Samuel CoskeyDepartment of Mathematics, Boise State University, 1910 University Dr, Boise, ID 83725-1555.
https://scoskey.github.io/
Mon, 19 Jun 2017 20:32:56 +0000Mon, 19 Jun 2017 20:32:56 +0000Jekyll v3.4.3The 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 (link coming soon)<!--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-thesisIntroduction to linear algebra<p>Math 301, Spring 2017<!--more--></p>
<p><em>Catalog description</em>: Linear algebra from a matrix perspective with applications from the applied sciences. Topics include the algebra of matrices, methods for solving linear systems of equations, eigenvalues and eigenvectors, matrix decompositions, vector spaces, linear transformations, least squares, and numerical techniques.</p>
Tue, 27 Dec 2016 00:00:00 +0000
https://scoskey.github.io/1617s-301
https://scoskey.github.io/1617s-301linear-algebracourseCalculus I<p>Math 170, Spring 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>
Tue, 27 Dec 2016 00:00:00 +0000
https://scoskey.github.io/1617s-170
https://scoskey.github.io/1617s-170calculuscourseSolving polynomial equations by radicals<p>A senior thesis by Mack Fox, Fall 2016<!--more--></p>
<p><em>Abstract</em>: This article will show the derivation of closed form radical expressions for polynomials of degree $n\leq4$. For degree one and two polynomials, it is simplistic to show solvability by radicals. For degree three and four polynomials however, these derivations can be quite complex. Due to this, much greater detail is shown throughout those sections. We will also introduce the reader to aspects of Group and Field theory which will serve as a stepping stone to Galois theory. We will use Galois theory to show that for polynomials of degree $n\geq5$, no closed form radical expression for the roots exists.</p>
Sat, 24 Dec 2016 00:00:00 +0000
https://scoskey.github.io/solving-polynomial-equations-by-radicals
https://scoskey.github.io/solving-polynomial-equations-by-radicalsset-theorysenior-thesisAn examination of the euclidean algorithm<p>A senior thesis by Ethan Stieha, Fall 2016<!--more--></p>
<p><em>Introduction</em>: The Euclidean Algorithm was first published in 300 B.C. yet still remains widely useful in solving the greatest common divisor of two computationally large natural numbers. The algorithm provides a step by step process to reduce natural numbers into remainders derived from the division theorem with the same common divisors. While the algorithm itself is rather simple, it has several unique behaviors that make it fascinating to study. As years pass, mathematicians consistently rely on the Euclidean algorithm to be well-conditioned, and provide accurate computational results.</p>
<p><em>Summary</em>: The thesis defines and illustrates the algorithm. It uses experimental methods to investigate the likelihood of each outcome of the algorithm. It then uses both experimental and rigorous methods to examine the case when the outcome is 1, that is, the two inputs are relatively prime.</p>
Sat, 24 Dec 2016 00:00:00 +0000
https://scoskey.github.io/an-examination-of-the-euclidean-algorithm
https://scoskey.github.io/an-examination-of-the-euclidean-algorithmset-theorysenior-thesisThe set splittability problem<p>With P. Bernstein, C. Bortner, S. Li, and C. Simpson. (<a href="https://arxiv.org/abs/1611.01542">arχiv</a>)<!--more--></p>
<p><em>Abstract</em>: The set splittability problem is the following: given a finite collection of finite sets, does there exits a single set that selects half the elements from each set in the collection? (If a set has odd size, we allow the floor or ceiling.) It is natural to study the set splittability problem in the context of combinatorial discrepancy theory and its applications, since a collection is splittable if and only if it has discrepancy $\leq1$.</p>
<p>After introducing the concepts and their background, we show that the set splittability problem is NP-complete. We in fact establish this for the generalized version called the $p$-splittability problem, in which one seeks to select the fraction $p$ from each set instead of half. Next we investigate several criteria for splittability and $p$-splittability, giving a complete characterization of $p$-splittability for three sets and of splittability for four sets. Finally we show that when there are sufficiently many elements, unsplittability is asymptotically much more rare than splittability.</p>
Fri, 04 Nov 2016 00:00:00 +0000
https://scoskey.github.io/splittability
https://scoskey.github.io/splittabilitysplittingpublicationBorel complexity theory in mathematics<p>A talk for the BSU math graduate seminar, September 2016<!--more--></p>
<p><em>Abstract</em>: Given a mathematical problem it is natural to wonder how complicated it is, but it is hard to imagine how to make this question rigorous. Borel complexity theory is an area of set theory which provides a framework to measure the complexity of classification problems in mathematics. We will introduce this theory, and show how it has been applied to classification problems in group theory, graph theory, and functional analysis.</p>
Sun, 25 Sep 2016 00:00:00 +0000
https://scoskey.github.io/borel-complexity-theory-in-mathematics
https://scoskey.github.io/borel-complexity-theory-in-mathematicsclassificationpresentationOn the set splittability problem<p>A talk for the Boise math department’s Algebra, Geometry and Cryptography seminar, September 2016<!--more--></p>
<p><em>Abstract</em>: The set splittability problem asks whether, given a collection of finite sets, there exists a single set that selects exactly half the elements from each set in the collection. If a set has odd size, we may select either the floor or the ceiling of half its elements. The question is naturally a part of combinatorial discrepancy theory, since a collection is splittable if and only if its discrepancy is at most 1. In this talk we will show that the set splittability problem is NP complete. On the other hand, we will give several partial solutions to the problem for small collections and other special collections. This work was completed during our REU program in collaboration with P. Bernstein, C. Bortner, S. Li, and C. Simpson.</p>
Mon, 19 Sep 2016 00:00:00 +0000
https://scoskey.github.io/the-set-splittability-talk
https://scoskey.github.io/the-set-splittability-talksplittingpresentationClassifying automorphisms of countable trees<p>Boise Set Theory Seminar, September 2016<!--more--></p>
<p><em>Abstract</em>: We summarize some of the results from Kyle Beserra’s master’s thesis. In Serre’s study of trees and their automorphisms, he observed that the automorphisms all lie in one of three classes: invert an edge, shift a bi-infinite path, or fix a subtree pointwise. But of course there are many types of automorphisms within each of these classes. So it is natural to ask just how complex is the classification of tree automorphisms? And what is the complexity of each of Serre’s three classes? We can make these questions formal using the language Borel complexity theory. In this talk we answer the question for regular trees.</p>
Fri, 09 Sep 2016 00:00:00 +0000
https://scoskey.github.io/classifying-automorphisms-of-countable-trees
https://scoskey.github.io/classifying-automorphisms-of-countable-treesclassification,conjugacy,treespresentation