Jekyll2018-04-13T02:39:28+00:00http://scoskey.org/Samuel CoskeyDepartment of Mathematics, Boise State University, 1910 University Dr, Boise, ID 83725-1555.
Conjugacy for homogeneous ordered graphs2018-04-12T00:00:00+00:002018-04-12T00:00:00+00:00http://scoskey.org/ordered-graphs<p>With Paul Ellis. (<a href="http://arxiv.org/abs/1804.04609">arχiv</a>)<!--more--></p>
<p><em>Abstract</em>: We show that for any countable homogeneous ordered graph $G$, the conjugacy problem for automorphisms of $G$ is Borel complete. In fact we establish that each such $G$ satisfies a strong extension property called ABAP, which implies that the isomorphism relation on substructures of $G$ is Borel reducible to the conjugacy relation on automorphisms of $G$.</p>With Paul Ellis. (arχiv)Classification of scattered linear orders2018-03-06T00:00:00+00:002018-03-06T00:00:00+00:00http://scoskey.org/classification-of-scattered-linear-orders<p>Boise Set Theory Seminar, March 2018<!--more--></p>
<p><em>Abstract</em>: A linear order is called scattered if the rational order doesn’t embed into it. Scattered linear orders admit a derivative operation and an ordinal rank. In this talk we introduce some machinery needed to study the complexity of the classification of scattered linear orders of a given countable rank.</p>Boise Set Theory Seminar, March 2018Hyperbinary numbers and fraction trees2018-02-06T00:00:00+00:002018-02-06T00:00:00+00:00http://scoskey.org/calkin-wilf<p>With Paul Ellis and Japheth Wood. (<a href="https://www.mathteacherscircle.org/news/mtc-magazine/s2018/touching-infinity/">article</a>)<!--more--></p>
<p><em>Abstract</em>: Questions about infinity are fascinating, and can lead into deep mathematical topics in set theory. The mathematics of infinite sets wasn’t clearly understood until Cantor defined cardinal numbers in the late 19th century, stating that two sets are the same size if there is a one-to-one correspondence between them. One surprising result from set theory, first proved by Cantor in 1873, is that there are precisely as many rational numbers (fractions) as there are counting numbers. Over one hundred years later, mathematicians Neil Calkin and Herbert S. Wilf published a more elegant proof of this fact.</p>
<p>This article is the result of our work to develop the ideas in the Calkin-Wilf proof, so that they would be accessible to the teachers in our three different Math Teachers’ Circles. We designed an investigation into the hyperbinary numbers (itself a 19th century topic that predates Cantor’s work on cardinality) and developed the Tree of Fractions, much in the style of Calkin and Wilf. We asked teachers to make observations, ask questions, and convince each other of the veracity of their claims.</p>With Paul Ellis and Japheth Wood. (article)Martin’s axiom and some applications2018-01-13T00:00:00+00:002018-01-13T00:00:00+00:00http://scoskey.org/martins-axiom-and-applications<p>Boise Set Theory Seminar, January 2018<!--more--></p>
<p><em>Abstract</em>: In this talk I presented the notation and machinery of forcing, the statement of Martin’s axiom, and some well-known applications in the area of Baire category and measure theory.</p>Boise Set Theory Seminar, January 2018Foundations of analysis2018-01-01T00:00:00+00:002018-01-01T00:00:00+00:00http://scoskey.org/1718s-314<p>Math 314, Spring 2018 (<a href="http://scoskey.org/m314">site</a>)<!--more--></p>
<p><em>Catalog description</em>: The real number system, completeness and compactness, sequences, continuity, foundations of the calculus.</p>Math 314, Spring 2018 (site)Foundations of geometry2018-01-01T00:00:00+00:002018-01-01T00:00:00+00:00http://scoskey.org/1718s-311<p>Math 311, Spring 2018 (<a href="http://scoskey.org/m311">site</a>)<!--more--></p>
<p><em>Catalog description</em>: Euclidean, non-Euclidean, and projective geometries from an axiomatic point of view.</p>Math 311, Spring 2018 (site)Classification of countable models of PA and ZFC2017-11-14T00:00:00+00:002017-11-14T00:00:00+00:00http://scoskey.org/classification-of-countable-models-of-pa-and-zfc<p>Boise Set Theory Seminar, November 2017<!--more--></p>
<p><em>Abstract</em>: In 2009 Roman Kossak and I showed that the classification problems for countable models of arithmetic (PA) is Borel complete, which means it is complex as possible. The proof is elementary modulo Gaifman’s construction of so-called canonical I-models. Recently Sam Dworetzky, John Clemens, and I adapted the method to show that the classification problem for countable models of set theory (ZFC) is Borel complete too. In this talk I’ll give the background needed to state such results, and then give an outline of the two very similar proofs.</p>Boise Set Theory Seminar, November 2017Borel complexity theory and classification problems2017-10-09T00:00:00+00:002017-10-09T00:00:00+00:00http://scoskey.org/borel-complexity-theory-and-classification-problems<p>Oregon mathematics department colloquium, Eugene, October 2017 (<a href="http://math.boisestate.edu/~scoskey/slides/bct-slides.pdf">slides</a>)<!--more--></p>
<p><em>Abstract</em>: Borel complexity theory is the study of the relative complexity of classification problems in mathematics. At the heart of this subject is invariant descriptive set theory, which is the study of equivalence relations on standard Borel spaces and their invariant mappings. The key notion is that of Borel reducibility, which identifies when one classification is just as hard as another. Though the Borel reducibility ordering is wild, there are a number of well-studied benchmarks against which to compare a given classification problem. In this talk we will introduce Borel complexity theory, present several concrete examples, and explore techniques and recent developments surrounding each.</p>Oregon mathematics department colloquium, Eugene, October 2017 (slides)Equivalence relations and classification problems, parts 1 and 22017-09-19T00:00:00+00:002017-09-19T00:00:00+00:00http://scoskey.org/equivalence-relations-and-classification-problems<p>Boise Set Theory Seminar, September 2017<!--more--></p>
<p><em>Abstract</em>: Many classification problems in mathematics may be identified with an equivalence relation on a standard Borel space. In earlier talks we have been introduced to the notion of Borel reducibility of equivalence relations, as well as to some of the most important equivalence relations studied. In this talk we will introduce several natural classification problems and identify where they lie in the Borel reducibility order.</p>Boise Set Theory Seminar, September 2017On the classification of automorphisms of trees2017-09-11T00:00:00+00:002017-09-11T00:00:00+00:00http://scoskey.org/trees<p>With Kyle Beserra. (<a href="https://arxiv.org/abs/1709.02467">arχiv</a>)<!--more--></p>
<p><em>Abstract</em>: We identify the complexity of the classification problem for automorphisms of a given countable regularly branching tree up to conjugacy. We consider both the rooted and unrooted cases. Additionally, we calculate the complexity of the conjugacy problem in the case of automorphisms of several non-regularly branching trees.</p>With Kyle Beserra. (arχiv)