(f)\oplus P(\tilde{\mathcal A})\).</p>
<p>We show that if \(\mathcal A\) is positive enumerable bi-transformable to \(\mathcal B\) (if there are positve enumerable functors \(F:\mathcal A\to\mathcal B\) and \(G:\mathcal B\to \mathcal A\) such that \(F\) and \(G\) are “enumerable pseudo-inverses”) then \(\mathcal A\) and \(\mathcal B\) have the same enumeration degree spectrum and that positive enumerable functors are independent from the already established enumerable and computable functors.</p>
<p>We submitted the article to a conference and hope to expand it in the future, adding more preservation results and investigating syntactic equivalences.</p>Dino Rosseggerdino@math.berkeley.eduNew preprint on arXiv!2020-10-05T00:00:00+00:002020-10-05T00:00:00+00:00https://drossegger.github.io/news/spectra-analyticcomp<script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>
<p>My newest preprint <a href="https://arxiv.org/abs/2010.00755">Degree spectra of analytic complete equivalence relations</a> just appeared on arXiv.</p>
<p>This article investigates the bi-embeddability and elementary bi-embeddability relation on countable structures both from the viewpoint of descriptive set theory and computable structure theory. Two structures are <em>bi-embeddable</em> if either is isomorphic to an substructure of the other. They are <em>elementary bi-embeddable</em> if either is isomorphic to a elementary substructure of the other.</p>
<p>Our main results are the following.</p>
<p><strong>Theorem 1.</strong> <em>The elementary bi-embeddability relation on graphs is a \(\pmb \Sigma^1_1\) complete equivalence relation under Borel reducibility.</em></p>
<p>The proof of this result uses Marker extensions with so called minimal models. A structure \(\mathcal A\) is <em>minimal</em> if it does not contain an elementary substructure.</p>
<p>Let \(\sim\) be an equivalence relation on a class of structures. Then given
a structure \(\mathcal A\) we define the \(\sim\) spectrum of \(\mathcal A\) as the
set of all sets Turing equivalent to a structure \(\sim\) equivalent to \(\mathcal A\), i.e.,
\[ DgSp_\sim(\mathcal A)=\{ X\subseteq \omega: \exists \mathcal B \sim \mathcal A \ X\equiv_T \mathcal A\}.\]</p>
<p><strong>Theorem 2.</strong> <em>Every bi-embeddability spectrum of a graph is the jump spectrum of an elementary bi-embeddability spectrum of a graph, i.e., for every \(\mathcal G\), there is a graph \(\mathcal G'\) such that</em>
\[ DgSp_\approx(\mathcal G’)=\{ X: X’\in DgSp_\approxeq(\mathcal G)\}.\]</p>
<p>This result is proven by analyzing the reduction given to prove Theorem 1. We show that this reduction induces a computable bi-transformation with a suitable class of structures.</p>Dino Rosseggerdino@math.berkeley.eduMy newest preprint Degree spectra of analytic complete equivalence relations just appeared on arXiv.New Homepage Online2020-08-06T00:00:00+00:002020-08-06T00:00:00+00:00https://drossegger.github.io/news/homepage-online<p>I just created this new homepage with all my professional information. I hope that you find it interesting and useful. I appreciate feedback, just write me an e-mail.</p>Dino Rosseggerdino@math.berkeley.eduI just created this new homepage with all my professional information. I hope that you find it interesting and useful. I appreciate feedback, just write me an e-mail.