Samuel CoskeyDepartment of Mathematics, Boise State University, 1910 University Dr, Boise, ID 83725-1555.
https://scoskey.github.io/
Wed, 15 Nov 2017 04:34:25 +0000Wed, 15 Nov 2017 04:34:25 +0000Jekyll v3.6.2Classification 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>
Tue, 14 Nov 2017 00:00:00 +0000
https://scoskey.github.io/classification-of-countable-models-of-pa-and-zfc
https://scoskey.github.io/classification-of-countable-models-of-pa-and-zfcclassification,papresentationBorel 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>
Mon, 09 Oct 2017 00:00:00 +0000
https://scoskey.github.io/borel-complexity-theory-and-classification-problems
https://scoskey.github.io/borel-complexity-theory-and-classification-problemsclassifiactionpresentationEquivalence relations and classification problems, parts 1 and 2<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>
Tue, 19 Sep 2017 00:00:00 +0000
https://scoskey.github.io/equivalence-relations-and-classification-problems
https://scoskey.github.io/equivalence-relations-and-classification-problemsclassificationpresentationOn the classification of automorphisms of 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>
Mon, 11 Sep 2017 00:00:00 +0000
https://scoskey.github.io/trees
https://scoskey.github.io/treesclassification,conjugacy,automorphisms,treespublicationReal and linear analysis<p>Math 515, Fall 2017 (<a href="http://github.com/scoskey/m515">site</a>)<!--more--></p>
<p><em>Catalog description</em>: Lebesgue measure on the reals, construction of the Lebesgue integral and its basic properties. Advanced linear algebra and matrix analysis. Fourier analysis, introduction to functional analysis.</p>
Sat, 12 Aug 2017 00:00:00 +0000
https://scoskey.github.io/1718f-515
https://scoskey.github.io/1718f-515real-analysiscourseHonors calculus I<p>Math 170H, Fall 2017<!--more--></p>
<p><em>Catalog description</em>: Definitions of limit, derivative, and integral. Computation of the derivative, including logarithmic, exponential and trigonometric functions. Applications of the derivative, approximations, optimization, mean value theorem. Fundamental theorem of calculus, brief introduction to the applications of the integral and to computations of antiderivatives. Intended for students in engineering, mathematics and the sciences.</p>
Sat, 12 Aug 2017 00:00:00 +0000
https://scoskey.github.io/1718s-170
https://scoskey.github.io/1718s-170calculuscourseThe classification of countable models of set theory<p>With John Clemens and Samuel Dworetzky. (<a href="https://arxiv.org/abs/1707.04660">arχiv</a>)<!--more--></p>
<p><em>Abstract</em>: We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well-founded models of ZFC.</p>
Mon, 17 Jul 2017 00:00:00 +0000
https://scoskey.github.io/zfcmod
https://scoskey.github.io/zfcmodclassification,set-theorypublicationOn the classification of vertex-transitive structures<p>With John Clemens and Stephanie Potter. (<a href="https://arxiv.org/abs/1707.02383">arχiv</a>)<!--more--></p>
<p><em>Abstract</em>: We consider the classification problem for several classes of countable structures which are “vertex-transitive”, meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above $E_0$ in complexity.</p>
Mon, 10 Jul 2017 00:00:00 +0000
https://scoskey.github.io/vertex-transitive
https://scoskey.github.io/vertex-transitiveclassificationpublicationThe classification problem for models of set theory<p>A master’s thesis by Samuel Dworetzky, Spring 2017 (<a href="http://scholarworks.boisestate.edu/td/1253/">link</a>)<!--more--></p>
<p><em>Abstract</em>: Models of set theory are ubiquitous in modern day set theoretic research. There are many different constructions that produce countable models of ZFC via techniques such as forcing, ultraproducts, and compactness. The models that these techniques produce have many different characteristics; thus it is natural to ask whether or not models of ZFC are classifiable. We will answer this question by showing that models of ZFC are unclassifiable and have maximal complexity. The notions of complexity used in this thesis will be phrased in the language of Borel complexity theory.</p>
<p>In particular, we will show that the class of countable models of ZFC is Borel complete. Most of the models in the construction as it turns out are ill-founded. Thus, we also investigate the sub problem of identifying the complexity for well-founded models. We give partial results for the well-founded case by identifying lower bounds on the complexity for these models in the Borel complexity hierarchy.</p>
Mon, 15 May 2017 00:00:00 +0000
https://scoskey.github.io/the-classification-problem-for-models-of-zfc
https://scoskey.github.io/the-classification-problem-for-models-of-zfcclassification,zfcmasters-thesisOn the classification of vertex-transitive structures<p>A master’s thesis by Stephanie Potter, Spring 2017 (<a href="http://scholarworks.boisestate.edu/td/1279/">link</a>)<!--more--></p>
<p><em>Abstract</em>: When one thinks of objects with a significant level of symmetry it is natural to expect there to be a simple classification. However, this leads to an interesting problem in that research has revealed the existence of highly symmetric objects which are very complex when considered within the framework of Borel complexity. The tension between these two seemingly contradictory notions leads to a wealth of natural questions which have yet to be answered.</p>
<p>Borel complexity theory is an area of logic where the relative complexities of classification problems are studied. Within this theory, we regard a classification problem as an equivalence relation on a Polish space. An example of such is the isomorphism relation on the class of countable groups. The notion of a Borel reduction allows one to compare complexities of various classification problems.</p>
<p>The central aim of this research is determine the Borel complexities of various classes of vertex-transitive structures, or structures for which every pair or elements are equivalent under some element of its automorphism group. John Clemens has shown that the class of vertex-transitive graphs has maximum possible complexity, namely Borel completeness. On the other hand, we show that the class of vertex- transitive linear orderings does not.</p>
<p>We explore this phenomenon further by considering other natural classes of vertex- transitive structures such as tournaments and partial orderings. In doing so, we discover that several other complexities arise for classes of vertex-transitive structures.</p>
Mon, 15 May 2017 00:00:00 +0000
https://scoskey.github.io/on-the-classification-of-vertex-transitive-structures
https://scoskey.github.io/on-the-classification-of-vertex-transitive-structuresclassification,vertex-transitivemasters-thesis